❯♥rstà t ♥♦

♣rt♠♥t ♦ t♠ts r♦ ♥rqs

♦t♦r ♦rs ♥ t♠t ♥s

P ♥ t♠ts

Prt♦r♣r② ♠♦s rt♦♥s

r♦ss s♦♥ ♥ r♥ ♥stt②

♥③ ♦rs♥

s♦r Pr♦ ♦♥♥

♦s♦rs Pr♦ss r r♦♣♣Pr♦ r♥t stts

♦♦r♥t♦r Pr♦ ❱r str♦♣tr♦

❳❳❳ ②

P ss

s♦rs

Pr♦ ♦♥♥

♣rt♠♥t ♦ t♠ts r♦ ♥rqs❯♥rstà t ♥♦

Pr♦ss r r♦♣♣

♣rt♠♥t ♦ t♠ts ♥ ♦♠♣tr ♥s❯♥rstà t Pr♠

Pr♦ r♥t stts

♥sttt té♠tqs ss Prs P❯♥rsté Prs r♦t

♣rt♠♥t ♦ t♠ts r♦ ♥rqs❱ sr ♥ ♥♦ t②

i

♥♦♠♥ts

t♥ ♠② s♦r ♦♥♥ ♦ ♠ ♥ts ♥ s r♥ ts ②rs s♣ ♠♥t♦♥ ♦s t♦ r r♦♣♣ ♦r r ♠♥♥ ssst♥ trss ♥♥ ♥♥t ♣t♥ t♥ r♥t stts ♦r ♦r r ♦r♥ t♠ ♥ Prs♦♥ ts ♥trst♥ t♦♣ t♥ s♣♣ ♦♥ ♦s ♣ ♥♦ ♥ ♥ ♣rss ♠ rt t♦ t ♥♥♦♥ rrs ♦r tr r rs♦♥♥ ♦♥strt ♦♠♠♥ts ♦ s♦ t♦ t♥ P♦ s♥ ♦ s s♦♥♠ ♦ t♦ t st② t♥ ♠② ♦ ♠t ♥ r♥ ♥ ♦♥♥ sr ♥♦t ♦♥② ♥ ♦ ♥ ts ②rs

iii

♠♠r②

Prt♦r♣r② ♠♦s ♦♠♦♥♦s ♥ s♣ ♦r t s♣t s♦♥ ♣② ♥trr♦ ♥ ts tss ♥ r♦♠ ♠t♠t ♣♦♥t ♥stt stt②♥ s②st♠s ♦ ♦r♥r② r♥t qt♦♥s ♥ ♦ ♣rt r♥t qt♦♥s ♦♣r♦ t②♣

rst t ♣rt♦r♣r② ♠♦ sr ② s②st♠ ♦ t♦ s ♥ str♦♥ t ♦♥ t ♣r② r♦t ♥ ♣rt♦r♣♥♥t tr♦♣ ♥t♦♥ r t♥ ♥t♦ ♦♥t ♠♥ str♥t ♦ ts ♣rt s tt ts ♥t♦♥s r♥♦t s♣ ② ♥②t ①♣rss♦♥s t ♦♥② rtr③ ② s♦♠ ♦♦②♠♥♥ ♣r♦♣rts tr♠♥♥ tr s♣s ♥ t ss ♦ ts ♣r♦♣rts r t♦ ♣r♦r♠ t stt② ♥②ss ♦ t s②st♠ s♥ t ♣rt♦♥ ♥②♥ ♠sr ♦ t ♣rt♦r ♥trr♥ s rt♦♥ ♣r♠trs s②st♠♠ts ♦♠♥s♦♥t♦ rt♦♥s ♣♦♥ts s s ♦♥♦♥s ♥ s♣♣♦♥t t s ♦rt t♦ ♥♦t tt t② r ♥♣♥♥t ♦ t ♣rtr ①♣rss♦♥♦ t ♠♦ ♥t♦♥s ♥♠r ♥stt♦♥ s rtr rr ♦♥ ♦♦s♥♦r t ♠♦ qt♦♥s s♦♠ ♥②t ①♣rss♦♥s ♥♦♥ ♥ trtr sts t ss♠ ♣r♦♣rts ♥ s♥ t♦♥t ♦♥t♥t♦♥ t t♦♦♦①s ♥stt♦♥ ♥ t♦♥ s s♦♥ t ♣rs♥ ♦ ♦ rt♦♥s tttr♠♥ t s♣♣r♥ ♦ ♠t ②s tr♦ t ♦r♠t♦♥ ♦ ♦♠♦♥ ♥tr♦♥ ♦rts ♥♦♥ s♦♠ qr♠ ♣♦♥ts ♦r♦r tt rtr ♦♠♥s♦♥t♦ rt♦♥ ♣♦♥t ♥r③♦♣ ♦tr t ts♣ ♥ t ♦♥♦♥s rt♦♥ ♣♦♥ts ts tr t②♣s ♦ ♦♠♥s♦♥t♦ rt♦♥s r t ♦♥② ♠ss ② ♣♥r s②st♠ ♦ ♦r♥r② r♥tqt♦♥s

s♦♥ ♣rt ♦ ts tss ♦ss ♦♥ t st② ♦ t♦ ♣rt♦r♣r② ♠♦st s♦♥ tt st② ♥ st ♠t t♦ ss t②♣s ♦ ♥t♦♥ rs♣♦♥ss ♥ t rt♦♥ ♣rt ♥ ♣rs♥t r♦sss♦♥ tr♠ ♥ t t♦ tr♦♣s r ♦♥sr ♣r②s ♥ ♣rt♦rs r rtr ♥t♦ sr♥♣rt♦rs ♥ ♥♥ ♣rt♦rs ♦r♠r r ♣rt♦rs t ♥ t ♣rt♦♥♣r♦ss t ttr r rst♥ ♥s ♥ strt r♦♠ s②st♠ ♦ tr♣rt r♥t qt♦♥s t st♥r ♥r s♦♥ ♥ tr♠s ♦ ♣♥♥ t ♦t❱♦trr rt♦♥ tr♠ r♦ qs st②stt ♣♣r♦①♠t♦♥ ♥ ♣ t s②st♠ ♦ t♦ Ps t ♣r② ♥ t♦t ♣rt♦r ♥sts s♥♥♦♥s ♥ ♥ ♦♥t②♣ ♥t♦♥ rs♣♦♥s ♣♣rs t♦tr t r♦sss♦♥ tr♠ ♥ t ♣rt♦r qt♦♥ t s ♣r♦ tt ts ss ♦ ♣rt♦r♣r② ♠♦s ♥ ♥♦t rs t♦ r♥ ♥stt② ♥ ♠♦② t strt♥ ♠♦

iv

♥srt♥ ♦♠♣tt♦♥ ♠♦♥ ♣rt♦rs ❲t ts ♥ ♥ ♣ tr qsst②stt ♣♣r♦①♠t♦♥ t s②st♠ ♦ t♦ Ps ♦r ♣r② ♥ t♦t ♣rt♦r♥sts rtr③ ② ♥t♦♥♥st②♣ ♥t♦♥ rs♣♦♥s ♥ r♦sss♦♥ tr♠ ♥ t ♣rt♦r qt♦♥ ❲ ♦♦ ♦r ♦♥t♦♥s ♦♥ t ♣r♠trs s t♦ r♥ ♥stt② ♥ ♦♠♣r ts r♥ ♥stt②r♦♥s t t ♦♥s ♦t♥ ♥ t r♦sss♦♥ tr♠ s ssttt ② ♥r s♦♥

v

ss♥t♦

st ts rr ♠♦ r♥③ ♣r♣rt♦r trttt ♥③♠♥t ♥s♦ s♣③♠♥t ♦♠♦♥♦ sss♠♥t ♦♥sr♥♦ s♦♥ s♣③ ♣♥t♦ st ♠t♠t♦ ♣rt♥t♦ ♥♦♥♦ ♦♥srt sst♠ q③♦♥ r♥③ ♦r♥r q③♦♥ r♥③ rt ♣r③ t♣♦ ♣r♦♦

♥ ♣rt♦r ♥ ♣r♠ ♣rt ♥ stt♦ ♥ ♠♦♦ ♣r♣rt♦rs♣③♠♥t ♦♠♦♥♦ rtt♦ q③♦♥ r♥③ ♦r♥r ♥ s♦♥♦♣rs ♥ ♦♥sr③♦♥ ♥ tt♦ ♦rt ♥ rst ♣r ♥ rs♣♦st♥③♦♥ ♣rt♦r ♣♥♥t ♣♥t♦ ♦r③ ♦ st♦ rs ♥ tt♦ ♥③♦♥ sr♦♥♦ qst ♣r♦ss ♥♦♥ ♥♥♦ ♥s♣rss♦♥ s♣t ♠ s♦♥♦rttr③③t s♦♦ ♥ ♣r♦♣rtà ♦♠♥ ♥③♦♥ s♣ t③③t ♥ ttrtr ♣r♦♣rtà s♦♥♦ s♥t ♣r ttr ♥s qtt sst♠♦♥ rr♦ sst♥③ qr ♦r♦ ♣r♦♣rtà sttà ♠♥t rtr ②♣♥♦ t③③♥♦ ♣r♠tr ♦r③♦♥ rttr③③♥♦ ♣r♦ss♦ ♣r③♦♥ ♠♦♦ ♣rs♥t ♣♥t ♦r③♦♥ ♦♠♥s♦♥ q ♥ ♦r③♦♥ ♦♥♦♥s ♥ t♣♦ s♣ ♥♦♥ t ♣rt♦r r③③③♦♥ st ♣r ♥③♦♥ ♠♦♦ ♦ st♦ é stt♦♣r♦st♦ ♥♠r♠♥t ss♥♦ ♥s♣rss♦♥ ♣r ♥③♦♥ rst ♣r ♣r ♥③♦♥ tr♦ s♦s♥♦ ♣r♦♣rtà ♦♥srt t③③♥♦ s♦tr ♦♥t♥③♦♥ t♦♥t ♣r t st♦ ♠♦strt♦ tr♦r♣rs♥③ ♦r③♦♥ ♦ tr♠♥♥♦ s♣r③♦♥ ♠t ♠♥t ♦r♠③♦♥ ♦rt ♦♠♦♥ tr♦♥ ♥♦tr é stt♦ ♥t♦ ♥♦r③♦♥ ♦♣ ♥r③③t ♥ tr♦ ♣♥t♦ ♦r③♦♥ ♦♠♥s♦♥ ♦r③♦♥ ♦♠♥s♦♥ ♥t s♦♥♦ ttt s♦ q ♠♠ss ♥sst♠ q③♦♥ r♥③

s♦♥ ♣rt ts rt ♥ s♦ st♦ sst♠ ♣r♣rt♦r♦♥ s♦♥ ♥ ♥♦♥♦ ♦tt ♥ ♥ ♦♣♣♦rt♥♦ ♠t t♣ rs♣♦st♥③♦♥ ss ♦♠ tr♠♥ rtt♦ ♥ tr♠♥ s♦ ♥♦♥ ♥r ♥tt♦ ♥♦♥♦ ♦♥srt tr♦ ♣r ♣rt♦r st t♠s♦♥♦ ss ♥ ss sr♥ ♣rt♦rs ♥♥ ♣rt♦rs ♣r♠ s♦♥♦ ♣rt♦r tt♠♥t ♠♣♥t ♥ ♣r③♦♥ ♠♥tr s♦♥ ♥♦♥ s♦♥♦tt ♥ t ♣r♦ss♦ r ♥ sst♠ ♦♠♣♦st♦ tr q③♦♥ r♥③ rt ♣r③ ♥ s♦♥ é ♠♦③③t ♥ ♠♦♦ ss♦ ♠♥t♥ tr♠♥ ♥r ♥ ♦r♠ ♣♥♦ ♥tr③♦♥ tr ♣r ♣rt♦r é♥③♠♥t t♣♦ ♦t❱♦trr ♥t ♥ ♣♣r♦ss♠③♦♥ qs st②stt é ♣♦ss rrr sst♠ ♣rt♥③ ♦tt♥♥♦ ♥ sst♠ P

vi

♥ ♣r ♣r ♥ ♣r t♦ttà ♣rt♦r ♥ rs♣♦st ♥③♦♥ é t♣♦ ♦♥ ♥ ♣rt♦r ♣r♣♥♥t ♣rs♥t ♥ ♥♦♥♥rtà♥ tr♠♥ s♦♥ st ss ♠♦ ♥♦♥ à ♦♦ ♥sttà r♥ ❱♥ q♥ ♦♥srt ♥ ♠♦♦ tr q③♦♥ ♥ ♦♠♣t③♦♥ tr ♣rt♦r ♣r♠tt rr ♠♥t ♥♣♣r♦ss♠③♦♥ qs st②stt♥ sst♠ ♣r♣rt♦r ♦♥ rs♣♦st ♥③♦♥ t♣♦ ♥t♦♥♥s♥ tr♠♥ r③♦♥ ♥♦r ♥ ♥♦♥♥rtà ♥ tr♠♥ s♦♥ ❱♥♦♥♦q♥ rt ♦♥③♦♥ s ♣r♠tr ♣r♠tt♦♥♦ r ♥sttà r♥ ♦♥r♦♥tt rstt s ♥ s♦ s♦♥ ♥r ♥ q♦ ♥♦♥♥r

Contents

♥tr♦t♦♥

Pr♠♥rs rt♦♥ ♦ qr ♥ ♦♥t♥♦st♠ ②♥♠ s②st♠s

♥♣r♠tr rt♦♥s ♦♣r♠tr rt♦♥s

s♦♥ r♥ ♥stt② r♥ ♥stt② ♥②ss r♥ ♥stt② ♥②ss t r♦sss♦♥

Prt ②♥♠s ♦ ♣rt♦r♣r② ♠♦s t str♦♥ t ♦♥

t ♣r② ♥ ♣rt♦r♣♥♥t tr♦♣ ♥t♦♥s

♥tr♦t♦♥

♣r②♣rt♦r ♠♦ ♥ t qr♠ stt② ♥②ss s ss♠♣t♦♥s ♥ ♠♦ qt♦♥s ttrt ♥r♥t st ♦♥♦①st♥ qr ♥ tr stt② ♣r♦♣rts ♦①st♥ qr

♣ ♦ φ(x, h) ♦t♦♥s t♦ φ(x, h) = p/p0

tt② ♣r♦♣rts ♦ ♦①st♥ qr♠ stts tt E∗1 t s ♦ rx ≥ r0 tt E∗1 t s ♦ rx < r0 tt E∗3

viii Contents

♠r st② ♦rs ♦ t s②st♠

♦♥♥ r♠rs

♣♣♥① ts ♦♥ s♦♠ ♣r♠trs s♣ rt♦♥ ♣♦♥t st ♦ ♠♥ s②♠♦s

Prt ♦t rt♦♥s♦♥ ♣rt♦r♣r② s②st♠s ♥♦♥ t

♦♥t②♣ ♥ t ♥t♦♥♥s ♥t♦♥ rs♣♦♥ss

♥tr♦t♦♥

stt♦♥ ♦ ss ♥t♦♥ rs♣♦♥ss rt♦♥ ♦ t ♦♥t②♣ ♥t♦♥ rs♣♦♥s

♠♥s♦♥③t♦♥ rt♦♥ ♦ t ♥t♦♥♥s ♥t♦♥ rs♣♦♥s

♠♥s♦♥③t♦♥ ♦r♦s rsts ♦ ♦♥r♥

Pr♦♦ ♦ ♦r♠ Pr♦♦ ♦ Pr♦♣♦st♦♥ Pr♦♦ ♦ ♦r♠

r♥ ♥stt② ♥②ss ♦♥t②♣

♦♠♦♥♦s qr♠ stts r♥ ♥stt② ♥②ss

♥t♦♥♥s ♦♠♦♥♦s qr♠ stts r♥ ♥stt② t ♥r s♦♥ tr♠s r♥ ♥stt② t r♦sss♦♥ r♥ ♥stt② r♦♥s ♥r s r♦ss s♦♥

♦♥♥ r♠rs

Contents ix

♣♣♥① ❱r② ♦r♠t♦♥

r♥s

List of Figures

s♣ rt♦♥ ♣♦♥t t♥ rt♦♥ ♣♦♥t ♦♥♦♥s rt♦♥ ♣♦♥t

②♣ s♣s ♦ ♣r② r♦t ♥t♦♥s t str♦♥ t

P♦st② ♥r♥t ♥ ttrt st ♣ ♦ t ♥t♦♥ ψ(x) r♥ ♦ φ(x, h) ♦r x ∈ (ǫ, 1) ♥ r♥t s ♦ h ♦t♦♥s t♦ φ(x, h) = p/p0 qr♠ sr ♥ t (h, p, x) s♣ ①st♥ r♦♥s ♦ ♦①st♥ qr♠ stts ♥ t (h, p) ♣♥ tt② tr♥ ♦ x∗j rss p ♦r r♥t s ♦ h tt② r♦♥s ♦ E∗1(h, p) ♥ t (h, p) ♣♥ rx ≥ r0 tt② r♦♥s ♦ E∗1(h, p) ♥ t (h, p) ♣♥ rx < r0 ♥stt② r♦♥s ♦ E∗3(h, p) ♥ t (h, p)

♦ ♥ ♦ rt♦♥ rs ♠t ②s ♦rs rss p Ps ♣♦rtrts ♦r h0 < h < h1 ♥ r♥t p ♦ ♥ ♦ rt♦♥ rs ♦r h > hR rx < r0 Ps r♠ ♦s t♦ t ♦♥♦♥s rt♦♥

r♥ ♥stt② r♦♥s ♦r ♥r ♥ r♦sss♦♥

List of Tables

♠r ♦ s♦t♦♥s ξj(h) t♦ qt♦♥ ψ(x) = 1/h ♠♠r② ♦ ①st♥ ♦ s♦t♦♥s t♦ Φ(x, h) = p/p0 ♠♠r② ♦ stt② ♣r♦♣rts ♦ qr♠ stts

Introduction

Prt♦r♣r② ♠♦s ♦♠♦♥♦s ♥ s♣ ♦r t s♣t s♦♥ ♣② ♥trr♦ ♥ ts tss ♥ ② ♠t♠t ♣♦♥t ♥stt stt②♥ s②st♠s ♦ ♦r♥r② r♥t qt♦♥s ♥ ♦ ♣rt r♥t qt♦♥s ♦♣r♦ t②♣

rst t ♣rt♦r♣r② ♠♦sr ② s②st♠ ♦ t♦ s♥ str♦♥ t ♦♥ t ♣r② r♦t ♥ ♣rt♦r♣♥♥t tr♦♣♥t♦♥ r t♥ ♥t♦ ♦♥t str♦♥ t ♠♦s ♥rr♦♥ ♣♥♦♠♥ ♥ s r♣rs♥t ② ♣r② r♦t ♥t♦♥ ♦♠s ♥t ♦rs♥t② s♠ s ♦ ♣r② ♦♠ss ♦ s♠t t t r♥t ♦r♠t♦♥s ♦ t ♣r② r♦t rt ♥ t s♥ ♦ ♣rt♦rs ♥ ♣r♦♣♦s♥ t trtr ♥② t♦rs ❬ ❪ s ♦r ♥st♥ ♠t♣t t♦r t♦ t ♦st tr♠ ♥tr♦ ♥♦tr rt ♦r t♣r② ♦♠ss ss t♥ t rr②♥ ♣t② ♦♥r♥♥ t tr♦♣ ♥t♦♥ srs t ♣rt♦r ♥t♦♥ rs♣♦♥s t♦ ♣r② ♥♥ ❬ ♣ ❪ t s ♥tr♦ ♥ ♣rt♦r♣r② ♠♦s t♦ t ♥t♦ ♦♥t t strt♦♥ ♠t♥ t♣rt♦♥ ♣r♦ss r♥t ♦r♠t♦♥s ♥ ♣r♦♣♦s ♥ t trtr s ♦r♥st♥ trr③ t ❬❪ ♥t♦♥ ♥s ♥ ♦t♦rs ❬ ❪♥ ts tss ts ♥t♦♥s r ♥♦t s♣ ② ♥②t ①♣rss♦♥s t ♦♥②rtr③ ② s♦♠ ♦♦② ♠♥♥ ♣r♦♣rts tr♠♥♥ tr s♣s♥ ♦ t ♠♥ ♠♥ts ♦ ts tss s tt ♦♥ t ss ♦ ts ♣r♦♣rts♦♥② t stt② ♥②ss ♦ t s②st♠ s ♥ ♣r♦r♠ s♥ t ♣rt♦♥♥② ♥ ♠sr ♦ t ♣rt♦r ♥trr♥ s rt♦♥ ♣r♠trs s②st♠ ♠ts ♦♠♥s♦♥t♦ rt♦♥s ♣♦♥ts s s ♦♥♦♥s ♥ s♣ ♣♦♥t t s ♦rt t♦ ♥♦t tt t② r ♥♣♥♥t ♦ t ♣rtr①♣rss♦♥ ♦ t ♠♦ ♥t♦♥s ♥♠r ♥stt♦♥ s rtr rr ♦♥♦♦s♥ ♦r t ♠♦ qt♦♥s s♦♠ ♥②t ①♣rss♦♥s ♥♦♥ ♥ t t

2 Introduction

rtr sts t ss♠ ♣r♦♣rts ♥ s♥ t♦♥t ♦♥t♥t♦♥t t♦♦♦① s ♥stt♦♥ ♥ t♦♥ s s♦♥ t ♣rs♥ ♦ ♦rt♦♥s tt tr♠♥ t s♣♣r♥ ♦ ♠t ②s tr♦ t ♦r♠t♦♥♦ ♦♠♦♥ ♥ tr♦♥ ♦rts ♥♦♥ s♦♠ qr♠ ♣♦♥ts ♦r♦r tt rtr ♦♠♥s♦♥t♦ rt♦♥ ♣♦♥t ♥r③♦♣♦tr t t s♣ ♥ t ♦♥♦♥s rt♦♥ ♣♦♥ts ts trt②♣s ♦ ♦♠♥s♦♥t♦ rt♦♥s r t ♦♥② ♠ss ② ♣♥r s②st♠ ♦♦r♥r② r♥t qt♦♥s

s♦♥ ♣rt ♦ ts tss ♦ss ♦♥ t st② ♦ t♦ ♠r♦s♦♣ ♥ tr♠s♦ t♠ ss ♣rt♦r♣r② ♠♦s t s♦♥ tt st② ♥ st ♠tt♦ ss t②♣s ♦ ♥t♦♥ rs♣♦♥ss ♥ t rt♦♥ ♣rt ♥ ♣rs♥t r♦sss♦♥ tr♠ ♥ t t♦ tr♦♣ s r ♦♥sr ♣r② ♥ ♣rt♦rs r rtr ♥t♦ sr♥ ♣rt♦rs ♥ ♥♥ ♣rt♦rs ♦r♠r r♣rt♦rs t ♥ t ♣rt♦♥ ♣r♦ss t ttr r rst♥ ♥s ♥ strt r♦♠ s②st♠ ♦ tr ♣rt r♥t qt♦♥s t st♥r ♥rs♦♥ ♥ tr♠s ♦ ♣♥ ♥ t ♦t❱♦trr rt♦♥ tr♠ r♦ qs st②stt ♣♣r♦①♠t♦♥ ♥ ♣ t s②st♠ ♦ t♦ Ps t ♣r② ♥t♦t ♣rt♦r ♥sts s ♥♥♦♥s ♥ ♥ ♦♥t②♣ ♥t♦♥ rs♣♦♥s♣♣rs t♦tr t r♦sss♦♥ tr♠ ♥ t ♣rt♦r qt♦♥ ♥ ♠♦② t strt♥ ♠♦ ♥srt♥ ♦♠♣tt♦♥ ♠♦♥ ♣rt♦rs ❲t ts ♥ ♥ ♣ tr qs st②stt ♣♣r♦①♠t♦♥ t s②st♠ ♦ t♦ Ps ♦r♣r② ♥ t♦t ♣rt♦r ♥sts rtr③ ② ♥t♦♥♥s ♥t♦♥ rs♣♦♥s ♥ r♦sss♦♥ tr♠ ♥ t ♣rt♦r qt♦♥ ❲ ♣rs♥t s♦r♦r♦s rst ♦ ♦♥r♥ ♦ t s♦t♦♥s ♦ ts s②st♠ t♦rs t s♦t♦♥♦ t rt♦♥r♦ss s♦♥ s②st♠ ❲ r s♦ ♥trst ♥ t r♥ ♥stt②♥②ss ♦ ts s②st♠s ♦r t rst s t s ♥♦♥ tt ♣rt♦r♣r② ♠♦st ♣r②♣♥♥t tr♦♣ ♥t♦♥ ♥ t rt♦♥ ♣rt ♥ st♥r ♥rs♦♥ ♥ ♥♦t rs t♦ r♥ ♥stt② ❬❪ ♥ t t r♦sss♦♥♠♦ ♥♦ ♣ttr♥s s♠ t♦ ♣♣r ♥r ♦♦② rs♦♥ ss♠♣t♦♥ ♦♥t s♦♥ ♦♥ts ♦r t s♦♥ s ♥ t ♥t♦♥ rs♣♦♥s s ♥t♦♥♥s ♦♦ ♦r ♦♥t♦♥s ♦♥ t ♣r♠trs s t♦ r♥ ♥stt② ♥ ♦♠♣r ts r♥ ♥stt② r♦♥s t t♦♥s ♦t♥ ♥ t r♦sss♦♥ tr♠ s ssttt ② ♥r s♦♥

tss s ♦r♥③ s ♦♦ ❲ rst ♥tr♦ s♦♠ s ♦♥♣t ♦♥ rt♦♥ t♦r② ♥ s♦♥r♥ ♥stt② s ♥ t sq Prt ♦♥t♥s tst② ♦ t ♥r ♣rt♦r♣r② ♠♦ t str♦♥ t ♥ t ♣r② r♦t♥ ♣rt♦r♣♥♥t tr♦♣ ♥t♦♥ ♥ t t s ss♠♣t♦♥s ♥ t♦r♥r② r♥t qt♦♥s sr♥ t ♦ ②♥♠s ♦ ♣rt♦r♣r② s②s

Introduction 3

t♠ r ♣rs♥t t♥ t stt② ♣r♦♣rts ♦ t ♥♦♥♦①st♥ qr♠stts r s♠♠r③ ♥ ♥ ①st♥ ♥ stt② ♥②ss ♦ t ♦①st♥ qr♠ stts s ♣r♦r♠ ss♠♥ t♦ rt♦♥ ♣r♠trs ♥② rsts ♦♥♠r s♠t♦♥s ♦t♥ ♦r s♦♠ ♦♥rt r③t♦♥ ♦ t ♠♦ ♥t♦♥sr ♣rs♥t t♦ strt t ♦rs ♦ t s②st♠

Prt ♣rs♥ts t stt♦♥ ♦ t ♦♥t②♣ ♥ ♥t♦♥♥s t②♣ ♥t♦♥ rs♣♦♥s ♥ t ♦♥t①t ♦ rt♦♥s♦♥ s②st♠s strt♥ r♦♠ s②st♠ ♦ tr ♣rt r♥t qt♦♥s ♦r ♣r② sr♥ ♣rt♦r♥ ♥♥ ♣rt♦r r♦♠ ts ♣♣r♦ s♦ r♦sss♦♥ tr♠ rss ♥ t♣rt♦r qt♦♥s ♥st ♦ st♥r ♥r s♦♥ ♥ t r♥ ♥stt② ♥②ss ♦ ts s②st♠ s ♣rs♥t

Where we go from here

st② ♦ t ♥r ♠♦ ♦ Prt ssts rtr ♠♣r♦♠♥ts rst ♣r ♥②ss ♦ ♦♠♥s♦♥t♦ rt♦♥ ♣♦♥ts s r② ♥♥ ♦tr② t♦ r t s②st♠ ♦♥ t ♥tr ♠♥♦ ♥ ♦rr t♦ ttr rtr③ ts♣♦♥ts ♣♦ss tr ♦r rt s♦ t♦ t s♦♥ ♣rt ♦ ts tss s t①t♥s♦♥ ♦ ♣rs♥t rsts t♦ t s ♦ s♣t s♦♥ ♥ ♥r s♦♥ ♥t qt♦♥s ♦ ts s②st♠ ♦ ♦t♥ ♥r ♦♥t♦♥s ♦r r♥ ♥stt②♦r ts ss ♦ ♣rt♦r♣r② ♠♦s

♦♥r♥♥ t Prt tr r sr rtr rsr rt♦♥s r♦♠ t♥♠r ♣♦♥t ♦ ♣ttr♥ s♠t♦♥s ♦♠♣t t t♦rt st②rtr♠♦r t ♥t♦♥ ♦ ♥ ♣♣r♦♣rt ♥♠r s♠ ♦r t ♥trt♦♥♦ t r♦sss♦♥ tr♠ s st ♥ ♦♣♥ qst♦♥ s ♦♥sq♥ ♦♠♣rs♦♥t♥ t ♦t♦♠s ♦ t r ♠♦ ♥ t ss ♦♥ s t♦♥rst♥ t r♦ ♦ t ♥t♦♥♥st②♣ ♥t♦♥ rs♣♦♥s r♦♠ t♠♦♥ ♣♦♥t ♦ strt♥ r♦♠ ♠♦r ♦♠♣① ♠r♦s♦♣ ♠♦ ❬ ❪♦ t♦ r♥t r♦sss♦♥ tr♠ ♥② t ssst♦♥ ♦ r♦sss♦♥ t rs♣t t♦ tr ♣r♦♣rts ♦ ♥♥♥ s♦♥r♥ ♥stt②♦ s ♥ ♥trst♥

♥ ♣s ♦ ts tss ♣rs♥t t ♣t♦♥s ♦t♦r ② t ♥t♥ t st ♦ ts ♥ ♣♦strs ♣rs♥t t ♦♥r♥s ♥ ♠♠r ♦♦s

Preliminaries

s ♣tr ♥tr♦s s♦♠ s ♦♥♣t ♦ rt♦♥ t♦r② ♥ r♥ ♥stt② s ♥ ts tss rst ♥ s♦♠ t②♣ ♦ rt♦♥s ♦ ♥t ♠♥s♦♥♦♥t♥♦st♠ ②♥♠ s②st♠s ♣♥♥ ♦♥ ♦♥ ♦r t♦ ♣r♠trs ❲ ♦♦t ♥♦tt♦♥ ♦ t ③♥ts♦s ♦♦ ❬❪ ♥ t ♠♥ ♦♥♣ts ♦ s♦♥r♥♥stt② ♥ rt♦♥s♦♥ s②st♠s r s♠♠r③ ❬ ❪ ♥ t s ♦ st♥r ♥r s♦♥ ♥ ♥ ①t♥s♦♥ ♦ ts t♦r② t♦ t s ♦ r♦sss♦♥s r♣♦rt ❬ ❪ s♦ ♥ ts s tr♠♥ t rt♦♥ trs♦ ss ② ♥r stt② ♥②ss

Bifurcation of equilibria in continuous-time dynamical

systems

♦♥sr ♦♥t♥♦st♠ ♣r♠tr♣♥♥t ②♥♠ s②st♠

x = f(x, α)

r x ∈ Rn s t t♦r ♦ t ♣s rs α ∈ Rm s ♣r♠tr ♥ f ss♠♦♦t t rs♣t t♦ ♦t x ♥ α

❱r②♥ t ♣r♠tr t ♣s ♣♦rtrt s♦ rs ♣♣r♥ ♦ ♥♦♥q♥t t♦♣♦♦ ♣s ♣♦rtrt ♥r rt♦♥ ♦ ♣r♠tr s rt♦♥ s rt♦♥ s ♥ ♦ t t♦♣♦♦ strtr ♦ t s②st♠ sts ♣r♠trs ♣ss tr♦ rt♦♥ rt s

rt♦♥s ♥ tt ♦♦♥ t t ♣s ♣♦rtrts ♦ t s②st♠ ♥ s♠ ♥♦r♦♦ ♦ t qr♠ ♣♦♥t r ♦ rt♦♥s ♦r rt♦♥s ♦ qr r r s♦ rt♦♥s tt ♥♥♦t tt ② ♦♦♥ t

6 Preliminaries

s♠ ♥♦r♦♦s ♦ qr♠ ♣♦♥ts rt♦♥s r ♦ ♦r tr r ♦ rt♦♥s ♥ rt♥ ♦ rt♦♥s r ♥♦ ♥s ss ♦♦♥ t t ♦ rt♦♥ ♣r♦s ♦♥② ♣rt ♥♦r♠t♦♥ ♦♥ t♦r ♦ t s②st♠

t α = α0 ♥ ♦♥sr ♠①♠ ♦♥♥t ♣r♠tr st strt♠♦♥t♥♥ α0 ♥ ♦♠♣♦s ② t♦s ♣♦♥ts ♦r t s②st♠ s ♣s ♣♦rtrttt s t♦♣♦♦② q♥t t♦ tt t α0 ♥ s strt ♥ t ♣r♠trs♣ Rm ♦t♥ t ♣r♠tr ♣♦rtrt ♦ t s②st♠ ♣r♠tr ♣♦rtrtt♦tr t ts rtrst ♣s ♣♦rtrts ♦♥sttt rt♦♥ r♠ ♥t s♠♣st ss t ♣r♠tr ♣♦rtrt s ♦♠♣♦s ② ♥t ♥♠r ♦ r♦♥s♥ Rm ♥s r♦♥ t ♣s ♣♦rtrt s t♦♣♦♦② q♥t s r♦♥sr s♣rt ② rt♦♥ ♦♥rs r s♠♦♦t s♠♥♦s ♥ Rm rs srs ♦♥rs ♥ ♥trst ♦r ♠t s ♥trst♦♥s s t ♦♥rs ♥t♦ sr♦♥s ♥ s♦ ♦♥ rt♦♥ ♦♥r② s ♥ ②s♣②♥ ♣s ♦t qr♠ ② t ♥ s♦♠ rt♦♥ ♦♥t♦♥str♠♥♥ t t②♣ ♦ ts rt♦♥ ❲♥ ♦♥r② s r♦ss t rt♦♥♦rs

♦♠♥s♦♥ ♦ rt♦♥ s t r♥ t♥ t ♠♥s♦♥ ♦ t ♣r♠tr s♣ ♥ t ♠♥s♦♥ ♦ t ♦rrs♣♦♥♥ rt♦♥ ♦♥r② q♥t② t rtr ♥♦r♠② t ♦♠♥s♦♥ s t ♥♠r ♦ ♥♣♥♥t ♦♥t♦♥str♠♥♥ t rt♦♥ s s t ♠♦st ♣rt ♥t♦♥ ♦ t ♦♠♥s♦♥s♥ t ♠s r tt t ♦♠♥s♦♥ ♦ rt♥ rt♦♥ s t s♠ ♥ ♥r s②st♠s ♣♥♥ ♦♥ s♥t ♥♠r ♦ ♣r♠trs

One-parameter bifurcations

♦♥sr ♦♥♣r♠tr ♦♥t♥♦st♠ ②♥♠ s②st♠

x = f(x, α), x ∈ Rn, α ∈ R,

r f s s♠♦♦t t rs♣t t♦ ♦t x ♥ α t x = x∗ ♥♦♥②♣r♦qr♠ ♥ t s②st♠ ♦r α = α∗ r r ♥r② ♦♥② t♦ ②s ♥ t ②♣r♦t② ♦♥t♦♥ ♥ ♦t tr s♠♣ r ♥♣♣r♦s ③r♦ ♥ λ1 = 0 ♦r ♣r ♦ s♠♣ ♦♠♣① ♥s rst ♠♥r② ①s ♥ λ1,2 = ±iω0, ω0 > 0 ♦r s♦♠ ♦ t ♣r♠tr ♦rr♥ ♦ t ③r♦ ♥ s r♥t t♦ tr♥srt ♦r t♦ ♦rt♦♥ rs t ♦rr♥ ♦ ♣r ♦ ♣r ♠♥r② ♥s s t♦ ♦♣ rt♦♥

1 ♥ qr♠ s ②♣r♦ tr r ♥♦ ♥s ♦ t ♦♥ ♠tr① ♦ t s②st♠t t t qr♠ t ③r♦ r ♣rt

Preliminaries 7

Transcritical bifurcation

t♦♣♦♦ ♥♦r♠ ♦r♠ ♦ ♥r ♦♥♠♥s♦♥ ②♥♠ s②st♠ ♣♥♥ ♦♥ ♦♥ ♣r♠tr ♥ tr♥srt rt♦♥ s t ♦♦♥

x = f(x, α) = αx− x2, x ∈ R, α ∈ R.

s s②st♠ s t♦ qr x1 = 0, x2(α) = α ♦r t ♦r♠r λ = fx(x1, α) =−α ♠♣s tt t qr♠ s st ♥ α > 0 ♥ ♥st ♥α < 0 ♦r t ttr λ = fx(x2, α) = α s♦ tt t s st ♥ α < 0 ♥ ♥st♥ α > 0 ♦r α = 0 x1 = x2 = 0 ♥♠② t qr ♦ λ = 0♥ t qr ①♥ tr stt② t t rt♦♥ ♣♦♥t

Fold bifurcation

♥ ♥♦♥s②♠♠tr s②st♠s t ♦rr♥ ♦ ③r♦ ♥ s rt ♦♥② t♦ ♦ rt♦♥ t♦♣♦♦ ♥♦r♠ ♦r♠ ♦ ♥r ♦♥♠♥s♦♥ s②st♠♥ ♦ ♦r t♥♥t ♦r s♥♦ rt♦♥ s t ♦♦♥ ♦♥sr t♦♥♠♥s♦♥ ②♥♠ s②st♠ ♣♥♥ ♦♥ ♦♥ ♣r♠tr

x = f(x, α) = α + x2, x ∈ R.

t α∗ = 0 ts s②st♠ s ♥♦♥②♣r♦ qr♠ x∗ = 0 t λ = fx(0, 0) = 0♦r α < 0 tr r t♦ qr ♥ t s②st♠

x1,2(α) = ±√−α,

t ♥t ♦♥ ♦ s st t ♣♦st ♦♥ s ♥st ♦r α > 0 trr ♥♦ qr ♥ t s②st♠ ❲♥ α r♦sss ③r♦ r♦♠ ♥t t♦ ♣♦st st t♦ qr st ♥ ♥st ♦ ♦r♠♥ t α = 0 ♥ qr♠t λ = 0 ♥ s♣♣r ♥ t (α, x)♣♥ t qt♦♥ f(x, α) = 0 ♥s ♥qr♠ ♠♥♦ s s♠♣② t ♣r♦ α = −x2

♦♥sr ♥♦ t ♥r t♦♠♥s♦♥ s②st♠

x = f(x, α), x = (x1, x2)T ∈ R2, α ∈ R,

t s♠♦♦t ♥t♦♥ f s t α = 0 t qr♠ x = 0 t ♦♥♥ λ = 0 ♦♥ ♠tr① J(α) ♥ rtt♥ s

J(α) =

(

a(α) b(α)

c(α) d(α)

)

8 Preliminaries

♥trs r s♠♦♦t ♥t♦♥s ♦ α ts ♥s r t r♦♦ts ♦ t rtrst qt♦♥

λ2 − tr Jλ+ det J = 0,

r tr J = tr J(α) = a(α) + d(α) ♥ det J = det J(α) = a(α)d(α) − b(α)c(α)♦

λ1,2(α) =1

2

(

tr J(α)±√

tr J(α)2 − 4 det J(α))

.

♦ rt♦♥ ♦♥t♦♥s ♠♣s

tr J(0) 6= 0, det J(0) = 0.

Andronov-Hopf bifurcation

rt♦♥ ♦rrs♣♦♥♥ t♦ t ♣rs♥ ♦ λ1,2 = ±iω0, ω0 > 0 s ♦♣♦r ♥r♦♥♦♦♣ rt♦♥ ♦♥sr t ♦♦♥ s②st♠ ♦ t♦ r♥tqt♦♥s ♣♥♥ ♦♥ ♦♥ ♣r♠tr

x1 = αx1 − x2 − x1(x21 + x22)

x2 = x1 + αx2 − x2(x21 + x22)

s s②st♠ s t qr♠ x1 = x2 = 0 ♦r α t t ♦♥ ♠tr①

J(α) =

(α −11 α

)

,

♥ ♥s λ1,2 = α ± i s qr♠ s st ♦s ♦r α < 0 ♥ ♥♥st ♦s ♦r α > 0 t t rt ♣r♠tr α = 0 t qr♠ s♥♦♥♥r② st ♥ t♦♣♦♦② q♥t t♦ t ♦s s♦♠t♠s t s ② ttrt♥ ♦s s qr♠ s srr♦♥ ♦r ♦ α > 0 ②♥ s♦t ♦s ♦rt ♠t ② tt s ♥q ♥ st ♠r② t s②st♠

x1 = αx1 − x2 + x1(x21 + x22)

x2 = x1 + αx2 + x2(x21 + x22)

♥r♦s t ♥r♦♥♦♦♣ rt♦♥ t α = 0 ♥ t qr♠ t t♦r♥ s t s♠ stt② ♦r α 6= 0 t s st ♦r α < 0 ♥ ♥st ♦rα > 0 ♦♥trr② t♦ t ♣r♦s s②st♠ ts stt② t t rt ♣r♠tr s ♦♣♣♦st t s ♥♦♥♥r② ♥st t α = 0 ♥ tr s ♥ ♥st ♠t② ♦r α < 0 s♣♣rs ♥ α r♦sss ③r♦ r♦♠ ♥t t♦ ♣♦sts

Preliminaries 9

♦♥sr t ♥r ♦r♠

x1 = αx1 − ωx2 + l1x1(x21 + x22)

x2 = ωx1 + αx2 + l1x2(x21 + x22)

r ω, l1 ∈ R ♥ ω > 0 ♥ rrt t s②st♠ ♥ ♣♦r ♦♦r♥ts

ρ = αρ+ l1ρ3

φ = ω.

s st ♦r♠ s♦s tt t trt♦r② s♣rs r♦♥ t ♦r♥ t ♦♥st♥t ♥r♦t② ω t st♥ r♦♠ t ♦r♥ rs ♥ ♦r♥ t t rst s t ♥♦r♠ ♦r♠ ♦ t ♣t♦r s t stt② ♦ t ②♣♥s ♣♦♥ t s♥ ♦ l1 t rst ②♣♥♦ ♦♥t ♦r♦r t ♦♣rt♦♥ s s♣rrt l1 < 0 ♥ srt l1 > 0 ♥ tr♠s ♦ ♦s♦♥s tsrt♦♥ ♥♦s ♥ qr♠ ♥ ② ♦r sr♥s t♦ ♣♦♥t♥ t ♦s♦♥ ♦rs ♥ t s♣rrt s st ② s ♥ ts ♥tr♦r♥ ♥st ♦s ❲♥ t ♣r♠tr s r t ② sr♥s ♥t t ♦st t qr♠ ♥ tr t ♦s♦♥ ♦♥② st qr♠ r♠♥s ②♦♥trst ♥ t srt s t ② s ♥st ♥ s t ♦♥r② ♦ t s♥♦ ttrt♦♥ ♦ t st qr♠ ♥s t ② s tr t ♦s♦♥ trs ♦♥② r♣r

t s ♥♦ ♥ t ♦♥t♦♥ ♦r ♦♣ rt♦♥ ♦♥sr ♥♦ t ♥rt♦♠♥s♦♥ s②st♠

x = f(x, α), x = (x1, x2)T ∈ R2, α ∈ R,

t s♠♦♦t ♥t♦♥ f s t α = 0 t qr♠ x = 0 t ♥sλ1,2 = ±iω0, ω0 > 0 ♦♥ ♠tr① J(α) ♥ rtt♥ s

J(α) =

(a(α) b(α)c(α) d(α)

)

t s♠♦♦t ♥t♦♥s ♦ α s ts ♠♥ts ts ♥s r t r♦♦ts ♦ trtrst qt♦♥

λ2 − tr Jλ+ det J = 0,

r tr J = tr J(α) = a(α) + d(α) ♥ det J = det J(α) = a(α)d(α) − b(α)c(α)♦

λ1,2(α) =1

2

(

tr J(α)±√

tr J(α)2 − 4 det J(α))

.

♦♣ rt♦♥ ♦♥t♦♥ ♠♣s

tr J(0) = 0, det J(0) = ω20 > 0.

10 Preliminaries

Two-parameter bifurcations

♦♥sr t♦♣r♠tr ♦♥t♥♦st♠ ②♥♠ s②st♠

x = f(x, α)

r x = (x1, x2, . . . , xn)T ∈ Rn s t t♦r ♦ t ♣s rs α = (α1, α2)

T ∈R2 s t t♦r ♦ ♣r♠trs ♥ f s s♥t② s♠♦♦t ♥t♦♥ ♦ (x, α)♣♣♦s tt t α = α∗ t s②st♠ s ♥ qr♠ x = x∗ ♦r trt ♦ ♦r ♦♣ rt♦♥ ♦♥t♦♥s r sts ♥ ♥r② tr s rt♦♥ r B ♥ t (α1, α2)♣♥ ♦♥ t s②st♠ s ♥ qr♠①t♥ t s♠ rt♦♥

♦r ♣rs② ss♠ tt t α = α∗ = (α1∗, α2∗) t s②st♠

x = f(x, α), x ∈ R, α = (α1, α2)T ∈ R2,

s ♥ qr♠ x = x∗ t ♥ λ = fx(x∗, α∗) = 0 s②st♠ ♦ t♦sr ♥♦♥♥r qt♦♥s ♥ R3

f(x, α) = 0

fx(x, α) = 0

♥r② ♥s s♠♦♦t ♦♥♠♥s♦♥ ♠♥♦ Γ ⊂ R3 ♥ ♣rtr r♣ss♥ tr♦ t ♣♦♥t (x∗, α1∗, α2∗) ♣♦♥t (x, α) ∈ Γ ♥s ♥ qr♠♣♦♥t x t ③r♦ ♥ t t ♣r♠tr α st♥r ♣r♦t♦♥π : (x, α) → α ♠♣s Γ ♦♥t♦ r Bt ♥ t ♣r♠tr ♣♥ ♦ rt♦♥r ♥ ♦ rt♦♥ ts ♣ ♦♥ ts r

♦♥sr ♥♦ ♣♥r s②st♠

x = f(x, α), x = (x1, x2)T ∈ R2, α = (α1, α2)

T ∈ R2,

♥ t α = α∗ = (α1∗, α2∗)T ♥ qr♠ x∗ = (x1∗, x2∗)

T t ♣r ♦♥s ♦♥ t ♠♥r② ①s λ1,2 = ±iω∗ s②st♠ ♦ tr sr ♥♦♥♥rqt♦♥s ♥ R4 t ♦♦r♥ts (x1, x2, α1, α2)

f(x, α) = 0,

tr J(x, α) = 0,

♥s r Γ ⊂ R4 ♣ss♥ tr♦ (x∗, α∗) ♣♦♥t ♦♥ t r s♣s♥ qr♠ ♦ t s②st♠ t λ1,2 = ±iω∗, ω∗ > 0 s ♦♥ s det J(x, α) > 0.

Preliminaries 11

st♥r ♣r♦t♦♥ ♦ Γ ♦♥t♦ t (α1, α2)♣♥ ②s t ♦♣ rt♦♥♦♥r② BH

t t ♣r♠trs (α1, α2) r s♠t♥♦s② t♦ tr rt♦♥ rΓ t ♦♦♥ ♥ts ♠t ♣♣♥ t♦ t ♠♦♥t♦r ♥♦♥②♣r♦ qr♠ t s♦♠ ♣r♠tr s ①tr ♥s ♥ ♣♣r♦ t ♠♥r② ①sts ♥♥ t ♠♥s♦♥ ♦ t ♥tr ♠♥♦ W c ♦r s♦♠ ♦ t ♥rt②♦♥t♦♥s ♦r t ♦♠♥s♦♥ ♦♥ rt♦♥ ♥ ♦t

rst ♦♥sr t ♦ rt♦♥ r Bt t②♣ ♣♦♥t ♦♥ ts r♥s ♥ qr♠ t s♠♣ ③r♦ ♥ λ1 = 0 ♥ ♥♦ ♦tr ♥s♦♥ t ♠♥r② ①s rstrt♦♥ ♦ t s②st♠ t♦ ♥tr ♠♥♦W c s t♦r♠

ξ = aξ2,

r ② ♥t♦♥ t ♦♥t a s ♥♦♥③r♦ t ♥♦♥♥rt ♦ rt♦♥♣♦♥t ❲ t r s ♥ tr t ♦♦♥ s♥rts ♥ ♠t

• ♥ t♦♥ r ♥ λ2 ♣♣r♦s t ♠♥r② ①s ♥ Wc ♦♠st♦♠♥s♦♥

λ1 = 0, λ2 = 0.

s r t ♦♥t♦♥s ♦r t ♦♥♦♥s ♦r ♦③r♦ rt♦♥♦ ts rt♦♥ ♥ n ≥ 2 r② t ♦♥♦♥s rt♦♥ ♥ s♦ ♦t ♦♥ ♦♣ rt♦♥ r s ω0 ♣♣r♦s ③r♦t ts ♣♦♥t t♦ ♣r② ♠♥r② ♥s ♦ ♥ ♦ ③r♦♥

• ♦ ①tr ♦♠♣① ♥s λ2,3 rr t t ♠♥r② ①s ♥W c ♦♠str♠♥s♦♥

λ1 = 0, λ2,3 = ±iω0,

♦r ω0 > 0 s ♦♥t♦♥s ♦rrs♣♦♥ t♦ t ♦♦♣ rt♦♥ ♦s②ts rt♦♥ ♥ ♦r ♦♥② n ≥ 3

• ♥ λ1 = 0 r♠♥s s♠♣ ♥ t ♦♥② ♦♥ ♦♥ t ♠♥r② ①sdimWc = 1 t t ♥♦r♠ ♦r♠ ♦♥t a ♥ss

λ1 = 0, a = 0.

s r t ♦♥t♦♥s ♦r s♣ rt♦♥ s ♣♦ss ♥ s②st♠s tn ≥ 1

12 Preliminaries

❲ ♦♥sr ♥♦ ♦♣ rt♦♥ r BH t t②♣ ♣♦♥t ♦♥ ts rt s②st♠ s ♥ qr♠ t s♠♣ ♣r ♦ ♣r② ♠♥r② ♥sλ1,2 = ±ω0, ω0 > 0 ♥ ♥♦ ♦tr ♥s t Reλ = 0 ♥tr ♠♥♦W c s t♦♠♥s♦♥ ♥ ts s ♥ tr r ♣♦r ♦♦r♥ts (ρ, φ) ♦r t rstrt♦♥ ♦ t s②st♠ t♦ ts ♠♥♦ s ♦rt② q♥t t♦

ρ = l1ρ3,

φ = 1,

r ② ♥t♦♥ t rst ②♣♥♦ ♦♥t l1 6= 0 t ♥♦♥♥rt ♦♣♣♦♥t ❲ ♠♦♥ ♦♥ t r ♥ ♥♦♥tr t ♦♦♥ ♣♦ssts

• ♦ ①tr ♦♠♣①♦♥t ♥s λ3,4 ♣♣r♦ t ♠♥r② ①s ♥W c ♦♠s ♦r♠♥s♦♥

λ1,2 = ±iω0, λ3,4 = ±iω1,

t ω0,1 > 0 s ♦♥t♦♥s ♥ t ♦♣♦♣ rt♦♥ t s ♣♦ss♦♥② n ≥ 4

• rst ②♣♥♦ ♦♥t l1 ♠t ♥s λ1,2 = ±iω0 r♠♥ s♠♣♥ tr♦r dimW c = 2

λ1,2 = ±iω0, l1 = 0.

t ts ♣♦♥t srt ♥r♦♥♦♦♣ rt♦♥ tr♥s ♥t♦ s♣rrt♦♥ ♦r rs ❲ ts ♥t t♥ rt♦♥ ♦r ♥r③ ♦♣rt♦♥ t s ♣♦ss n ≥ 2 s ♦r t s♣ rt♦♥ t t♥ rt♦♥ ♥♥♦t tt ② ♠r② ♠♦♥t♦r♥ t ♥s

s rt♦♥ ♣♦♥ts ♥ ♠t ♥ ♥r t♦♣r♠tr s②st♠s ♠♦♥ ♦♥ ♦♠♥s♦♥♦♥ rt♦♥ rs ♦ ts rt♦♥s s rtr③ ② t♦ ♥♣♥♥t ♦♥t♦♥s r r ♥♦ ♦tr ♦♠♥s♦♥t♦ rt♦♥s ♥ ♥r ♦♥t♥♦st♠ s②st♠s t ♦♦s tt ♥ ♣♥r s②st♠♦♥② tr ♦♠♥s♦♥t♦ rt♦♥ r ♣♦ss t t♥ t s♣ ♥ t♦♥♦♥s ❲ r♣♦rt ♠♦r ts ♦r ts tr t②♣s

Preliminaries 13

Cusp bifurcation

♦♥sr t s②st♠

x = f(x, α), x ∈ R, α = (α1, α2)T ∈ R2,

t s♠♦♦t ♥t♦♥ f s t α∗ = 0 ♥ qr♠ x∗ = 0 s②st♠s s♣ rt♦♥ ♥ (x∗, α∗)

λ = fx(x∗, α∗) = 0, a =1

2fxx(x∗, α∗) = 0.

ss♠ s♦ tt t ♦♦♥ ♥rt② ♦♥t♦♥s r sts

fxxx(0, 0) = 0, (fα1fxα2

− fα2fxα1

)(0, 0) = 0.

♥ tr r s♠♦♦t ♥rt ♦♦r♥t ♥ ♣r♠tr ♥s tr♥s♦r♠♥t s②st♠ ♥t♦

η = β1 + β2η ± η3,

tt s t t♦♣♦♦ ♥♦r♠ ♦r♠ ♦r t s♣ rt♦♥ s s②st♠ ♦♥sr♦r ♥st♥ t ♣s ♥ r♦♠ ♦♥ t♦ tr qr ♦ rt♦♥ ♦rst rt♦♥ r T = (β1, β2) : 4β

32 − 27β2

1 = 0 ♦♥ t (β1, β2)♣♥ ♥ ②t ♣r♦t♦♥ ♦♥t♦ t ♣r♠tr ♣♥ ♦ t r Γ

β1 + β2η − η3 = 0,

β2 − 3η2 = 0.

r T s t♦ r♥s T1 ♥ T2 ♠t t♥♥t② t t s♣ ♣♦♥t(0, 0) s r rst♥ s t ♣r♠tr ♣♥ ♥t♦ t♦r♦♥s ♥ r♦♥ ♥s t tr r tr qr t♦ st ♥

♦♥ ♥st ♥ r♦♥ ♦ts t tr s s♥ qr♠ s st ♥ ♦ rt♦♥ t rs♣t t♦ t ♣r♠tr β1 ts ♣ r♦ss tr T1 ♦r T2 t ♥② ♣♦♥t ♦tr t♥ t ♦r♥ t r T1 s r♦ssr♦♠ r♦♥ t♦ r♦♥ t rt st qr♠ ♦s t t ♥st♦♥ ♥ ♦t s♣♣r s♠ ♣♣♥s t♦ t t st qr♠ ♥ t♥st qr♠ t T2 ♣♣r♦ t s♣ ♣♦♥t r♦♠ ♥s r♦♥ tr qr ♠r t♦tr ♥t♦ tr♣ r♦♦t ♦ t rt♥ s ♦ t ♥♦r♠♦r♠

♥ s ② t♦ ♣rs♥t ts rt♦♥ s t♦ ♣♦t t qr♠ ♠♥♦ ♦t ♥♦r♠ ♦r♠

14 Preliminaries

rt♦♥ r♠ ♥ t ♣r♠tr s♣ ♦ ♦♥♠♥s♦♥ s♣ ♣♦♥t ❬❪

M = (η, β1, β2) : β1 + β2η ± η3 = 0,♥ R3 st♥r ♣r♦t♦♥ ♦ M ♦♥t♦ t (β1, β2) ♣♥ s s♥rts ♦ ♦t②♣ ♦♥ t ♦ rt♦♥ r ①♣t ♥ t ♦r♥ r s♣ s♥rt②s♦s ♣ ♦t tt t ♦ rt♦♥ r s s♠♦♦t r②r ♥ s ♥♦♦♠tr s♥rt② t t s♣ ♣♦♥t t t s t ♣r♦t♦♥ tt ♠s t ♦♣r♠tr ♦♥r② ♥♦♥s♠♦♦t

s♣ rt♦♥ ♠♣s t ♣rs♥ ♦ t ♣♥♦♠♥♦♥ ♥♦♥ s ②strss tstr♦♣ ♠♣ t♦ r♥t st qr♠ s ② t s♣♣r♥♦ tr st qr♠ ♦ rt♦♥ s t ♣r♠trs r②

Bautin (generalized Hopf) bifurcation

♦♥sr t ♣♥r s②st♠

x = f(x, α), x = (x1, x2)T ∈ R2, α = (α1, α2)

T ∈ R2,

t s♠♦♦t ♥t♦♥ f s t α∗ = 0 ♥ qr♠ x∗ = 0 s②st♠ s t♥ rt♦♥ ♥ (x∗, α∗) t qr♠ s ♣r② ♠♥r② ♥s

Preliminaries 15

♥ t rst ②♣♥♦ ♦♥t ♥ss

λ1,2 = ±iω0, ω0 > 0, l1 = 0.

♥ t ♣rs♥ ♦ ts rt♦♥ ♥ ①♣t t ①st♥ ♦ t♦ ♠t ②s♦r ♥r② ♣r♠tr s ♦r♦r t② ♦ ♥ s♣♣r ♦♥ r♠♥t♥ r♦♠ t ♦♠♥s♦♥ ♣♦♥t

♦r ♣rs② t ♥♦r♠ ♦r♠ ♥ ♣♦r ♦♦r♥ts (ρ, φ) s t♦ ♥♣♥♥tqt♦♥s

ρ = ρ(β1 + β2ρ2 − ρ4),

φ = 1.

s♦♥ qt♦♥ srs r♦tt♦♥ t ♥t ♥r ♦t② s②st♠♠ts tr qr♠ ρ = 0 ♥ ♣♦st qr ♦ t rst qt♦♥ sts②

β1 + β2ρ2 − ρ4 = 0

♥ sr rr ♠t ②s s qt♦♥ ♥ ③r♦ ♦♥ ♦r t♦ ♣♦sts♦t♦♥s ②s s ♥ s♥ ♥ r ts s♦t♦♥s r♥ r♦♠ ttr ♦♥ ♦♥ t ♥

H = (β1, β2) : β1 = 0♥ ♦ ♥ s♣♣r t t ♣r♦

T = (β1, β2) : β22 + 4β1 = 0, β2 > 0.

♥ H ♦rrs♣♦♥s t♦ t ♦♣ rt♦♥ qr♠ s st ♦r β1 < 0♥ ♥st ♦r β1 > 0 rst ②♣♥♦ ♦♥t l1(β) = β2 r♦r tt♥ rt♦♥ ♣♦♥t β1 = β2 = 0 s♣rts t♦ r♥s H− ♥ H+ ♦rrs♣♦♥♥ t♦ ♦♣ rt♦♥ t ♥t ♥ t ♣♦st ②♣♥♦ ♦♥t st ♠t ② rts r♦♠ t qr♠ r♦ss H− r♦♠ t t♦ rt ♥ ♥st ② ♣♣rs r♦ss H+ ♥ t ♦♣♣♦st rt♦♥ ②s♦ ♥ s♣♣r ♦♥ t r T ♦rrs♣♦♥♥ t♦ ♥♦♥♥rt ♦ rt♦♥ ♦ t ②s rs t ♣r♠tr ♣♥ ♥t♦ tr r♦♥s ♥ r♦♥ ♥ t ♣r♠tr ♣♥ t s②st♠ s s♥ st qr♠ ♥ ♥♦ ②s

r♦ss♥ t ♦♣ rt♦♥ ♦♥r② H− r♦♠ r♦♥ t♦ r♦♥ ♠♣s t♣♣r♥ ♦ ♥q ♥ st ♠t ② srs ♥ ♥tr r♦♥ r♦ss♥ t ♦♣ ♦♥r② H+ rts ♥ ①tr ♥st ② ♥s t rst♦♥ t qr♠ r♥s ts stt② ♦ ②s ♦ ♦♣♣♦st stt② ①st♥s r♦♥ ♥ s♣♣r t t r T tr♦ ♦ rt♦♥ tt s s♥ st qr♠

16 Preliminaries

rt♦♥ r♠ ♥ t ♣r♠tr s♣ ♦ t♥ ♣♦♥t ❬❪

Bogdanov-Takens (double-zero) bifurcation

♦♥sr t ♣♥r s②st♠

x = f(x, α), x = (x1, x2)T ∈ R2, α = (α1, α2)

T ∈ R2,

t s♠♦♦t ♥t♦♥ f s t α∗ = 0 ♥ qr♠ x∗ = 0 s②st♠ s ♦♥♦♥s rt♦♥ ♥ (x∗, α∗) t qr♠ s t♦ ③r♦ ♥s

λ1,2 = 0.

rt♦♥ ♦♥t♦♥ ♠♣s tt

det J(x∗, α∗) = 0, tr J(x∗, α∗) = 0.

s ♦r t t♥ rt♦♥ t ♦♥♦♥s rt♦♥ s rs t♦ ♠t② rt♦♥ ♥♠② t ♣♣r♥ ♦ t ♦♠♦♥ ♦rt ♦r ♥r② ♣r♠tr s

Preliminaries 17

t ♥ ♣r♦ tt ♥② ♥r ♣♥r t♦♣r♠tr s②st♠ x = f(x, α)♥ t α = 0 ♥ qr♠ tt ①ts t ♦♥♦♥s rt♦♥ s♦② t♦♣♦♦② q♥t ♥r t qr♠ t♦ ♦♥ ♦ t ♦♦♥ ♥♦r♠♦r♠s

η1 = η2,

η2 = β1 + β2η1 + η21 ± η1η2.

qr ♦ t s②st♠ r ♦t ♦♥ t ♦r③♦♥t ①s η2 = 0 ♥ sts②t qt♦♥

β1 + β2η1 + η21 = 0,

♥ t♥ ③r♦ ♥ t♦ r r♦♦ts sr♠♥♥t ♣r♦

T = (β1, β2) : 4β1 − β22 = 0

♦rrs♣♦♥s t♦ ♦ rt♦♥ β2 6= 0 t♥ t ♦ rt♦♥ s ♥♦♥♥rt♥ r♦ss♥ T r♦♠ rt t♦ t ♠♣s t ♣♣r♥ ♦ t♦ qr E1 ♥E2 ♣♦♥t β = 0 s♣rts t♦ r♥s T− β2 < 0 ♥ T+ β2 > 0 ♦ t♦ r t ♣ss tr♦ T− ♠♣s t ♦s♥ ♦ st ♥♦ E1 ♥ s ♣♦♥t E2 r♦ss♥ T+ ♥rts ♥ ♥st ♥♦ E1 ♥ sE2 rt ①s β1 = 0 s ♥ ♦♥ t qr♠ E1 s ♣r ♦♥s t ③r♦ s♠ λ1 + λ2 = 0 ♦r ♣rt

H = (β1, β2) : β1 = 0, β2 < 0,

♦rrs♣♦♥s t♦ ♥♦♥♥rt ♦♣ rt♦♥ t ♣♣r ①s s ♥♦♥rt♦♥ ♥ ♦rrs♣♦♥♥ t♦ ♥tr s ♦♣ rt♦♥ s rs t♦ st ♠t ② l1 < 0 ② ①sts ♥r H ♦r β1 < 0 qr♠E2 r♠♥s s ♦r ♣r♠tr s t♦ t t ♦ t r T ♥ ♦s♥♦t rt r r ♥♦ ♦tr ♦ rt♦♥s ♥ t ②♥♠s ♦ ts s②st♠♦♦♥ t r ♥ r♦♥ tr r ♥♦ qr ♥ ts ♥♦ ♠t ②s

♥tr♥ r♦♠ r♦♥ ♥t♦ r♦♥ r♦ss♥ t ♦♠♣♦♥♥t T− ♦ t ♦ r②s t♦ qr s ♥ st ♥♦ ♥ t ♥♦ tr♥s ♥t♦ ♦s♥ ♦ss stt② s r♦ss t ♦♣ rt♦♥ ♦♥r② H st ♠t ②s ♣rs♥t ♦r ♣r♠tr s ♦s t♦ t t ♦ H ♦s ♥♦t rt ♥r♦♥ ♥ ♣♣r♦s t s tr♥♥ ♥t♦ ♦♠♦♥ ♦rt t P ♥ ♥

♥st ♥♦ ♥ s ①st♥ ♦r t ♣r♠tr s ♥ r♦♥ ♦♥ s♣♣r t t ♦ r T+

18 Preliminaries

rt♦♥ r♠ ♥ t ♣r♠tr s♣ ♦ ♦♥♦♥s ♣♦♥t ❬❪

Diffusion-driven instability in reaction-diffusion systems

t♦♥s♦♥ qt♦♥s r ② s ♦r ♠♦♥ ♠ rt♦♥s ♦♦ s②st♠s ♣♦♣t♦♥ ②♥♠s ♥ ♥r rt♦r ♣②ss ② r ♦ t♦r♠

ut = D∆u+ f(u, λ)

r u = (u1, . . . , uk) r♣rs♥ts r♦s sst♥s ♥ ♠ rt♦♥ ♦r s♣s♦ ♦♦ s②st♠ λ ∈ Rp s t♦r ♦ ♦♥tr♦ ♣r♠trs ∆ s t ♣♦♣rt♦r ♥ t s♣t rs ♥ D ∈ Rk×k s ♦t♥ ♦♥ ♥ ♦rrs♣♦♥st♦ t s♦♥ rts t② sr s♦♥ ♦ r♥t sst♥s ♦r s♣s ♥t♦♥ f : Rk ×Rp → Rk s t♦r ♦ s♠♦♦t ♥♦♥♥r ♥t♦♥s ♥ r♣rs♥tst rt♦♥ ♠♦♥ t sst♥s

♦♦♥ t ♠ rt♦♥s r♥ sst tt ♥r rt♥ ♦♥t♦♥s ♠s ♥ rt ♥ s ♥ s ② s t♦ ♣r♦ st② stt

Preliminaries 19

tr♦♥♦s s♣t ♣ttr♥s t♦ s♦♥r♥ ♥stt② ♦ t st②stt

rt♦♥s♦♥ s②st♠ ①ts s♦♥r♥ ♥stt② s♦♠t♠s r♥ ♥stt② t ♦♠♦♥♦s st② stt s st t♦ s♠ ♣rtrt♦♥s ♥t s♥ ♦ s♦♥ t ♥st t♦ s♠ s♣t ♣rtrt♦♥s ♥ s♦♥ s♣rs♥t ♥tr♣② t♥ ♥trt♦♥ t ♥♦♥♥r ♥ts ♥ s♦♥ sr ♥ r♥ t s♣t② ♥♦♠♦♥♦s ♥stt② ts ♠♥s♠ tr♠♥t s♣t ♣ttr♥ tt ♦s ♥ t ♦♦♥ ♥ssr② ♥ s♥t♦♥t♦♥s ♦r s♦♥r♥ ♥stt② ♦ t st② stt ♥ t s ♦ ♥rs♦♥ ♦♥ D ♥ s♦ ♥ t s ♦ r♦ss s♦♥ ♠♦r ♦♠♣① s♦♥tr♠s

Turing instability analysis

❲ r t ♥ssr② ♥ s♥t ♦♥t♦♥s ♦r s♦♥r♥ ♥stt②♦ t st② stt ♥ t ♥tt♦♥ ♦ s♣t ♣ttr♥ ♦r ♥r s②st♠ ♦ t♦rt♦♥s♦♥ qt♦♥s

❲ ss♠ tt t ♣r♦ss ♣♣♥s ♥ ♦♥ r♦♥ Ω ⊂ RN ss♠ t♦ s♠♦♦t C2 ♦♥ ♥ ♦♥♥t r t ♠♦s ♥s r♠♦♥ ❲ s ♦♥♥trt ♦♥ t s ♥ t ♠♦s s s♣ss ts ♦♥ s♦♥ ♦t② t♥ ♥♦♥rt r♦♥ s♣♣♦s t♦ trst rt♦♥s♦♥ s②st♠s ♥ ♠♥s♦♥③ ♥ s t♦ t t♥r ♦r♠

ut −∆u = γf(u, v),

vt − d∆v = γg(u, v),

r t ♥♥♦♥s r u := u(x, t) ♥ v := v(x, t) t ∈ R, x ∈ Ω f, g rs♠♦♦t ♥♦♥♥r ♥t♦♥s ♥ r♣rs♥t t rt♦♥ ♠♦♥ t sst♥s ♦r t♥trt♦♥ ♠♥s♠s t♥ s♣s d > 0 s t rt♦ ♦ s♦♥ ♦♥ts♥ γ r♣rs♥ts t rt str♥t ♦ t rt♦♥ tr♠s ❲ s ss♠ tt♠♦s s♣s r ♦♥♥ ♥ t r♦♥ Ω s♦ tt t ① ♦ ♥st② ♦ s♣s t t ♦♥r② ∂Ω s s s t ♦♠♦♥♦s ♠♥♥ ♦♥r②♦♥t♦♥s ♥♦ ♥st② ① ♦ s♣s t t ♦♥r②

n(x) · ∇xu = 0, n(x) · ∇xv = 0, x ∈ ∂Ω,

r ∂Ω s t ♦s ♦♥r② ♦ t rt♦♥ s♦♥ ♦♠♥ Ω ♥ n s t♥t ♦tr ♥♦r♠ t♦ ∂Ω ♥ t♦♥ ♦♥sr ♥t ♦♥t♦♥s

20 Preliminaries

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω.

♥ r ♦♥r♥ t s♦♥r♥ ♥stt② r ♥trst ♥ ♥r♥stt② ♦ ♦♠♦♥♦s st② stt tt s s♦② s♣t② ♣♥♥t ♥ tt ♥♦ s♣t rt♦♥ u ♥ v sts② t s②st♠

ut = γf(u, v),

vt = γg(u, v);

♥ ♦♠♦♥♦s st② stt (u0, v0) s st② stt ♦r t rt♦♥ ♣rt t ♥♦s♣t rt♦♥ ♥♠② t s♦t♦♥ ♦

f(u, v) = 0, g(u, v) = 0.

♥ssr② ♦♥t♦♥ t♦ r♥ ♥stt② s tt ♥ s♥ ♦ ♥② s♣trt♦♥ t ♦♠♦♥♦s st② stt ♠st ♥r② st rst tr♠♥t ♦♥t♦♥s ♦r ts t♦ ♦ ♥rs♥ ♦t t st② stt (u0, v0) st

w =

(u− u0v − v0

)

,

♥ t ♦♠♦♥♦s s②st♠ ♦♠s ♦r |w| s♠

wt = γJw, J =

(

fu fv

gu gv

)

|(u0,v0)

.

❲ ♥♦ ♦♦ ♦r s♦t♦♥s ♦ t ♦r♠

w ∝ eλt

r λ s t ♥ st② stt w = 0 s ♥r② st Reλ < 0 s♥♥ ts s t ♣rtrt♦♥ w → 0 s t → ∞ ♥s λ r t s♦t♦♥s♦ det(γJ − λI) = 0 s q♥t t♦

λ2 − γ(fu + gv)λ+ γ2(fugv − fvgu) = 0.

♥r stt② tt s Reλ < 0 s r♥t ♥ ♦♥②

tr J = fu + gv < 0, det J = fugv − fvgu > 0,

♥ ♥r ♠♣♦s rt♥ ♦♥str♥ts ♦♥ t ♣r♠trs

Preliminaries 21

♦ ♦♥sr t rt♦♥ s♦♥ s②st♠ ♥ ♥ ♥rs r♦♥ tst② stt s w = 0 t♦ t

wt = γJw +D∆xw, D =

(

1 0

0 d

)

.

♦ s♦ ts s②st♠ ♦ qt♦♥s st t♦ t ♦♥r② ♦♥t♦♥s rst ♥W (x) t♦ t t♠♥♣♥♥t s♦t♦♥ ♦ t s♣t ♥ ♣r♦♠ ♥②

∆xW + k2W = 0, n(x)∇xW = 0

♦r x ♦♥ ∂Ω r k s t ♥ t ♦♠♥ s ♦♥♠♥s♦♥ ♦r ①♠♣ t ♥tr [0, a] tt

W ∝ cos(nπx/a) r n s ♥ ♥tr ♥ ♥ ts s s k = nπ/a ♥1/k s ♠sr ♦ t ♣ttr♥ ♥ t s ♣r♦♣♦rt♦♥ t♦ t ♥tω = 2π/k = 2a/n t ♥ k s ♥♠r r♦♠ ♥♦ ♦♥ rrt♦ k s t ♥♠rs ❲t ♥t ♦♠♥s tr s srt st ♦ ♣♦ss♥♠rs s♥ n s ♥ ♥tr ♦♥sr t♦♠♥s♦♥ ♦♠♥ ♦r①♠♣ t rt♥ [0, Lx]× [0, Ly] tt W ∝ cos(k1πx/L1) cos(k2πy/L2)t k1, k2 ∈ N ♥

k2 =

(k1π

Lx

)2

+

(k2π

Ly

)2

.

t Wk(x) t ♥♥t♦♥ ♦rrs♣♦♥♥ t♦ t ♥♠r k ♥♥t♦♥ Wk stss ③r♦ ① ♦♥r② ♦♥t♦♥s s t ♣r♦♠ s ♥r ♥♦ ♦♦ ♦r s♦t♦♥s w(x, t) ♥ t ♦r♠

w(x, t) =∑

k

ckeλtWk(x),

r t ♦♥st♥ts ck r tr♠♥ ② ♦rr ①♣♥s♦♥ ♦ t ♥t ♦♥t♦♥s ♥ tr♠s ♦ Wk(x) ♥ λ s t ♥ tr♠♥s t♠♣♦r r♦tsttt♥ ts ♦r♠ ♥t♦ t ♥r③ rt♦♥s♦♥ qt♦♥ ♥ ♠♥t♥eλt t ♦r k

λWk = γJWk −Dk2Wk.

❲ ♦♦ ♦r ♥♦♥tr s♦t♦♥s ♦r Wk s♦ t λ r tr♠♥ ② t r♦♦ts ♦ trtrst ♣♦②♥♦♠

det(λI − γJ +Dk2) = 0,

22 Preliminaries

s q♥t t♦ ♥ t ♥s ♦ t ♠tr① Mk = γJ − Dk2 ❲ tt ♥s λ(k) s ♥t♦♥s ♦ t ♥♠r k s t r♦♦ts ♦

λ2 + λ[k2(1 + d)− γ(fu + gv)] + h(k2) = 0,

rh(k2) = dk4 − γ(dfu + gv)k

2 + γ2 det J.

♦t tt t ♦♥t ♦ λ s −tr Mk h(k2) = detMk s♥ tr Mk < 0

♦r k t ♦♥② ② t♦ ♥ ♥st st② stt t♦ s♣t str♥ss tt detMk < 0 ♦r s♦♠ k t s ♦♥sr t qt♦♥ ♦r λ st②stt (u0, v0) s ♥st t♦ s♣t str♥s Reλ(k) > 0 ♦r s♦♠ k 6= 0♥ tr J = (fu + gv) < 0 ♥ k2(1 + d) > 0 ♦r k 6= 0 t ♦♥② ② s tth(k2) < 0 ♦r s♦♠ k ♥ rqr det J > 0 ♦r t ♦♠♦♥♦s st②stt t ♦♥② ♣♦sst② ♦r h(k2) t♦ ♥t s tt (dfu + gv) > 0 ♥(fu + gv) < 0 ts ♠♣s tt d 6= 1 ♥ fu ♥ gv ♠st ♦♣♣♦st s♥s ♦ rtr rqr♠♥t t♦ t♦s ♠♣t ② t ♥r stt② ♦ t ♦♠♦♥♦sst② stt s dfu + gv > 0 s♦ tt d 6= 1 s ♥qt② s ♥ssr② t ♥♦ts♥t ♦r Reλ > 0 ♦r h(k2) t♦ ♥t ♦r s♦♠ ♥♦♥③r♦ k t ♠♥♠♠hmin ♠st ♥t ♠♥tr② r♥tt♦♥ t rs♣t t♦ k2 s♦s tt

hmin = γ2[

det J − (dfu + gv)2

4d

]

, k2min = γdfu + gv

2d,

ts t ♦♥t♦♥ tt h(k2) < 0 ♦r s♦♠ k2 6= 0 s

(dfu + gv)2 − 4d det J > 0.

t rt♦♥ ♥ hmin = 0 r♦♠ t ♥t♦♥ ♦ hmin rqr det J =(dfu + gv)

2/4d s q♥t t♦

d2f 2u + 2(2fvgu − fugv)d+ g2v = 0,

♥ s♦ ♦r ① ♣r♠trs ts ♥s rt s♦♥ ♦♥t rt♦ dc(> 1)s t ♣♣r♦♣rt r♦♦t ♦ ts qt♦♥ rt ♥♠r kc s t♥ ♥ ②

k2c = γdcfu + gv

2dc= γ

√

det J

dc.

❲♥r h(k2) < 0 t qt♦♥ ♦r λ s s♦t♦♥ s ♣♦st ♦r ts♠ r♥ ♦ ♥♠rs tt ♠ h < 0 ❲t d > dc t r♥ ♦ ♥st♥♠rs k21 < k2 < k22 s ♦t♥ r♦♠ t ③r♦s k21 ♥ k22 ♦ h(k2) = 0 s

Preliminaries 23

k21,2 =γ

2d

(

(dfu+ gv)±√

(dfu + gv)2 − 4d det J)

.

♦♥sr t s♦t♦♥ w t ♦♠♥♥t ♦♥trt♦♥s s t ♥rss r t♦s♠♦s ♦r Reλ(k2) > 0 s♥ ♦tr ♠♦s t♥ t♦ ③r♦ ①♣♦♥♥t② ❲tr♠♥ t r♥ k21 < k2 < k22 r

w(x, t) ∝∑

k1

k2ckeλ(k2)tWk(x)

♦r r t

♠♠r③♥ t ♦♥t♦♥s ♦r t ♥rt♦♥ ♦ s♣t ♣ttr♥s ② t♦s♣srt♦♥s♦♥ ♠♥s♠s r

tr J = fu + gv < 0,

det J = fugv − fvgu > 0,

dfu + gv > 0,

(dfu + gv)2 − 4d(fugv − fvgu) > 0,

r♠♠r♥ tt rts r t t t ♦♠♦♥♦s st② stt(u0, v0)

Turing instability analysis with cross-diffusion

♥t② ♥♦♥♥r s♦♥ tr♠s ♦r r♦sss♦♥ ♣♣r t♦ ♠♦ r♥t ♣②s ♣♥♦♠♥ ♥ r♥t ♦♥t①ts ♣♦♣t♦♥ ②♥♠s ♦♦②♥ ♠ rt♦♥s ❬ ❪ r♦sss♦♥ tr♠s s♦ ♥tr♦ ♥t r♥t ♦ t ♥st② ♦ ♦♥ s♣s ♥s ① ♦ ♥♦tr s♣s ♥ ts♦♥t①t t ♥r t♦s♣s rt♦♥r♦sss♦♥ s②st♠ s

ut −∆x(a(u, v)) = f(u, v)

vt −∆x(b(u, v)) = g(u, v),

r t ♥♥♦♥s r u := u(x, t) ♥ v := v(x, t) t ∈ R, x ∈ Ω f, g rs♠♦♦t ♥♦♥♥r ♥t♦♥s ♥ r♣rs♥t t rt♦♥ ♠♦♥ t sst♥s ♦r t♥trt♦♥ ♠♥s♠s t♥ s♣s a, b r s♠♦♦t ♥t♦♥s ♣♥♥ ♦♥♦t u, v s♦ ♥ ts s ♦♥sr ♦♠♦♥♦s ♠♥♥ ♦♥r② ♦♥t♦♥s

n(x) · ∇xu = 0, n(x) · ∇xv = 0, x ∈ ∂Ω,

24 Preliminaries

r ∂Ω s t ♦s ♦♥r② ♦ t rt♦♥ s♦♥ ♦♠♥ Ω ♥ n s t♥t ♦tr ♥♦r♠ t♦ ∂Ω ♥ ♥ ♥t ♦♥t♦♥s

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω.

r st② t r♥ rt♦♥ ♥ t ♣rs♥ r♦sss♦♥ ♦♦♥ ttr♠♥♥ t trs♦ s s ② ♥r stt② ♥②ss ♦♦♥ t ♥r s ♦♥sr ♥ ♦♠♦♥♦s st② stt (u0, v0) s st ♦r trt♦♥ ♣rt ♥rs s②st♠ ♥ t ♥♦r♦♦ ♦ (u0, v0) s

wt = Jw +D∆xw, w =

(u− u0v − v0

)

,

r ♥♦

J =

(

J11 J12

J21 J22

)

:=

(

fu fv

gu gv

)

|(u0,v0)

, D =

(

d11 d12

d21 d22

)

:=

(

au av

bu bv

)

|(u0,v0)

.

♥ ts s t s♦♥ ♠tr① s ♥♦ ♦♥r ♦♥ ♠tr① s ♥ t ♣r♦sst♦♥ s ♥r② ♥♦♥③r♦ ♦♦♥ ♠♥ts ♣♣r ♥ t ♠tr① Drtr♠♦r t r♦sss♦♥ ♦♥ts ♦♠ t② ♦♥st♥t tr s tr♠♥ ② t ♦♠♦♥♦s st② stt ♦ t s②st♠ t r♦sss♦♥tr♠ s ♣♦st dkj > 0 t♥ t ① ♦ s♣s k s rt t♦r rs♥s ♦ t ♦♥♥trt♦♥ ♦ s♣s j rs dkj < 0 ♠♣s tt t ① srt t♦r ♥rs♥ s ♦ t ♦♥♥trt♦♥ ♦ s♣s j

❲ r ♥trst ♥ t ♥s ♦ t ♠tr①

Mk = J −Dk2 =

(

J11 − d11k2 J12 − d21k

2

J21 − d21k2 J22 − d22k

2

)

.

r♥ ♥stt② sts ♥ ♥ t st ♦♥ ♦ t ♦♦♥ ♦♥t♦♥s s♦t ♦r s♦♠ k

tr Mk < 0, detMk > 0,

rtr Mk = tr J − tr D,

detMk = detD k4 − (d11J22 − d12J21 − d21J12 + d22J11)k2 + det J.

♥ ♥r t s♥ ♦ t ♠♥ts ♦ t s♦♥ ♠tr①D ♥ t♥ ♦ t tr ♥t tr♠♥♥t s ♥♦t ♣rsr ss♠ tt ♦t ♠♥ ♥r③ s♦♥

Preliminaries 25

♦♥ts d11, d22 r ♣♦st t ♦♥t♦♥ ♦♥ t tr ♦ Mk s ②s sts♥ t ♦♠♦♥♦s st② stt s ♥r② st ♦r t rt♦♥ ♣rt ♥♦♥② t ♦t♦♥ ♦ t s♦♥ ♦♥t♦♥ s rs t♦ s♦♥r♥ ♥stt②s t ♦♥t♦♥ ♦r s♦♥r♥ ♥stt② ♦rs ♥

detMk = detD k4 − (d11J22 − d12J21 − d21J12 + d22J11)k2 + det J < 0.

♥ ♥ ♥r s♥ ♦ t tr♠♥♥t ♦ t s♦♥ ♠tr① D s ♥♦t ♣rsr ♦r ♦r ♥st♥ ♦r ♠ s②st♠s tr♠♦②♥♠s ♠♣♦ss ♥t♦♥ t ♦♥str♥t tt ♥s ♦ t s♦♥ ♠tr① ♠st r ♥♣♦st ♠♣s tt tr D > 0 ♥ detD > 0 ❬❪ s♦ ss♠ ttdetD > 0 t♥

d11J22 − d12J21 − d21J12 + d22J11 > 0

s ♥ssr② t ♥♦t s♥t ♦♥t♦♥ ♦r r♥ ♥stt② ♦♥st ♦t r♥ ♥stt② ♦rs ♥ detMk < 0 ♦s

(d11J22 − d12J21 − d21J12 + d22J11)2 > 4 det J detD.

♠♠r③♥ s ♥ t ♥r s ♥ tr D < 0 ♥ detD > 0 t ♦♥t♦♥s♦r t ♥rt♦♥ ♦ s♣t ♣ttr♥s ② t♦s♣s rt♦♥s♦♥ ♠♥s♠sr

tr J < 0,

det J > 0,

d11J22 − d12J21 − d21J12 + d22J11 > 0,

(d11J22 − d12J21 − d21J12 + d22J11)2 > 4 det J detD.

ts ♦♥t♦♥s ♦s t♥ t qr♠ (u0, v0) s r♦sss♦♥r♥ ♥stt② r♥ ♥stt②

Prt

②♥♠s ♦ ♣rt♦r♣r② ♠♦s t

str♦♥ t ♦♥ t ♣r② ♥

♣rt♦r♣♥♥t tr♦♣ ♥t♦♥s

1

Introduction

s Prt s t t ②♥♠s ♦ ♣rt♦r♣r② s②st♠ sr ♥ tr♠s♦ ♠♣ ♣r♠tr ♠♦ ❬ ❪ ♥ t ♠♦r♣ strtr ♦ t♣♦♣t♦♥s s ♥t ♥ ♣rtr t st strtr ♦r ♥sts ♥ ♠ts sr ♣♣ ts s t ♣rt♦r ♥ ♣r② ♣♦♣t♦♥s ♠② rtr③② st ♦♥ stt r r♣rs♥t♥ tr ♥♥ ♥ tr♠s ♦ ♦♠sss♣t♥t ♦r♦r ss♠ ♠t ♥ ♦♥tr♦ ♥r♦♥♠♥t ♦r ♥st♥ r♥♦s ♥ t♠♣rtr ♥ ♠t② r ♠♥t♥ ♣♣r♦①♠t② ♦♥st♥ts♦ tt ♥ ♦♥sr t♠ ♥♣♥♥t ♦♦♦ ♣r♠trs ♥ ♥t ts♣t strt♦♥ ♦ t ♥s

♥ ts r♠♦r t ♦ ②♥♠s ♦ ♣rt♦r♣r② s②st♠ s sr ② s②st♠ ♦ t♦ ♦r♥r② r♥t qt♦♥s ♥ s ♠♥② rtr③ ② t♦r♠t♦♥s ♦ t ♣r② r♦t rt ♥ t s♥ ♦ ♣rt♦rs ♥ ♦ t ♣r②♦♥s♠♣t♦♥ rt ② ♣rt♦rs t ts rt ①♣rss s

♣r② r♦t rt = rXG(X),

♣r② ♦♥s♠♣t♦♥ rt = Y F (X, Y ),

r X ♥ Y r ♣r② ♥ ♣rt♦r ♥♥s r ss♠ tt t s ♠sr ♥ tr♠s ♦ ♦♠sss♣t ♥t ♠♦ ♥t♦♥s G(X) ♥ F (X, Y ) rs♣ rts ♥ tr s♣s tr♠♥ t t②♣ ♦ ♣r② r♦t r s ♥ ♥r t♠①♠♠ s♣ r♦t rt ♥ ♣rt♦♥ ♣r♦sss rs♣t② ② str♦♥②♣♥ ♦♥ t s ss♠♣t♦♥s ♠ ♦♥ t ♦♦♦ ♣r♦sss t♦ s♠t♥ tr s♣ s ♦t♥ ♥♥♦♥ ts ♦♥② s♦♠ ♦ tr qtt ♣r♦♣rts ♥ s♣ ♥ ts s r ♥ t ♥r③ ♦r ♣rt② s♣ ♠♦❬ ❪ rt♥s ♦♥sr strtr t ♥♦s ♠ ss s♣

1 ♦st ♦ t ♦♥t♥ts ♦ ts Prt ♣♣r ♦♥ ♦♥♥r ♥②ss ❲♦r ♣♣t♦♥s ❬❪

30 1 Introduction

ss♠♣t♦♥ ♦t t ♥tr ♦ t ♣rt♦♥ ♣r♦ss ♥ t t②♣ ♦ r♦t t♥ ② s♣ ♠♦ r t ♠♥ ♥t♦♥s r s♣ ♦t r② t s♣♣r♠tr ①♣rss♦♥ ♦♣ s tt ♠♦s s♣ ♥ ts r② ♥r ② ♣ ♦ ♦sr ♣♣r♦①♠t♦♥ t♦ rt② t♥ ♠♦r tt② s♣ ♦♥ss♣② ② t ♠♦ tt♥ ♣♦♥t ♦

♠♥ ♣r♣♦s ♦ ts ♦r s t♦ ♥stt t ②♥♠ ♦rs ♦ ♣rt♦r♣r② s②st♠ ♥ t ♠♦ ♥t♦♥s sr♥ t ♦♦ ♣r♦sss♦rr♥ ♥ t ♦♥sr tr♦♣ ♥ r ♥♦t s♣ ② ♥②t ①♣rss♦♥st ② s♦♠ rtrst ♣r♦♣rts tr♠♥♥ tr s♣s t s♦♥ ttts s s ♦r t ①st♥ ♥ stt② ♥②ss ♦ t qr♠ stts ♦t s②st♠ ♥ ①st♥ ♥ stt② ♦♥t♦♥s ♦ t qr♠ stts ♥ sts ♥ ♥r r♠♦r ♥ tr♠s ♦ s♦♠ r ♣r♠trs ❯♥♦rt♥t② t stt② ♥②ss ♦ ♠t ②s ♥♥♦t s② ♣r♦r♠ ♦♦♥ ts♥r ♣♣r♦ ♥ t ♠♦ ♥t♦♥s t♦ s♣ t♦ ♦ rtr ♥ tqtt ♥②ss tr t♦rs ❬ ❪ ♣rs♥t qt s♠r ♣♣r♦ t♦ t♥stt♦♥ ♦ t ②♥♠s ♥ ♥r③ ♠♦s ♥ t ♣r♦sss tt rt♥ ♥t♦ ♦♥t r ♥♦t rstrt t♦ s♣ ♥t♦♥ ♦r♠s ♥ s♦♠ ss trr ♥♦ ②♣♦tss ♦♥ t ♥r ♥t♦♥s s ♥ ts ♦r tt ♣♣r♦ ♦st♦ st② t ②♥♠ ♣r♦♣rts ♦ ♥r③ ♠♦s ♥ t r♠♦r ♦ ♦rt♦♥ t♦r② ♥ t ♦♥trr② t ② ss♠♣t♦♥ s tt t st ♦♥ st②stt ①sts ♥ t ♦ s②♠♣t♦t stt② ♥②ss ♦ tt ♥♥♦♥ st② stts t♥ ♣r♦r♠ r ♣♦♥t ♦ s r♥t ss♠♣t♦♥s ♦♥ t ♣r② r♦t♥t♦♥ ♥ ♦♥ t ♥t♦♥ rs♣♦♥s r rqr ♥ ♦rr t♦ t ♥t♦ ♦♥t♦♦ ♣r♦♣rts ♥ s♦ t♦ ♠t t ♥♠r ♦ t qr♠ stts ♠tt② t s②st♠ rtr♠♦r t stt② ♥ rt♦♥ ♥②ss s st ♦r qr♠ stt

♥ t ♦♣♠♥t ♦ ♣rt♦r♣r② t♦r② r♥t ♦r♠t♦♥s ♦ t ♣r②r♦t rt ♥ s♥ ♦ ♣rt♦rs ♥ ♣r♦♣♦s ♥ t trtr ♥ tss ♣rt♦r♣r② ♠♦ ♦ ♦t❱♦trr G(X) = 1 t r♦t ♦ t♣r② s ①♣♦♥♥t s ss♠♣t♦♥ ♦s ♦r s♠ X ♥ tr♦♣ rt♦♥s ♥t s②st♠ r str♥ ♥ ♠♦st t ♣r② s ♦♥s♠ ② t ♣rt♦r ②s♥r② qt rq♥t stt♦♥ ♥ ♥tr s ♦ s ♥ rt③ s♥ t♥ts t ♦rr♦♥ t t rstrt♦♥ ♥ ♣♦♣t♦♥ r♦t t♦ t♠ts ♦ t tts rr②♥ ♣t② ♥ ts s G(X) s♦ ♦♠ ♥t♦r s♥t② r X ♦ ts ♥ ❱♦trr ♠s ❬❪ ♣r♦♣♦s t s ♦ t♦st ♠♦

G(X) = 1− X

K+,

1 Introduction 31

r K+ s t rr②♥ ♣t② ♦ t tt ❬❪ ♥♦tr ss ♠♦t♥ ♥t♦ ♦♥t t ♦rr♦♥ t s t ♦♠♣rt③ ♠♦ ❬ ❪G(X) = log(K+/X) ♥♦t tt ♥ ts s r ♥ ♦s ♥♦t r♣rs♥t t♠①♠♠ s♣ r♦t rt t ♠♥s♦♥ s ♦r t r♦t rt tss♦ ♦♠♦♦r♦ ❬❪ ♥ s r② ♥r ♣rt♦r♣r② ♠♦ ♦♥sr ♦♥② t♦rr♦♥ t ♥ t ♣r② r♦t ② ss♠♥ G(X) ♥♦t s♣ ② ♥② ♥②t ①♣rss♦♥ t rtr③ ② t ♠♦♥♦t♦♥t② ♦♥t♦♥

G′(X) < 0, G(0) = 1, limx→+∞

G(X) < 0.

♥ t st s t ♣rt♦r♣r② ②♥♠s rst♥ r♦♠ t ♦ ♦ ♠♦♥♦t♦♥ G(X) ♥♠② ♦♥sr♥ ♦♥② t ♦rr♦♥ t ♦♥ t ♣r② r♦t ♥ ② ♥stt ② ♠♥② t♦rs ♦r ♠♦r ts s ❬❪ ♥ rr♥s tr♥

♥ t ♦tr ♥ t s ♥♦♥ tt ♥tr ♣♦♣t♦♥s ♠② ①t tr ♥t ♦r ♣♦st ♦rrt♦♥ t♥ G(X) ♥ X ♣♥♥ ♦♥ t r♥ ♦ ♥♥ ❬❪ ❬❪ ❬❪ ♣ ❬❪ ♣ ♥ ♣rtr ts ts r ♦♥ ♥ ♣♦♣t♦♥s ♦ s① ♦r♥s♠s ♥♦r t t♠ ♦r ♥ ♠t ♣❬❪ t trt♠♥t r♦♠ t ♦♦ ♣♦♥t ♦ r♥t t②♣s ♦ ts♥♦♥♠♦♥♦t♦♥ rt♦♥s♣s t s♦ t ❬❪ s ♥ ♥ t ❬❪

♥② t♦ t②♣s ♦ ts ♥ ♣r② ♣♦♣t♦♥ ♥ ♦♥sr ♥ ttrtr s ♦r ♥st♥ ❬ ❪ t ♥ t str♦♥ ♦r rt t ♣♥♥ ♦♥ t t tt t ♣r② r♦t♥t♦♥ G(X) s ♥♦♥♥t ♦r ♥t rs♣t② ♦r s♠ X ❬❪ ♦r♥t♦ ❬❪ tr s ♥ t t♥ t ♣r② r♦t ♥t♦♥ ♠st ♣♦sts♦♣ t X = 0 ♥ t♦♥ ♥ s ♦ str♦♥ t t ♥t♦♥ G(X) ♦♠s♥t ♦r s♥t② s♠ s ♦ X s t ♣r♦♠ ♦ ♠♥♠♠ ♣♦♣t♦♥s③ rss ❬❪ ❬❪ ♣ ♥ t s ♦rt ♥♦t♥ tt t ①t♥t♦♥ trs♦sr ♦t♥ t t♦ q♥t② ❬❪ ♣ ❬❪ ❬❪

♦ s♠t t t ♠♥② t♦rs ❬ ❪ s ♠t♣t t♦r t♦ t ♦st tr♠ (1−X/K+) ♦tr t♦rs ❬ ❪ s ♥ t tr♠ ♥ rrtt♥ ♥ t s ♦ str♦♥ ts ♠t♣t t♦rs t r ③r♦s s♦ t ♠♦s st ♥ ❬❪ ①♣t ♦♥♥♠② ❬❪ ♦r s♥ s♣s ②♥♠s s♠t t t ♥ t s♠ ②♥ ts ♠♦s t r♦t ♥t♦♥ G(X) ♥ ①♣rss s

G(X) = g( X

K+

)

=a0 + a1 X/K

+ − (x/K+)2

b0 + b1 X/K+=

(a0 +X/K+) (1−X/K+)

b0 + b1 X/K+,

32 1 Introduction

r ai, bi r ♠♥s♦♥ ♦♥ts t a1 = 1 − a0 b0 > 0 b1 = 0, 1 ♥t♦♥ G(X) ♥ ♥ s ♦♥ ♣♦st ③r♦ ♥ X = K+ ❲♥ a0 ≥ 0t s♦♥ ③r♦ s ♥♦♥♣♦st ♥ G(X) s♠ts t trs♥ a0 < 0 t s♦♥ ③r♦ s ♣♦st ♥ G(X) s♠ts str♦♥ t ❲♥♦t tt s♦ t ♣r② ① ♠♦ ♦♥sr ♥ ❬❪ ♥ ❬❪ ♥ sr ② ♥t♦♥ G(X) s ♥ t b0 = 0 ♥ b1 = 1 t ♦♥sr② ❩♦ t ❬❪ s ♥st ♠♦ ② rs♥ t ts♥ ♣r② r♦trt s♥ ♠♦♥♦t♦♥ ♦♥t②♣ t♦r t ♥ ❬❪ ♥t♦♥s G(x) ♥ ② r ♥♦t ♦♥ ♦r ♥rs♥ X

limX→+∞

|G(X)| = +∞.

♦r t ♦rr♦♥ t ♥ t♥ ♥t♦ ♦♥t s♦ ♥ tr♠s ♦ ♦♥♥t♦♥s s rst ♦♥ ♠♦ t G(X) ♥ s ♦♦s

G(X) s ♥ ♦r X ≤ XM ; G(X) = G(XM) ♦r X > XM > K+.

s♦♥ ♦♥ ♠♦ ♥ ♦r♠t ♥ tr♠s ♦ rt rt s ♥rs♥ ♦r s♠ X ♥ t♥ rs♥ ♥ ♥s♥ s X → +∞ t♦tr t ♦♥st♥t t rt ♣♦ss ♥②t r③t♦♥ ♦ ts ♠♦ s ♥ ②

G(X) = g( X

K+

)

∝[(

X

K+

)γ

exp

(

γ1−X/K+

X0/K+

)

− 1

]

,

r 0 < γ ≤ 1 ♥ X0 s t ♠①♠♠ ♣♦♥t ♦r G(X) ts ①♣rss♦♥ ♥ tr♠s ♦t t♦ ③r♦s K− ♥ K+ ♦ G(X) s

X0

K+= −1− ǫ

log ǫwith ǫ =

K−

K+.

♥ s♥ ♦ ♣rt♦rs t ♦t♦♥ ♦ t ♣r② ♣♦♣t♦♥ s ♦r♥ ② tqt♦♥

dX

dt= rXG(X).

t ♥t♦♥ G(X) sr♥ t str♦♥ t s ♥ s ♥ t♥ tr qr♠ stts ♦r t qt♦♥ Xeq = 0, K−, K+ rstt② ♣r♦♣rts r ♥♣♥♥t ♦ t ♦ ♦ t s♣ s♣ ♦G(X) 0, K+

r s②♠♣t♦t② st K− s ♥st ②♣ ♦rs ♦ t r♦t♥t♦♥s G(X) s ♥ ♥ ♥ t ♦rrs♣♦♥♥ tr♥s ♦ XG(X) rs♦♥ ♥ r

1 Introduction 33

G(X)

0K−

K+

0

1

0K− K+

0

1

0K− K+

0

1

XG(X)

0K−

K+

0

1

0K− K+

0

1

0K− K+

0

1

(3.5)

(3.6)

(3.7)

②♣ ♦rs ♦ t r♦t ♥t♦♥s G(X) s ♥ ♥ t ♥ t ♦rrs♣♦♥♥ tr♥s ♦ XG(X) rt

♥r ♥t♦♥ ♥ F (X, Y ) s t tr♦♣ ♥t♦♥ ♥ srst ♣rt♦r ♥t♦♥ rs♣♦♥s t♦ ♣r② ♥♥ ❬ ♣ ❪ t s ♥tr♦ ♥♣rt♦r♣r② ♠♦s t♦ t ♥t♦ ♦♥t t strt♦♥ ♠t♥ t ♣rt♦♥ ♣r♦ss ♦ ♦♦② ♠♥♥ ♥tr♣rtt♦♥ s♦♠ qtt ss♠♣t♦♥s♦♥ F (X, Y ) ♦t t ♣♥♥ ♦♥ X ♥ Y t♦ rqr

F (X, Y ) > 0 X > 0, Y ≥ 0, F (0, Y ) = 0, limX→+∞

F (X, Y ) < +∞∂F

∂X> 0,

∂F

∂Y< 0.

t rst t tr♦♣ ♥t♦♥ s ss♠ t♦ ♣♥♥t ♦♥② ♦♥ ♣r② ♥♥❬ ❪ ♥ ❬ ♣ ❪ ♦r♦r t ♦♥t②♣ ❬❪ ♦r t t②♣ ❬ ❪ ♠♦s r ♥tr♦ t♦ s♠t t strt♦♥ t ♦ t ♣rt♦♥♣r♦ss s ♦r♠t♦♥ ♦ F ♦♥② ♥ tr♠s ♦ X s rs t♦ t ♣r♦①s ♦♥r♠♥t ♥ ♦♦ ♦♥tr♦ ❬ ❪ ♥ t s ♥ t♦ ♥rt t♦t♦♠ ♦ t ①t♥t♦♥ ♦ t t♦ ♣♦♣t♦♥s ❬❪ t♦t t♥ ♥t♦ ♦♥t str♦♥ t ♥ t ♣r② r♦t ❬❪

34 1 Introduction

tr t ♥♦t♦♥ ♦ ♣r♣t t② ♦ ♦♦ s ♥tr♦ t s ssttt t tr♦♣ ♥t♦♥ s♦ ①♣rss ♥ tr♠s ♦ t rt♦ X/Y ♦ ♣r② t♦♣rt♦r ♥♥ ❬ ❪

F (X, Y ) = bf

(PX

Y

)

,

r b s t ♠①♠♠ ♣r② ♦♥s♠♣t♦♥ rt ♥ P s rrr t♦ s t ♥②♦ t ♣rt♦♥ ♣r♦ss s ♦r♠t♦♥ s♦s t ♦♠♥t♦♥ ♣r♦①s ♥♥ t♦♥ ♦s t♦ sr ①♣r♠♥t ♦srt♦♥s ♥ ♣rtr t ①t♥t♦♥♦ ♣rt♦r ♦r ♦t ♣r② ♥ ♣rt♦r ♣♦♣t♦♥s t♦t rs♦rt♥ t♦ t t ♦r ♥ ts ♣♣r♦ t tr♦♣ ♥t♦♥ s s♥rt② ♥ t ♦r♥❳ ❨ s ♣r♦♠ s s♦ ② s♦♠ t♦rs ② t ♦♣ ♠t♦ ❬❪ ♦r② t♠ rs♥ ❬❪ ♦♠ ♦tr t♦rs ❬❪ ♠♦ t rt♦ ❳❨ ② ♥ s♠ ♦♥st♥t A ♥ t ♥♦♠♥t♦r s♦ tt t② r♦t f(PX/(Y + A)) ♥ts tr ❬❪ ♦ t t ♣r♦♠ t♦ ts t♦♥ ♠② ♣♣rt t♦ st② ♦♦② trr③ ❬❪ s ♥ ①♣♦♥♥t ♦ ts ♦r♠ ♥ s♥t♦♥ rs♣♦♥s tr♠ ♦ t②♣ t s ♦rt ♥♦t♥ tt t t♦t ①t♥t♦♥s ♣♦ss ♦t♦♠ ♥ A = 0 t ♥♥♦t ♦t♥ t ts ♥♦♥s♥rtr♦♣ ♥t♦♥ ♦r ♥② A > 0 s trt ♥ t ♥ ❬❪ ♥② P ♥ s ss♠ t♦ ♦♥st♥t ② s♦♠ t♦rs ♥ tr ♠♦s ❬ ❪ ♥ t ♦tr♥ t ♦ ss♠ ♣rt♦r♣♥♥t ❬ ❪ P (Y ) s t♦ ♥rst Y t strt♦♥ t ♦r ♥rs♥ Y t♦ t ♣rt♦r ♥trr♥r♥ ♦r♥ ♥ P (Y )/Y s t♦ rs t Y t♦ sts② ∂F/∂Y < 0 ♦ttt ♥r t ss♠♣t♦♥ tt P (Y ) ∝ Y s Y → 0 t tr♦♣ ♥t♦♥ s ♥♦ ♠♦rs♥r ♥ t ♦r♥ r♥t ♦r♠t♦♥s ♥ ♣r♦♣♦s ♥ t trtr ♦rf(P (Y )X/Y ) trr③ t ❬❪ ♣r♦♣♦s ♥ t②♣ ♦r♠t♦♥ ♦r ♦t f(·)♥ P (·) ♥t♦♥ ♥s ♥ ♦t♦rs ❬ ❪ ♣r♦♣♦s ♦♥t②♣ ♦r ♦t f(·) ♥ P (·)

♥ ts Prt st② t♦s♣s ♦♥s① ♣r②♣rt♦r ♠♦ t s♥t② ♥r ♦r♠t♦♥ rr♥ ♦t t r♦t ♦ t ♣r② ♥ t ♥trt♦♥t♥ ♣r② ♥ ♣rt♦r ♥r ♣r♦♣rts ♦ G(X) ♥ F (X, Y ) ss♠♥ ts ♦r r r ♥t♥ ❲ ♦♥sr ♥♦♥♠♦♥♦t♦♥ G(X) t♦ ♥t ts ♦ ♦t ♥rr♦♥ ♥ ♦rr♦♥ ♦ t ♣r② r♦t ♥t♦♥G(X) s ♥♦t s♣ ② ♥ ♥②t ①♣rss♦♥ t ② t ♠♥ ♣r♦♣rts ♦ t♥t♦♥s ♥ ♥ r st t♦ ♠♦ str♦♥ ♦r rt t ♦r♥ t♦ ❬ ❪ G(X) s♦ ♥t ♥ ♥rs♥ ♦r s♥t② s♠ X ♣♦st t♥ K− rrr t♦ s t ♠♥♠♠♣♦♣t♦♥ s③ ❬❪ ❬❪ ♣ ♥ K+ ♦t♥ rrr t♦ s t rr②♥ ♣

1 Introduction 35

t② ♦ t ♥r♦♥♠♥t ❬ ♣ ❪ ♥ ♥t ♥ ♥♦♥♥rs♥ ♦r s♥t②r X ♣rt♦r♣♥♥t tr♦♣ ♥t♦♥ ♥ ②

F (X, Y ) = bf

(P0X

K0 +H0Y

)

,

r K0 s rr♥ ♦♠ss P0 ♥ H0 r ♠♥s♦♥ ♣r♠trs P0 ♥♦tst ♣rt♦♥ ♥② ♥ H0 s ♠sr ♦ t ♣rt♦r ♥trr♥ ♣r♦ssr♥ t tr♦♣ ♥t♦♥ f(·) ♣rr ♥t t s s t♦ ♥tr♦ st♦♥ r♠♥t ♥st ♦ s♥ t♦r♠♥ts ♥t♦♥ ts ♦s s t♦ ♦r♠tts ♣r♦♣rts ♥ ♠♦r ♦♥s ♦r♠ ♦r sr ② s ♥t♦♥ss♦s s♦♠ ♥trst♥ trs t ♣♣r♥ ♦ rtr qr♠ sttst r♥t stt② ♣r♦♣rts ♦♥sq♥t② t ②♥♠s ♦ t s②st♠ ♦①t rr st ♦ ♦t♦♠s ♣♥♥ ♦♥ t r♥ ♦ s ♦ t ♦♦♣r♠trs ♥tr♦ ♥ t ♠♦ qt♦♥s

s Prt s ♦r♥③ s ♦♦s ♥ ♣tr t s ss♠♣t♦♥s ♥ qt♦♥s ♦r t ♦ ②♥♠s ♦ ♣rt♦r♣r② s②st♠ r ♣rs♥t t qt♦♥sr rtt♥ ♥ tr♠s ♦ t ♠♥s♦♥ rs X/K+ ♥ Y/K+ t r♦t ♦t ♣r② ts ♥t♦ ♦♥t str♦♥ t ♥ t ♥trt♦♥s t♥ ♣r②♥ ♣rt♦r r tr♠♥ ② tr♦♣ ♥t♦♥ s ♥ ♥ t stt②♣r♦♣rts ♦ t ♥♦♥♦①st♥ qr♠ stts r s♠♠r③ ♥ ♥ ①st♥♥ stt② ♥②ss ♦ t ♦①st♥ qr♠ stts s ♣r♦r♠ ♣r♠trsrt t♦ P0 ♥ H0 ss♠ s rt♦♥ ♦♥s ♥ ♣tr rsts ♦♥♠r s♠t♦♥s ♦t♥ ♦r s♦♠ ♦♥rt r③t♦♥ ♦ t ♠♦ ♥t♦♥sr ♣rs♥t t♦ strt t ♦rs ♦ t s②st♠ rsts ♦♥r♠ ♥②t ♣rt♦♥s ♥ tr♦ t ♦♥ s♦♠ s♣ts ♦ t ②♥♠s ♦ t s②st♠♥ ♣tr s♦♠ ♦♥♥ r♠rs ♥ ♦♥ ♥ rsts r ♦♠♠♥t ♦♥t rr♥ t♦ t ①st♥ trtr t t ♥ ♦ ts Prt ♥ ♣♣♥① ts ♦♥ s♦♠ r ♣r♠trs ♥ ♦♥ ♥ ♣♣♥① ♠♦r t ♦tt tt s♣ rt♦♥ ♣♦♥t r r♣♦rt ♦r t rrs ♦♥♥♥ ts②♠♦s s ♥ ts Prt ♥ ♦t ♥ ♣♣♥①

2

The prey-predator model and the equilibrium

stability analysis

♥ ts ♣tr ♥tr♦ t ♠♦ qt♦♥s ♦ t ♦♥sr ♣rt♦r♣r②s②st♠ r t ♠♦ ♥t♦♥s sr♥ t ♦♦ ♣r♦sss ♦rr♥ ♥t ♦♥sr tr♦♣ ♥ r ♥♦t s♣ ② ♥②t ①♣rss♦♥s t ② s♦♠rtrst ♣r♦♣rts tr♠♥♥ tr s♣s t s♦♥ tt ts s s ♦r t ①st♥ ♥ stt② ♥②ss ♦ t qr♠ stts ♦ t s②st♠ss♠♥ t♦ rt♦♥ ♣r♠trs

2.1 Basic assumptions and model equations

❲ ♦♥sr t ②♥♠s ♦ ♣rt♦r♣r② s②st♠ sr ♥ tr♠s ♦ ♠♣♣r♠tr ♠♦ ❬ ❪ ♥ t ♠♦r♣ strtr ♦ t ♣♦♣t♦♥s s♥t ♥ ♣rtr t st strtr ♦r ♥sts ♥ ♠ts s r ♣♣ ts s t ♣rt♦r ♥ ♣r② ♣♦♣t♦♥s ♠② rtr③ ② st♦♥ stt r r♣rs♥t♥ tr ♥♥ ♥ tr♠s ♦ ♦♠sss♣t ♥t♦r♦r ss♠ ♠t ♥ ♦♥tr♦ ♥r♦♥♠♥t ♦r ♥st♥ r♥♦s ♥ t♠♣rtr ♥ ♠t② r ♠♥t♥ ♣♣r♦①♠t② ♦♥st♥ts♦ tt ♥ ♦♥sr t♠ ♥♣♥♥t ♦♦♦ ♣r♠trs ♥ ♥t ts♣t strt♦♥ ♦ t ♥s ❲t ts ss♠♣t♦♥ ♥ t♥ ♥t♦ ♦♥t t ①♣rss♦♥ ♦r t tr♦♣ ♥t♦♥ t ♥ qt♦♥s ♦r t♦ ②♥♠s ♦ t t♦ tr♦♣ s ♥ ♦♥tr♦ ♥r♦♥♠♥t r rtt♥ s

dX

dt= rxXG (X)− bY f

(P0X

K0 +H0Y

)

dY

dt= cbY f

(P0X

K0 +H0Y

)

−mY

38 2 The prey-predator model and the equilibrium stability analysis

r X ♥ Y r ♣r② ♥ ♣rt♦r ♥♥s r ss♠ tt t s ♠sr ♥ tr♠s ♦ ♦♠sss♣t ♥t ♥ t ♠♦ ♥t♦♥s G(X) ♥ f(X, Y )r s♣ rts ♥ tr s♣s tr♠♥ t t②♣ ♦ ♣r② r♦t rx s ♥ ♥rt ♠①♠♠ s♣ r♦t rt ♦ t ♣r② ♥ ♣rt♦♥ ♣r♦sss rs♣t② ♣r♠trs b s t ♠①♠♠ s♣ rt ♦ ♣r② ♦♥s♠♣t♦♥m s t s♣rt ♦ ♣rt♦r ♠♦rtt② c s ♦♥rs♦♥ t♦r ♠srs t ♦♥rs♦♥ ♥② r♦♠ ♣r② t♦ ♣rt♦r ♦♠ss K0 s rr♥ ♦♠ss P0 ♥ H0 r♠♥s♦♥ ♣r♠trs P0 ♥♦ts t ♣rt♦♥ ♥② ♥ H0 s ♠sr ♦t ♣rt♦r ♥trr♥ ♣r♦ss t

x =X

K+, y =

Y

K+, p = P0

K+

K0

, h = H0K+

K0

,

t♥ t ♣r♦s s②st♠ ♥ rrtt♥ ♥ tr♠s ♦ ♠♥s♦♥ rs ♥♦♥sr♥ s♦ t ♥t ♦♥t♦♥s

dx

dt= rxxg (x)− byf

(px

1 + hy

)

dy

dt= cbyf

(px

1 + hy

)

−my

x(0) = x, y(0) = y,

r p ♥ h r ♠♥s♦♥ ♣r♠trs rrr t♦ s ♣rt♦♥ ♥② ♥♣rt♦r ♥trr♥ r♥ t ♣rt♦♥ ♣r♦ss ♥t♦♥s g(·) ♥ f(·) s♦sts② s♦♠ rrt② ♥ ♥r ss♠♣t♦♥s tt ② ♦♦ ♦♥srt♦♥st s ss♠ tt

∃ ǫ : 0 < ǫ < 1, g(ǫ) = g(1) = 0; g(s)(s− ǫ)(1− s) > 0, s 6= ǫ, 1;

f(0) = 0, lims→+∞

f(s) = 1, f ′(s) > 0, s > 0.

rtr t ♣r♠ ♥ts t rt t rs♣t t♦ t r♠♥t ss♠♣t♦♥ ♠♣s tt t ♥t♦♥ g s ♦♥② t♦ ③r♦s ǫ ♥ 1 tt t s♣♦st ♥ ǫ < x < 1 ♥ ♥t ♥ 0 < x < ǫ ♦r x > 1 ♣r♠tr ǫ st rt♦ t♥ ♠♥♠♠ ♥ ♠①♠♠ ♣♦♣t♦♥ s③

ǫ = K−/K+,

s♦ tt t r♣rs♥t t ♠♥♠♠ ♣♦♣t♦♥ s③ 1 s t rr②♥ ♣t② ♦t ♠♥s♦♥ s②st♠ ❲t ts ss♠♣t♦♥s s g s st t♦ ♠♦ str♦♥

2.1 Basic assumptions and model equations 39

t ♦r♥ t♦ ❬ ❪ ❲t t ss♠♣t♦♥ rt♥ ♥t♦ ♦♥t t strt♦♥ ♥ t ♣rt♦♥ ♣r♦ss ♦r♦r s♦♠ rtrt♥ ss♠♣t♦♥s ♦♥ t s♠♦♦t♥ss ♦ ts ♥t♦♥s r rqr t♦ ♠t t♥♠r ♦ qr♠ stts ♦ ♥ t♦ ♠ t stt② ♥②ss trt❲ ss♠ st ♦♥ ♠①♠♠ ♥ g(s) s ♣♦st st ♦♥ ♥t♦♥ ♣♦♥t ♦sg(s) ♥ g(s) s ♣♦st ♥ ♥rs♥ r t♥ ♥t ♦♥①t② ♥g(s) s ♣♦st ♥ rs♥ ♥ r t♥ ♥t ♦♥①t② ♦r f(s) s♦♥t♦♥s ♥ rtt♥ s

∃ ξ0 : ǫ < ξ0 < 1, g(ξ0) = 1, g′(ξ0) = 0; g′(s)(ξ0 − s) > 0, ǫ < s < 1, s 6= ξ0,

∃ η0 : ǫ < η0 < ξ0, [sg(s)]′′s=η0

= 0, [sg(s)]′′ (η0 − s) > 0, ǫ < s < ξ0, s 6= η0;

[sg′(s)]′ < 0, ξ0 < s ≤ 1;

[f(s)

s

]′

< 0, s > 0.

♠rs

♦♥t♦♥ ♦♥ t ♣r②r♦t ♥t♦♥ rqrs tt ♦♥② ♦♥ ♠①♠♠♣♦♥t ξ0 ①sts ♥ (0, 1) r t ♥t♦♥ s ♣♦st ♥ t ♥t♦♥ g(s)s ♥♦r♠③ s♦ tt g(ξ0) = 1 ♥ ♦rr t♦ rx s t ♠①♠♠ s♣r♦t rt s ♠♥s tt tr s ♥ ♦♣t♠ ♣♦♣t♦♥ s③ ♦rrs♣♦♥♥t♦ ♠①♠♠ rt ♦ r♦t rx ♦r s♠r t♥ ξ0 t ♥ ♦ r② ♥ t♦♦ s♣rs ♦r rtr s t ♦♠♣tt♦♥ ♦r rs♦rs♦♠s ♥t

①♣rss♦♥ ♥ ♦♥t♦♥s ♥ ♥ rtt♥ s

[sg(s)]′′ =1

s

[s2g′(s)

]′= 2g′(s) + sg′′(s), [sg′(s)]′ = g′(s) + sg′′(s).

s ♦r ξ0 < s ≤ 1 ♥♠② ♥ g(s) s ♣♦st ♥ rs♥ ♦♥t♦♥ s str♦♥r t♥ ♦r♦r ts ♦♥t♦♥s ♦ t ♥t♦♥ sg(s)t♦ ♥ ♥t♦♥ ♣♦♥t ♥ (ε, ξ0) s ♥♠♥t ♦r t sr♣t♦♥ ♦t t ♥ t② ♦ ♦♥sr s t ♠♥♠ ss♠♣t♦♥s ♦♥t r♦t ♥t♦♥ t♦ sr str♦♥ t

♦♥t♦♥s ♥ r r t♥ g′′(s) < 0 ♥ f ′′(s) < 0 rs♣t② ♥ t ♥ g′(s) < 0 s t s ♥ t ♥tr (ξ0, 1] tt

40 2 The prey-predator model and the equilibrium stability analysis

g′′(s) < 0 ⇒ [sg′(s)]′ < 0 rtr♠♦r t ♥ ♣r♦ tt ♦♥t♦♥s ♥ ♠♣s tt t ♥t♦♥ f ♦s ♥♦t ♥② ♥t♦♥ ♣♦♥t ♥ fs ♥t ♦♥t② s ①s ♦r ♥st♥ t ♦♥t②♣ tr♦♣♥t♦♥ ♥ t r♠♥t ♦ ts ♥t♦♥ s px/(1 + hy) ts ♠♥s tt ♦rs♠ ♣r② ♥sts ♦r r ♥♠r ♦ ♣rt♦rs tr t s ♥ t♣rt♦r s ♥ tr♥t s♦r ♦ ♦♦ ♦r t ♣r② s ♥♠r ♦ strs♥ss t♦ t ♣rt♦r r ① ❬❪

rqr ♣r♦♣rts r ♦r ♥st♥ ② t ♦♦♥ ♠♦♥t♦♥s ② s ♥ trtr s rr♥s ♥ ❬❪

Pr② r♦t g(s) = g0(s− ǫ)(1− s),

g(s) = g0

(

s exp

(1− s

ξ0

)

− 1

)

,

r♦♣ ♥t♦♥s f(s) =s

1 + s,

f(s) = 1− exp(−s),

r g0 s s tt g(ξ0) = 1 ♥♠② g0 = 4/(1−ǫ)2 ♥ ♥ g0 = [ξ0 exp((1−ξ0)/ξ0)−1]−1 ♥ t ξ0 = −(1−ǫ)/ log ǫ ♥t♦♥ s t ♣♥ ♠♦❬❪ s r♦♠ ❬❪ s t ♦♥t②♣ tr♦♣ ♥t♦♥ ❬❪ ♥ s t ♠♦ ❬ ❪ t s ♦rt ♥♦t♥ tt ♦♦s♥ t ①♣rss♦♥ ♦r t tr♦♣ ♥t♦♥ f s t♦ t ♥t♦♥♥s ♥t♦♥ rs♣♦♥s

f

(px

1 + hy

)

=px

1 + px+ hy.

2.2 Attractive invariant set

❲ r ♥trst ♥ ♣♦st s♦t♦♥s t♦ t s ♣♦ss t♦ s♦ tt s♦t♦♥s♥tt♥ ♥ R+

2 r ♦♥ ♥ ♥t② ♥tr ♥ ttrt♥ st tt t s s♦♥♥ r

♦r♠ ❯♥r t ss♠♣t♦♥s ♥ t ♦s st

Ω =

(x, y) ∈ R+2 | 0 ≤ x ≤ 1, 0 ≤ x+

y

c≤ 1 +

rxm

s ♣♦st② ♥r♥t ♥ ♦r ♥t stts E = (x, y) ∈ R+2 t trt♦r②

(x(t), y(t)) ♥t② ♥trs ♥t♦ Ω s t→ +∞

2.2 Attractive invariant set 41

1 + rx/m

1 x

y

0 ε

rx/m

P♦st② ♥r♥t ♥ ttrt st ♦r s②st♠

Pr♦♦ ♣r♦♦ s strt♦rr ♣♣t♦♥ ♦ t ♦♠♣rs♦♥ t♦r♠ ♦rs ♥ ♠s s ♦ st ♦♥s ♦r t rt ♥ s ♦ ❬❪

rst tt t ①s r trt♦rs t♥ t② ♥ ♥♦t r♦ss ts♥srs tt t s♦t♦♥s strt♥ r♦♠ ♣♦st ♥t ♦♥t♦♥s r♠♥ ♣♦st ♥♣rtr ♦♦s ♥ ♥t ♦♥t♦♥s ♦♥ t y①s t rst qt♦♥ ♦ rs t♦ x = 0 ♥ t s♦♥ t♦ y = −my t♥ t y①s s trt♦r② ♣♦♥t♥t♦rs t ♦r♥ ♥♦♦s② ♦♦s ♥ ♥t ♦♥t♦♥ ♦♥ t x①s ts♦♥ qt♦♥ ♦ rs t♦ y = 0 ♥ t rst t♦ x = rxxg(x) t♥ t♣♦st x①s s ♥ ♥♦♥ ♦ tr trt♦rs s♣rt ② tr stt♦♥r② sttsx = 0, x = ε, x = 1

♦♥sr ♥♦ (x(t), y(t)) s♦t♦♥ ♦ t ♥t ♦♥t♦♥ x(0) = x0 > 0♥ y(0) = y0 > 0 t♦ t ♣♦stt② ♦ y ♥ ss♠♣t♦♥ tt

x = rxxg(x)− byf

(px

1 + hy

)

≤ rxxg(x).

❲ ♥ ♦♥sr t r s②st♠

˙x = rxxg(x)

x(0) = x0,

♦r x(t) = 1 s stt♦♥r② s♦t♦♥ r♦r 0 < x0 < 1 t♥ 0 < x(t) < 1♥s t♦ t ♦♠♣rs♦♥ t♦r♠ ♦r s ♣♣ t♦ s②st♠ ♥

42 2 The prey-predator model and the equilibrium stability analysis

x = rxxg(x)− byf

(px

1 + hy

)

x(0) = x0,

t 0 < x(t) ≤ x0 < 1 ♥ r y s t♦t s ♣♦st ♣r♠tr t ♦♦stt 0 < x(t) < 1 0 < x0 < 1

♦♥sr ♥♦ ♥t t (x0, y0) s tt

x0 +y0c

≤ 1 +rxm,

♥ ♥r ♦♠♥t♦♥s ♦ t r♥t qt♦♥s ♦ ♦r

x+y

c= rxxg(x)−m

y

c±mx = x(rxg(x) +m)−m

(

x+y

c

)

.

s 0 < x < 1 ♥ 0 < g(x) < 1

x+y

c+m

(

x+y

c

)

= x(rxg(x) +m) ≤ m( r

m+ 1)

.

Ptt♥ z = x+ y/c t♥ z0 = x0 + y0/c ♦t♥ tt

z +mz ≤ m(rxm

+ 1)

♥ ♠t♣ ② emt ♥ ♥trt r♦♠ 0 t♦ τ ♦t♥∫ τ

0

z +mzemtdt ≤ m(rxm

+ 1)∫ τ

0

emtdt

♦r q♥t② ∫ τ

0

d

dt(zemt)dt ≤

(rxm

+ 1)∫ τ

0

d

dt(emt)dt.

♥ ssttt♥ τ t t

z ≤ z0e−mt +

(rxm

+ 1)

(1− e−mt) = e−mt[

z0 −(rxm

+ 1)]

+(rxm

+ 1)

≤(rxm

+ 1)

.

r♦r tt

0 ≤ z(t) = x(t) +y(t)

c≤(rxm

+ 1)

♦♥ t ♣r♦♦ ♦ t ♣♦st② ♥r♥ ♦ Ω

2.3 Non-coexistence equilibria and their stability properties 43

❲ ♥t t♦ ♣r♦ tt Ω s s♦ ♥ ttrt st t ♥♦ ♦♦s ♥t t(x0, y0) ∈ R2

+ −Ω ♦♦♥ t ♦sr tt ˙x < 0 ♥ x > 1 t♥ ①x(0) = x0 > 1 s♦ ts s♦t♦♥ rss ♥ t♥ s ♠t ssr② sx = 1 s stt♦♥r② s♦t♦♥ ♦

limt→+∞

x(t) = 1.

♣♣②♥ t ♦♠♣rs♦♥ t♦r♠ ♦r s t♦ s②st♠s ♥ t x0 ≥x0 > 1

limt→+∞

x(t) = limt→+∞

x(t) = 1.

♥♦♦s② t♥s t♦ t ♦♠♣rs♦♥ t♦r♠ ♦r s ♣♣ t♦ s②st♠s

x+y

c= −m

(

x+y

c

)

+mx( r

m+ 1)

x(0) +y(0)

c= x0 +

y0c

♥

˙x+˙y

c= −m

(

x+y

c

)

+mx( r

m+ 1)

x(0) +y(0)

c= x0 +

y0c

t

x0 +y0c

≥ x0 +y0c

≥( r

m+ 1)

,

♦t♥ tt

limt→+∞

x(t) +y(t)

c= lim

t→+∞x(t) +

y(0)

c=( r

m+ 1)

.

2.3 Non-coexistence equilibria and their stability

properties

①st♥ ♥ stt② ♥②ss ♦ t ♥♦♥♥t qr♠ stts E =(xeq, yeq) ♦r t s②st♠ s ♣r♦r♠ ss♠♥ t ♣r♠trs h ♥ p srt♦♥ ♣r♠trs t r♠♥♥ ♣r♠trs r ①

44 2 The prey-predator model and the equilibrium stability analysis

♦r ♥② h ≥ 0 ♥ p > 0 t s②st♠ ♠ts s qr♠ stts t ♥ sttE0 = (0, 0) ♥ ♥r t ss♠♣t♦♥s ♦♥ g(·) t♦ ♥♦♥♦①st♥ stts

Eǫ = (ǫ, 0), E1 = (1, 0).

t ry t ♠①♠♠ s♣ r♦t rt ♦ t ♣rt♦r r♦♠ ♥ tt

ry = cb−m = cb(1− α), t α =m

cb.

ry < 0 α > 1 t♥ y′(t) < 0 ♦r ♥② x, y > 0 s ♦①st♥ qr♠stts ♥♥♦t ①st y(t) s ②s rs♥ ♥ s t → +∞ t ♥ s② s♥tt (x(t), y(t)) ♦♥rs t♦ tr E0 ♦r E1 ♣♥♥ ♦♥ t ♥t ♦♥t♦♥s ♥t ♦♦♥ ss♠ ry > 0

α < 1.

♥ ts s t ♦♦♥ ♠♣t♦♥ ♦s

ry > 0 =⇒ ∃ p0 = f−1(α) > 0,

♥ p0 tr♥s ♦t t♦ rt ♦ p ♦r t ①st♥ ♥ stt② ♦ tqr♠ stts ss♦t t t ♣♥s ♦♥② ♦♥ t rt♦ α p0 = p0(α)♥

dp0dα

> 0, p0(0) = 0, limα→1

p0(α) = +∞.

♦r♥ t♦ t rt♠♥r♦♠♥ ♦r♠ ❬❪ t ♦ stt② ♣r♦♣rts ♦t qr♠ stts Eι ι = 0, ǫ, 1 r tr♠♥ ② t ♥②ss ♦ t ♥s♦ t ♦♥ ♠tr① ss♦t t s②st♠

J(x, y) =

rxg(x) + rxxg′(x)− bp uf ′(pv) −bf(pv) + bph uvf ′(pv)

cbp uf ′(pv) cbf(pv)−m− cbph uvf ′(pv)

ru =

y

1 + hy, v =

x

1 + hy,

♥ t ♥ Eι ❲ ♦t♥

J(0, 0) =

rxg(0) 0

0 −m

, J(ǫ, 0) =

rxg′(ǫ) −bf(pǫ)

0 cbf(pǫ)−m

,

J(1, 0) =

rxg′(1) −bf(p)

0 cbf(p)−m

.

2.4 Coexistence equilibria 45

♥s t♦ t ♠tr① J(0, 0) s t♦ ♥t ♥s ♥♣♥♥t ♦ ts ♦ t ♣r♠trs h ♥ p r♦♠ t ♦♦s tt g′(ǫ) > 0 ♥ g′(1) < 0s♦ tt J(ǫ, 0) ♥ ♣♦st ♥ ♥t ♥ rs♣t②ss♠♥ tt ry > 0 s♦ tt ♦s tt

cbf(pǫ)−m > 0 ⇐⇒ p >p0ǫ,

cbf(p)−m > 0 ⇐⇒ p > p0.

s t sst♦♥ ♦ t ♥♦♥♦①st♥ qr♠ stts s t ♦♦♥

E0 s ♦② st ♥♦ ♦r ♥② p > 0

Eǫ s ②s ♥st t s s ♦r p < p0/ǫ t ♥st ♠♥♦ Wu

(Eǫ

)

②♥ ♦♥ t x①s ♥ ♥ ♥st ♥♦ ♦r p > p0/ǫ

E1 s ♦② st ♥♦ ♦r p < p0 ♥ s ♦r p > p0 t st ♠♥♦Ws

(E1

)②♥ ♦♥ t x①s

t s ♦rt ♥♦t♥ tt t ♥s ♥ t♥ t stt② ♣r♦♣rts ♦ t♥♦♥♦①st♥ stts tr♥ ♦t t♦ ♥♣♥♥t ♦ h ♦r♦r t ♦♦s ttt stt② ♣r♦♣rts ♦ E0 ♥ Eǫ r ♥♣♥♥t ♦ p t♦s ♦ E1 ♣♥♦♥ p

2.4 Coexistence equilibria

♦①st♥ qr♠ stts E∗(h, p) = (x∗(h, p), y∗(h, p)) r ♦♥ s t s♦t♦♥s ♦

rxxg(x)− byf

(px

1 + hy

)

= 0

cbf

(px

1 + hy

)

−m = 0,

s s t ♥trst♦♥s ♦ t ♥♥s ♦ s②st♠ rtt♥ s

y = βxg(x), f

(px

1 + hy

)

= α,

rβ =

rxc

m.

46 2 The prey-predator model and the equilibrium stability analysis

t s ♦rt ♥♦t♥ tt t rst ♥♥ ♥ s ♠♣ r ♥ t ♣s♣♥ s ♥♣♥♥t ♦ ♣r♠trs h ♥ p ❯♥r t ss♠♣t♦♥ ♥t ♥t♦♥ ♦ p0 tt t s♦♥ ♥♥ ♥ s t strt♥

p

p0x = 1 + hy,

♦s ♣♦st♦♥ ♥ s♦♣ ♣♥ ♦♥ ♦t ♣r♠trs h ♥ p sttt♥ y r♦♠t rst qt♦♥ ♥ ♦t♥ t qt♦♥ ♦r x

p

p0= φ(x, h) :=

1

x+ hβg(x).

❲ r ♥trst ♥ s♦t♦♥s t♦ x∗(h, p) ∈ (ǫ, 1) ♥ ♦rr t♦ y∗(h, p) > 0♥ t r♦♠ t ss♠♣t♦♥ g(x) > 0 ♦♥② ♥ (ǫ, 1) ♥ ts ♠♣s tt p > p0♥ t ♦♦♥ ss♠ h > 0 s h = 0 s ♥ st ♥ t ♥❬❪

2.4.1 Shape of φ(x, h)

s♣ ♦ φ(x, h) ♥ ♦♥sq♥t② t ♥♠r ♦ s♦t♦♥s t♦ ♥ (ǫ, 1)str♦♥② ♣♥s ♦♥ t ♣r♠tr h ❲

∂φ

∂x=

1

x2[−1 + hψ(x)] , t ψ(x) = βx2g′(x),

♥∂φ

∂h= βg(x) > 0, x ∈ (ǫ, 1).

r♦♠ t ss♠♣t♦♥ g′(x) < 0 ♥ t♥ ∂φ/∂x < 0 ♦r x ∈ (ξ0, 1) sφ(x, h) ♠② ♥♦♥♠♦♥♦t♦♥ t rs♣t t♦ x ♦♥② ♥ x ∈ (ǫ, ξ0) ♥ h 6= 0 ♣♦♥ts r ∂φ/∂x = 0 r s♦t♦♥s ♦ t qt♦♥

ψ(x) =1

h, h > 0.

❲ ψ(ǫ) > 0 ψ(ξ0) = 0 ♥ r♦♠ t ss♠♣t♦♥ ψ(x) s ♦♥② ♦♥♠①♠♠ η0 ♦r x ∈ (ǫ, ξ0) ♦r♥ t♦ t s♣ ♦ ψ(x) r ♥ (ǫ, ξ0)t ♦♦s tt tr r t♦ s♦t♦♥s ♦r qt♦♥ ♥ ψ(ǫ) ≤ 1/h < ψ(η0)♥ ♦♥ s♦t♦♥ ♥ 1/h < ψ(ǫ) t η1 t ♥q ♣♦♥t ♥ t r♥ (η0, ξ0)s tt ψ(η1) = ψ(ǫ) t s ♥

h0 =1

ψ(η0), h1 =

1

ψ(ǫ)

2.4 Coexistence equilibria 47

ψ(η0) = 1/h0

ψ(ǫ) = 1/h1

0 ξ0η1η0x

ǫ

♣ ♦ t ♥t♦♥ ψ(x)

♥ ♥♦t t ξ1(h) ♥ ξ2(h) t t♦ ♣♦ss s♦t♦♥s t♦ qt♦♥ ξ2(h)①sts ♦♥② ♦r h0 ≤ h ≤ h1 ξ1(h) ♦r h ≥ h0 ♥ s♠♠r③ t rsts♦t t ♥♠r ♦ s♦t♦♥s t♦ r♦♠ t ♦♦s s♦ tt t t♦

h♥♠r ♦ s♦t♦♥s ♣♦st♦♥ ♦♥

t♦ ∂φ/∂x = 0 t x ①s

0 ≤ h < h0 h = h0 ξ2(h0) = η0 = ξ1(h0)

h0 < h < h1 ǫ < ξ2(h) < η0 < ξ1(h) < η1h = h1 ǫ = ξ2(h1) < η0 < ξ1(h1) = η1h > h1 η1 < ξ1(h) < ξ0

♠r ♦ s♦t♦♥s ξj(h) t♦ qt♦♥ ♥ tr ♣♦st♦♥s ♦♥ t x ①s ♦r r♥tr♥s ♦ h

♥t♦♥s ξ1(h) ♥ ξ2(h) r ♠♦♥♦t♦♥

dξ1dh

> 0 h ≥ h0,dξ2dh

< 0 h0 ≤ h ≤ h1

♥ ♠♦r♦r limh→+∞

ξ1(h) = ξ0 t

φ0 = φ(η0, h0), φj(h) = φ(ξj(h), h), j = 1, 2.

48 2 The prey-predator model and the equilibrium stability analysis

r♦♠ t ①♣rss♦♥ ♦ ∂φ/∂x ♥ ψ(x) ♥ ♥ tt ξ2(h) s ♦♠♥♠♠ ♥ ξ1(h) s ♦ ♠①♠♠ ♦ φ(x, h) ♥ ξ2(h) < ξ1(h) t ♦♦stt

φ2(h) < φ1(h).

♥∂φ

∂h> 0,

(∂φ

∂x

)

x=ξj

= 0,

r♦♠dφj

dh=

(∂φ

∂h

)

x=ξj

+

(∂φ

∂x

)

x=ξj

dξjdh

t ♦♦s tt φj(h) r ♠♦♥♦t♦♥ dφj/dh > 0 ♦r♦r

1 < φ0 =1

η0+ βh0g(η0) = φ1(h0) = φ2(h0) <

1

ǫ, φ1(h1) > φ2(h1) =

1

ǫ,

♥

φ1(h) > φ(ξ0, h), φ1(h) → φ(ξ0, h) = βh+1

ξ0s h→ +∞.

st② φ1(h) ♠② rtr ♦r ss t♥ 1/ǫ ♦r h0 < h < h1 t hǫ t♥q s♦t♦♥ t♦ t qt♦♥ φ1(h) = 1/ǫ ♦ r t♦ ♣♦t t qtts♣ ♦ φ(x, h) ♦r x ∈ (ǫ, 1) ♥ r♥t r♥s ♦ h r s r sr♣rs♥tt ♦ t qtt ♦rs ♦ φ(x, h) ♥ ♥ t g(x)sts②♥ t ♣r♦♣rts ♥ ♥t♦♥ φ(x, h) strt②♥rs♥ ♥ h ♥ 0 ≤ h < h0 r s strt② rs♥ ♥ x ♥h0 < h < h1 rs t s♦s ♦ ♠♥♠♠ ♦r x t♥ ǫ ♥η0 ♥ ♦ ♠①♠♠ ♦r x t♥ η0 ♥ 1 ♥② ♥ h > h1 r φ(x, h) s ♦♥② ♦ ♠①♠♠

2.4.2 Solutions to φ(x, h) = p/p0

♥ ♥t♦ ♦♥t t rsts ♦ t ♣r♦s sst♦♥ ♥ tr♠♥ tr♥s ♦ p/p0 ♦r s♦t♦♥s t♦ t x ∈ (ǫ, 1) ♥ t r♦s♥trs ♦ t ♣r♠tr h ♥ φ(x, h) s♦s r♥t tr♥s rss x r ♦r 0 ≤ h < h0 ♦♥ s♦t♦♥ t♦ ♦r h0 < h < h1 ♠② r♦♠ ♦♥ t♦ tr s♦t♦♥s ♥ r strt t ♣♦st♦♥s ♦ t ♦♠♥♠♠ ♥ ♠①♠♠ ♦ φ(x, h) t rs♣t t♦ 1/ǫ tr♠♥ t r♦♥s♥ t ♣r♠tr s♣ (h, p) ♦ ①st♥ ♦ t s♦t♦♥s t♦ ♥ t ♦r0 < h < h0 ♦♥② ♦♥ s♦t♦♥ t♦ ♥♣♥♥t ♦ t ♦ p s t

2.4 Coexistence equilibria 49

ǫ η0 10

1

1/ǫ

x

h = 0

h = h0

0 < h < h0

0 ≤ h ≤ h0

ǫ η0 10

1

1/ǫ

x

h0 < h < hǫ

h0 < h < hǫ

ǫ η0 10

1

1/ǫ

x

hǫ < h < h1

hǫ < h < h1

xǫ η0 10

1

1/ǫh = h1

h > h1

h ≥ h1

r♥ ♦ φ(x, h) ♦r x ∈ (ǫ, 1) ♥ r♥t s ♦ h rr♦s ♥t t rt♦♥ ♦♥rs♥

♥ s♥ ♥ r rt t♦ t s♥r♦ st ♥ r r r♣rs♥ts t s♥r♦ ♦r h0 < h < hǫ st ♥ r r♦r ♥rs♥ p r♦♠ p0 t♦ p0/ǫ p 6= p0φi, i = 1, 2 ♥ ♦♥ tr ♦♥qr r r♣rs♥ts ♥st t stt♦♥ ♦r hǫ < h < h1 st♥ r r ♦r ♥rs♥ p r♦♠ p0 t♦ p0φ1 p 6= p0φ2 ♥ ♦♥ tr t♦ qr ♦r h > h1 ♠② ♦♥ ♦r t♦ s♦t♦♥s t♦ ♣♥♥ ♦♥ t ♦ p/p0 ♦♠♣r t♦ 1/ǫ s t ♥ s♥ ♥ r rt t♦ t s♥r♦ st ♥ r rsts r ♦t ♥ t♦♦♥ ♥ s♦♥ ♥ r

♥ t ♣r♠tr s♣ (h, p) r t rs p = p0φ1(h), h ∈ (h0, hǫ),♥ p = p0φ2(h), h ∈ (h0, h1), r stt♦♥r② rt♦♥ rs ♦r h ∈ (h0, h1)♥ ♥rs♥ p r♦♠ p0 ♥ p = p0φ2(h) t stts E∗2 ♥ E∗3 ♣♣r t

50 2 The prey-predator model and the equilibrium stability analysis

ǫ 10

1

1/ǫ

φ(x, h)

p/p0

x

0 ≤ h ≤ h0

ǫ ξ2 ξ1 10

1

φ2

φ1

1/ǫ

x

φ(x, h)

p/p0

h0 < h < hǫ

ǫ ξ2 ξ1 10

1

φ2

1/ǫφ1

x

φ(x, h)

p/p0

hǫ < h < h1

ǫ ξ1 10

1

1/ǫ

φ1

φ(x,h)

x

p/p0

h ≥ h1

♦t♦♥s t♦ φ(x, h) = p/p0 ♦r x ∈ (ǫ, 1)

t② ♦ ♥♦t ♦ t E∗1 trs ♥ p = p0φ1(h) t stts E∗1 ♥ E∗2

♦ ♥ s♣♣r t ♦ ♥♦t ♦ t E∗3 ♥ ts rs t tr♠♥♥t♦ t ♦♥ ♠tr① J(E∗j(h, p)) ss♦t t s ③r♦ ♥ t tt ♦♥ qr t s♦♥ ♥ t♦♥ s ♦r s t②♣ ♦ s♥♦ rt♦♥ ♥ t♦♥ t t♦ rt♦♥ rs p = p0φ1(h) ♥p = p0φ2(h) ♥trst t t rt ♣♦♥t B0 = (h0, p0φ0) t ♥q ♦♥ ♥ t♣r♠tr s♣ r t tr qr♠ stts ♦♥ t ts ♣♦♥t t t♦rt♦♥ rs sr ♦♠♠♦♥ t♥♥t s♥ x∗j = η0, j = 1, 2, 3 ♥ r♦♠ ∂φ1/∂h = ∂φ2/∂h = βg(η0) t♥ B0 s ♠t♠t s♣ ♥ t (h, p)♣♥ ♦r♦r ♥♦t tt t ♣♦♥t B0 r♦♠

∂φ

∂x(x∗j, h0) = 0 ♥

∂2φ

∂x2(x∗j, h0) = 0.

2.4 Coexistence equilibria 51

h p/p0♥♠r ♦ ♣♦st♦♥ ♦♥

s♦t♦♥s t♦ φ = p/p0 t x ①s

0 ≤ h < h0 1 < p/p0 < 1/ǫ ǫ < x∗1 < 1

h = h0

1 < p/p0 < φ0 η0 < x∗1 < 1p/p0 = φ0 η0 = x∗1 = x∗2 = x∗3

φ0 < p/p0 < 1/ǫ ǫ < x∗1 = x∗3 < η0

h0 < h < hǫ

1 < p/p0 < φ2 ξ1 < x∗1 < 1p/p0 = φ2 x∗3 = x∗2 = ξ2 < ξ1 < x∗1

φ2 < p/p0 < φ1 ǫ < x∗3 < ξ2 < x∗2 < ξ1 < x∗1 < 1p/p0 = φ1 x∗3 < ξ2 < x∗2 = x∗1 = ξ1

φ1 < p/p0 < 1/ǫ ǫ < x∗3 < ξ2

hǫ < h < h1

1 < p/p0 < φ2 ξ1 < x∗1 < 1p/p0 = φ2 x∗3 = x∗2 = ξ2 < ξ1 < x∗1

φ2 < p/p0 < 1/ǫ ǫ < x∗3 < ξ2 < x∗2 < ξ1 < x∗1 < 11/ǫ < p/p0 < φ1 ξ2 < x∗2 < ξ1 < x∗1 < 1

p/p0 = φ1 x∗2 = x∗1 = ξ1

h ≥ h1

1 < p/p0 ≤ 1/ǫ ξ1 < x∗1 < 11/ǫ < p/p0 ≤ φ1 ǫ < x∗2 ≤ ξ1 ≤ x∗1 < 1

p/p0 = φ1 x∗2 = x∗1 = ξ1

♥s ♦ p/p0 ♦r t ①st♥ ♦ s♦t♦♥s x∗j = x∗j(h, p) ∈ (ǫ, 1) t♦ qt♦♥ tr♥♠r ♥ tr ♣♦st♦♥s ♦♥ t x ①s ♣♥♥ ♦♥ h φj = φj(h), ξj = ξj(h)

ts rsts ♥t t ♣rs♥ ♦ s♣ s♥rt② ♦r♥ t♦ ❲t♥②st♦r② ❬❪ ♦r t qr♠ sr ♠♣t② ♥ ② p = p0φ(x, h) ♥ s♦♥♥ r ♥ t tr ♠♥s♦♥ s♣ (h, p, x) ♣r♦t♦♥ ♦♥ t (h, p)♣♥ ♦ s sr ♥ s♥ r t sts ♥ t (x, p)♣♥ ♦r ① h ♥ r♦♠ r ♦r q♥t② r♦♠ r ♥♣♣♥① ♠♦r t ♦t t tt s♣ rt♦♥ ♣♦♥t r r♣♦rt

♥ t (h, p) ♣♥ t ♥s

p

p0= 1,

p

p0=

1

ǫ,

p

p0= φj(h), j = 1, 2

r ♦♥r② ♥s ♦ ①st♥ r♦♥s r ♦r t ♦①st♥ qr♠stts E∗j(h, p) = (x∗j(h, p), y∗j(h, p))

♦r x∗j 6= ξ1(h), ξ2(h) r♦♠ qt♦♥ t ♦♦s tt

∂x∗j∂h

= −βg(x∗j)(∂φ

∂x

)−1

x∗j

,∂x∗j∂p

=

(

p0∂φ

∂x

)−1

x∗j

.

52 2 The prey-predator model and the equilibrium stability analysis

x

p

h

p0/ε

p0

0

0

1

h0

h1

❱ ♦ t qr♠ sr p = p0φ(x, h) ♥ t (h, p, x) s♣ t sts ♦ s sr♥ t (x, p) ♣♥ ♦r ① h ♥ r♦♠ r

♥ (∂φ/∂x) s ♥t ♦r x = x∗1, x∗3 ♥ ♣♦st ♦r x = x∗2 t♦♦♥ ♠♦♥♦t♦♥t② ♣r♦♣rts ♦r x∗j

∂x∗j∂h

> 0,∂x∗j∂p

< 0, j = 1, 3;∂x∗2∂h

< 0,∂x∗2∂p

> 0.

rt♦♥ ♣r♦ss s sr ♥ t ♥ t rt♦♥ r♠s r r x∗j r r♣♦rt rss p ♦r r♥t s ♦ h ♦rrs♣♦♥♥ t♦ t♦r r♥t r♥s ♦ h ♦ r ❲t rr♥ s♦ t♦ r ♥s ♥ t ♥ r ♦ t qr♠ stts ♦ ♥ s♣♣r ♥ s♣♦ts tr♥srt rt♦♥ ♣♦♥ts ♠r t P r ♦t ♦♥ t ♥sp = p0 ♥ p = p0/ǫ ♦ r s♥♦ rt♦♥ ♣♦♥ts P ♦♥ trs p = p0φ1(h) ♥ p = p0φ2(h)

♦①st♥ qr♠ stts ♥ t② ①st ♦♥ t r y =βxg(x), x ∈ (ǫ, 1), ♥ t ♣s s♣ ♦s② tr ♥♠r ♥ tr ♣♦st♦♥ ♣♥ ♦♥ t ♣r♠trs h ♥ p ♥ tr ♦r s tr♠♥ ② t

2.4 Coexistence equilibria 53

tr♥s ♦ x∗j(h, p), j = 1, 2, 3, rss h ♥ p

♠r

❱ ♥ t ♦♥r② ♥ h = h0, p ∈ (p0φ0, p0/ǫ) r t tr♠♥♥t ♦J(E∗1(h, p)) s ♣♦st s t♦♥ ♥ tr♦r t s ♥♦t ♦ rt♦♥♥ ♥ ts ♥ t tr♥ ♦ x∗1 rss p s r♣♦rt ♥ r ♥ s♦s♥ ♥t♦♥ ♣♦♥t t rt t♥♥t t p = p0φ0 ❲♥ p0φ0 < p < p0/ǫ tt

limh→h+

0

E∗3(h, p) = limh→h−

0

E∗1(h, p),

♥♠② t qr s♣ ♥♠s ts ♥ s ♥ ♥tr♦ ♥ ♦rr t♦ ♦ s♠♦♦t tr♥st♦♥ ♦ t qr♠ E∗3 ♥st ♦ ♥ r♣t ♥ r♦♠ E∗3

t♦ E∗1 ♦♥ t rt♦♥ ♥ p = p0φ1(h) s r

0 hǫ h1

E∗1,E∗2

B0

E∗3

p = p0φ1(h)

E∗1,E∗2,E∗3

p = p0φ2(h)

E∗1

p0/ǫ

p0φ0

h0

h

p0p0

p

①st♥ r♦♥s ♦ ♦①st♥ qr♠ stts ♥ t (h, p) ♣♥ s♣② ♥ r② r♥ts ♥♦ts r♥t ♥♠r ♦ qr ②♥ ♥ t r♦♥

54 2 The prey-predator model and the equilibrium stability analysis

x∗eq

h = 0

BP1

ǫ

0 p0φ0 p0/ǫpp0

BP

x∗1 = x∗3

x∗1

h = h0

x∗1

0 ≤ h ≤ h0

ξ1

x∗eq

ξ2

ε

0 p0 p0/εp0φ2 p0φ1 p

x∗3

x∗2

x∗1

1BP

LP

LP

BP

h0 < h < hǫ

x∗eq

1

ξ1

ξ2

ε

0 p0 p0φ2 p p0/ε p0φ1

BP

LP

LPBP

x∗2

x∗1

x∗3

hǫ < h < h1

1

ξ1x∗eq

ε

0p

p0/εp0 p0φ1

BP

LP

BP

x∗2

x∗1

h ≥ h1

tt② tr♥ ♦ x∗j rss p ♦r r♥t s ♦ h P tr♥srt rt♦♥ Ps♥♦ rt♦♥

2.5 Stability properties of coexistence equilibrium states

❲ ♥t ♥♦ st② t stt② ♣r♦♣rts ♦ t ♦①st♥ qr♠ stts♦♥ ♥ t ♣r♦s t♦♥ t E∗(h, p) = (x∗(h, p), y∗(h, p)) ♥r ♦①st♥ qr♠ stt ❲ r tt x∗(h, p) ♥ y∗(h, p) sts② t rt♦♥s tt r

y∗ = βx∗g(x∗),px∗

1 + hy∗= p0, f

(px∗

1 + hy∗

)

= α, α =m

cb, β =

rxc

m,

t x∗ = x∗(h, p) ❲ ♥

µ0 := µ(p0), 0 < µ0 < 1,

2.5 Stability properties of coexistence equilibrium states 55

r µ(s) = sf ′(s)/f(s) ts ♣r♦♣rts r sss ♥ ♣♣♥① r♦♠ t①♣rss♦♥ ♦ t ♦♥ ♠tr① s♣② ♥ t♦♥ ♥ s♠♣② t ♥trs♦ t ♦♥ ♠tr① t t E∗(h, p)

J11(E∗(p, h)) = rxg(x∗) + rxx∗g′(x∗)− b

py∗1 + hy∗

f ′

(px∗

1 + hy∗

)

= rxg(x∗) + rxx∗g′(x∗)− by∗

p0f′(p0)

f(p0)

f(p0)

x∗

= rxg(x∗) + rxx∗g′(x∗)− rxµ0g(x∗)

J12(E∗(p, h)) = −bf(

px∗1 + hy∗

)

+ bhpx∗y∗

(1 + hy∗)2f ′

(px∗

1 + hy∗

)

= −bf(p0) + bhy∗px∗

1 + hy∗

p0f′(p0)

f(p0)

f(p0)

px∗

= −rxβ

+ bhy∗p0µ0f(p0)

px∗

= −rxβ

+ rxhµ0p0pg(x∗)

J21(E∗(p, h)) = cbpy∗

1 + hy∗f ′

(px∗

1 + hy∗

)

= cby∗p0f

′(p0)

f(p0)

f(p0)

x∗

= rxcµ0g(x∗)

J22(E∗(p, h)) = cbf

(px∗

1 + hy∗

)

−m− cbhpx∗y∗1 + hy∗

f ′

(px∗

1 + hy∗

)

= −cby∗p0p0f

′(p0)

f(p0)

f(p0)

px∗

= −rxcµ0p0pg(x∗).

♥ t ♦♥ ♠tr① ss♦t t E∗(h, p) s

J(E∗(p, h)) = rx

(1− µ0)g(x∗) + x∗g′(x∗) −1/β + µ0g(x∗)hp0/p

cµ0g(x∗) −cµ0g(x∗)hp0/p

.

56 2 The prey-predator model and the equilibrium stability analysis

❲ ♥t st② t s♥ ♦ t tr ♥ t tr♠♥♥t ♦ ts ♠tr① r♥t♦♥s ♦ h, p, x∗ ♥♦t ②

T (h, p, x∗) = tr J(E∗(h, p)), D(h, p, x∗) = det J(E∗(h, p)).

② rt ♦♠♣tt♦♥ ♥ t♥ ♥t♦ ♦♥t ♥ t t qr♠rtt♥ ♥ t ♦r♠

hp0pg(x∗) =

1

βx∗

(

x∗ −p0p

)

,

♦t♥T (h, p, x∗) = rx (Λ(h, p)g(x∗) + x∗g

′(x∗)) ,

rΛ(h, p) = 1− µ0 − cµ0h

p0p,

♥

D(h, p, x∗) = −r2xcµ0g(x∗)

[

hp0pg(x∗) + h

p0px∗g

′(x∗)−1

β

]

= −r2xcµ0g(x∗)p0p

[

− 1

βx∗+ hx∗g

′(x∗)

]

= −rxmµ0p0g(x∗)

px∗[−1 + hβx2∗g

′(x∗)] = −rxmµ0p0g(x∗)x∗p

(∂φ

∂x

)

x=x∗

.

♥ (∂φ/∂x)x=x∗2> 0 t ♦♦s tt det J(E∗2(h, p)) < 0 ♦r h ♥ p ♥♦t ♦♥♥

t♦ t rt♦♥ rs ♦ r ts E∗2(h, p) s s ♣♦♥t trs♦r j = 1, 3 s♥ (∂φ/∂x)x=x∗j

< 0 t ♦♦s tt det J(E∗j(h, p)) > 0 ♦r h ♥ p♥♦t ♦♥♥ t♦ rt♦♥ rs ts E∗j(h, p), j = 1, 3 s ♦② s②♠♣t♦t②st ♥ ♦♥② T (h, p, x∗j) < 0 ❲ ♥t ♥♦ t♦ st② t s♥ ♦ t tr ♦t ♦♥ ♠tr① t t E∗1(h, p) ♥ E∗3(h, p)

♦♦♥ t t ♦r♠ s♥ ♥♦ tt ǫ ≤ x∗ ≤ 1 ♥ g(x∗) > 0 r ♥trst ♥ tr♠♥ t s♥ ♦ Λ(h, p) ♥ g′(x∗)

s rrs t qr♠ E∗1(h, p) ♥ ♠♣♦rt♥t r♦ ♦r ts stt② ♣r♦♣rtss ♣② ② t t♦ ♠♣t♦♥s

Λ(h, p) ≤ 0 ⇐⇒ p

p0≤ γh;

x∗1(h, p) ≥ ξ0 ⇐⇒ p

p0≤ φ(ξ0, h) = βh+

1

ξ0,

2.5 Stability properties of coexistence equilibrium states 57

rγ =

cµ0

1− µ0

.

♦ s♥r♦s ♠r ♣♥♥ ♦♥ t ♥trst♦♥ ♦ t t♦ strt ♥s

p

p0= γh,

p

p0= βh+

1

ξ0

♥ t (h, p) ♣♥ t ♥trst♦♥ ♦rs ♥ β < γ ♥

rx < r0 :=mµ0

1− µ0

.

s rrs t qr♠ E∗3(h, p) s♥ ♥♦ tt t ①sts ♥ t r♥h0 < h < h1 ♥ tt 0 < x∗3 < ξ2 < η0 < ξ0 ♠♣s g′(x∗3) > 0 tt ♦♥② t ♠♣t♦♥ s ♠♣♦rt♥t ♦r ts stt② ♣r♦♣rts

♥ t ♥①t sst♦♥s st② t s♥ ♦ t tr ♦ ♦♥ ♠trs ❲rt r t ①♣rss♦♥s ♦ t rts ♦ T (h, p, x∗) s ♥ t ♦♦♥

∂T

∂h= −rxcµ0

p0pg(x∗) < 0,

∂T

∂p= rxcµ0

p0h

p2g(x∗) > 0,

∂T

∂x∗= rxΛg

′(x∗) + (x∗g′(x∗))

′.

2.5.1 State E∗1: the case of rx ≥ r0

❯♥r t ss♠♣t♦♥ rx ≥ r0 ♦r q♥t② β ≥ γ tt ♥ t (h, p)♣♥

βh+1

ξ0> γh ∀h > 0,

s♦ tt t t♦ ♥s p/p0 = βh + 1/ξ0 ♥ p/p0 = γh ♦ ♥♦t ♥trst ❲ rr t♦r t♦ strt t rsts ♦ ts s ❲ r t♦ ♣r♦ tt tr ①sts ♦♣ rt♦♥ r p = p1(h) ♦♥ tr J(E∗1(h, p1(h)) = 0 ts r s♠♦♥♦t♦♥② ♥rs♥ ♥

p0 max1, γh < p1(h) < p0

(

βh+1

ξ0

)

.

58 2 The prey-predator model and the equilibrium stability analysis

♦r♠ ss♠ ♥ rx ≥ r0 ♥ ♦r ♥② h ≥ 0

∃ p1(h) : p0 max1, γh < p1(h) < p0

(

βh+1

ξ0

)

,

s tt

T (h, p1(h), x∗1(p1, h)) = 0, T (h, p, x∗1(h, p)) < 0 ♦r p ∈ [p0, p1(h)),

♥ T (h, p, x∗1(h, p)) > 0 ♥ t ①st♥ r♦♥ ♦ E∗1 ♦ t (h, p) ♣♥ rp > p1(h)

0 ≤ h ≤ h0, p1(h) < p <p0ǫ

♥ h0 < h, p1(h) < p < p0φ1(h).

♦r♦rdp1dh

> 0.

Pr♦♦ r♦♠ t ♠♣t♦♥s ♥ t ♦♦s tt ♥ t r♦♥ ♦ t(h, p) ♣♥ r

h ≥ 1

γ, 1 ≤ p

p0≤ γh

Λ ≤ 0 ♥ x∗1 > ξ0, g

′(x∗1) < 0,

♠♣② T (h, p, x∗1) < 0 ♦♥sr ♥♦ t r♦♥ r

max1, γh ≤ p

p0≤ βh+

1

ξ0< φ1(h).

♥ t ♦♥rs tt

T (h, p, x∗1) < 0 ♦rp

p0= max1, γh,

r tr p = p0, x∗1(h, p0) = 1 ♦r p = p0γh, ξ0 < x∗1(h, p0γh) < 1 ♥

T

(

h, p0

(

βh+1

ξ0

)

, ξ0

)

> 0 ♦rp

p0= βh+

1

ξ0.

r♦♠ t ♠♣t♦♥s

Λ > 0 ♥ x∗1 > ξ0, g′(x∗1) < 0, (x∗1g

′(x∗1))′< 0.

2.5 Stability properties of coexistence equilibrium states 59

♥ ♥t♦ ♦♥t ♥ t ♠♦♥♦t♦♥t② ♣r♦♣rt② ♦x∗1 ∂x∗1/∂p < 0 tt

∂T

∂x∗1< 0,

dT

dp=∂T

∂p+

∂T

∂x∗1

∂x∗1∂p

> 0.

t ♦♦s tt T (h, p, x∗1) ♦♥sr s ♥t♦♥ ♦ p ♦r ① h ≥ 0 s ♥t♦r p/p0 = max1, γh s ♥rs♥ t p ♦r max1, γh < p/p0 < βh+ 1/ξ0 ♥♣♦st ♦r p/p0 = βh+1/ξ0 s tr s st ♦♥ ③r♦ ♦ T (h, p, x∗1) ♥♦t ②p1(h) ♦r p0 max1, γh < p < p0(βh+ 1/ξ0)

♦r♦r ♥ t ①st♥ r♦♥ r ♦ E∗1 ♦ t (h, p) ♣♥ rp > p1(h)

Λ > 0 ♥ x∗1 < ξ0, g′(x∗1) > 0,

♠♣② T (h, p, x∗1) > 0♥② t ♠♦♥♦t♦♥t② ♣r♦♣rt② ♦ p1(h) ♦♦s r♦♠ ∂T/∂h < 0

♦r t ♦♥sr ♥t♦♥s g(·) ♥ f(·) ♥ ♥ t ♣r♠trs s♣ ♥ ♣♣♥① p1(h) tr♥s ♦t t♦ r② ♦s t♦ βh + 1/ξ0 ①st♥ r♦♥ ♦ E∗1(h, p) ♥ t (h, p) ♣♥ s ♥ tr sr♦♥s r t t r♦♥s ♥ r E∗1(h, p) s ♦② st ♥ t r②

r♦♥ r t s ♥stt ∆(h, p, x∗1) = T 2(h, p, x∗1) − 4D(h, p, x∗1) ♦r ① h ≥ 0 t p = p1(h)

T (h, p1(h), x∗1(h, p1(h))) = 0, ∆(h, p1(h), x∗1(h, p1(h))) < 0,

s♦ tt t ♦♥ ♠tr① s s♠♣ ♣r ♦ ♣r ♠♥r② ♥s ♦r♦r dT/dp > 0 ♦r max1, γh < p/p0 < βh + 1/ξ0 s p = p1(h) s ♦♣rt♦♥ r ♦r t qr♠ E∗1(h, p) t ♦tt ♦♥ ♥ r ♥♠t ②s ♠r ♦r p ♥ ♥♦r♦♦ ♦ p1(h) stt② ♦ t ♠t②s ♥ tr♠♥ ② t s♥ ♦ t rst ②♣♥♦ ♦♥t ss♦tt t s②st♠ ❬❪ ♣ ♦r♠ ❬❪ ♣ ♦r♠ ♦♥t ♠♦ ♥t♦♥s r ① ♣♥♥ ♦ ts ♦♥t ♦♥ t ♣r♠trsh ♥ p s r② ♥trt ♥ t♦rt ♥②ss s sst♥t ♥rt♥ s♥ s t♥ tr♠♥ ♥♠r② ♥ s♦ t t s♣ s♦tr ❬❪ ♦r s♣ s②st♠s rtr③ ② r♥t ♣r② r♦t ♥tr♦♣ ♥t♦♥s ♥ t st ♦ ♦♦♦ ♣r♠trs ♥ ♥ ♣♣♥① s♦t s ♦ t ♦♣ rt♦♥ r ♦♥ t ♠t ②s ①st s ♥ ♦♥♥♠r② rsts strt ♥ t ♥①t st♦♥

60 2 The prey-predator model and the equilibrium stability analysis

p0

0

p0/ξ0

T (p,h,x∗1) < 0

1/γ h1

p = p0(βh+ 1/ξ0)

p = p0γh

p = p1(h)

p = p0φ1(h)

3

21

T (p,h,x∗1) > 0B0

p0φ0

p0/ǫ

p

h0h

♦ stt② ♥ ♥stt② r♦♥s ♥ t (h, p) ♣♥ ♦ t ♦①st♥ qr♠ sttE∗1(h, p) ♥ t s rx ≥ r0 ♥ r② t r♦♥ ♥ T (h, p, x∗1) > 0

2.5.2 State E∗1: the case of rx < r0

stt♦♥ s s♦♠t ♠♦r ♥♦ t♥ ♥ t ♣r♦s s ❲ rr t♦r t♦ strt t rsts ♦ ts s ❯♥r t ss♠♣t♦♥ rx < r0 ♦rq♥t② β < γ tt ♥ t (h, p) ♣♥ t ♥ p/p0 = γh ♥trsts t♥ p/p0 = βh + 1/ξ0 ♥ t r p = p0φ1(h) t ♣♦♥ts R ♥ S rs♣t②t hR ♥♣♥♥t ♦ p ♥ hS ♣♥♥t ♦♥ p t h♦♦r♥ts ♦ t♥trst♦♥ ♣♦♥ts R ♥ S

hR =1

ξ0(γ − β)=

1

ξ0β(r0/rx − 1), hS = φ−1

1

(p

p0

)

.

s♦ ♥ ts s ♥ ♣r♦ tt tr ①sts ♦♣ rt♦♥ r p = p1(h)♦r h < hR s ♠♦♥♦t♦♥② ♥rs♥ r ♦r♠ ♥ r♦r♠t s ♦♦s

♦r♠ ss♠ ♥ rx < r0 ♥

♦r 0 ≤ h < hR, ∃ p1(h) : p0 max1, γh < p1(h) < p0

(

βh+1

ξ0

)

,

s tt

2.5 Stability properties of coexistence equilibrium states 61

T (h, p1(h), x∗1(p1, h)) = 0,

T (h, p, x∗1(h, p)) < 0 ♦r p ∈ [p0, p1(h)), h ≥ 0

T (h, p, x∗1(h, p)) > 0 ♦r 0 ≤ h ≤ h0, p1(h) < p <p0ǫ

♥

h0 < h < hR, p1(h) < p < p0φ1(h).

♦r♦rdp1dh

> 0.

rtr r♦♥ t ♣♦st tr s

hR < h < hS, γh <p

p0< φ1(h),

r hS s t ♥q s♦t♦♥ t♦ t qt♦♥ γh = φ1(h)

♣r♦♦ ♦ ♦r♠ s ♦♠tt s t ♦♦s t ♥s ♦ t ♣r♦♦ ♦ t♣r♦s ♦r♠

♥ ♥r t stt② ♣r♦♣rts ♦ E∗1(h, p) ♥♥♦t sts ♥ t sr♦♥ ♦ ①st♥ ♠r t r r② ♥ r ♥ ②

h > hR, βh+1

ξ0<

p

p0< minγh, φ1(h).

♥ ♥ ts r♦♥ t ♠♣t♦♥s ♥ ♦ ♥♦t ♦ ♥ ♥♦ ♥♦r♠t♦♥ ♦t t s♥ ♦ T (h, p, x∗1) t s ♣♦ss ♥rtss t♦ ts♥ ♦ t tr T (h, p, x∗1) ♦♥ t r p = p0φ1(h) ♥ t tr♥s ♦t tt ♦rh st② ♦ hS T (h, p, x∗1) > 0 ♥ limh→+∞ T (h, p, x∗1) = rx(1− µ0)(1− γ/β) s ♥t ♦♥② ♥ rx < r0 ♥ t♥ t♥s t♦ t ♠♦♥♦t♦♥t②♣r♦♣rts tr ①sts ♥q hBT r♥ T (hBT , p, x∗1) = 0 sts r t ♣rs♥ ♦ ♦♥♦♥s ♣♦♥t ♥trst♦♥ ♦ s♥♦ ♦♣ ♥ s♣rtr① ♦♠♦♥ ♦♦♣ r tt sss ♥ t ♥①tst♦♥ s♦ ♥ ts s t stt② ♦ t ♠t ②s s ♥ ♥♠r② st♥ t rsts sss ♥ t ♥①t st♦♥

2.5.3 State E∗3

s rrs t qr♠ E∗3(h, p) ♥♦ tt t ①sts ♥

h0 < h < h1, φ2(h) <p

p0<

1

ǫ,

62 2 The prey-predator model and the equilibrium stability analysis

0 1/γ

p = p0γh

p = p0φ1(h)

T (p,h,x∗1) > 0

T (p,h,x∗1) < 0

hS h1hR

p0

h0h

p0/ǫ

p0φ0

p

p0/ξ0p = p1(h)

p = p0(βh+ 1/ξ0)

B0

T (p,h,x∗1) < 0

h1hSh

hR

p

T (p,h,x∗1) > 0

p = p0γh

p = p0φ1(h)

S

R

p = p0(βh+ 1/ξ0)

♦ stt② ♥ ♥stt② r♦♥s ♥ t (h, p) ♣♥ ♦ t ♦①st♥ qr♠ sttE∗1(h, p) ♥ t s rx < r0 t r② ♥♦ts r♦♥s ♥ T (p, h, x∗1) > 0 r r② ♠rs r♦♥s♥ t stt② ♣r♦♣rts ♦ E∗1 ♥♥♦t ♥②t② tr♠♥ r s ♥ ♥r♠♥t ♦t s rt♥r r♦♥ ♦ r

2.5 Stability properties of coexistence equilibrium states 63

♥ tt 0 < x∗3 < ξ2 < η0 < ξ0 ♠♣s g′(x∗3) > 0 t ♦♦s tt ♦♥② t♠♣t♦♥ s ♠♣♦rt♥t ♦r ts stt② ♣r♦♣rts

♥ r♦♠ t ①st♥ ♦♥t♦♥s ♦ E∗3(p, h) ♥ ♠♣t♦♥ t ♦♦stt ♦r h0 ≤ h ≤ h1

max φ2(h), γh <p

p0<

1

ǫ=⇒ T (p, h, x∗3) > 0.

stt♦♥ ♦r E∗3 s strt ♥ r t h1 = 1/(γǫ) t ♥trst♦♥♣♦♥t t♥ t strt ♥s p = p0/ǫ ♥ p = p0γh

h1 ≥ h1,

r0rx

≤ ǫg′(ǫ),

t♥ E∗3(p, h) s ♥♦♥t♦♥② ♥st r ♦r ♥st♥ ts s♦s ♦r ♦♥t②♣ tr♦♣ ♥t♦♥ ♣♥ ♠♦ ♦r g(s) ♥ ♣r♠trss ♥ t rx > r0

h1 < h1,

r0rx> ǫg′(ǫ),

t♥ ♥ ♥r t stt② ♣r♦♣rts ♦ E∗3(h, p) ♥♥♦t sts ♥ tr♦♥ ♥ ②

h2 ≤ h ≤ h1, φ2(h) ≤p

p0≤ min

γh,1

ǫ

,

r h2 s t ♥q s♦t♦♥ t♦ t qt♦♥ γh = φ2(h) r r② r♦♥ ♥r ♥ ♥ ts r♦♥ t ♠♣t♦♥ ♦s ♥♦t ♦ ♥ ♥♦ ♥♦r♠t♦♥ ♦t t s♥ ♦ T (h, p, x∗3)

stt② ♣r♦♣rts ♦ qr♠ stts r s♠♠r③ ♥

64 2 The prey-predator model and the equilibrium stability analysis

T (h, p, x∗3) > 0

E∗3p = p0φ2(h)

p = p0γhB0

p0/ǫ

p0φ0

p

p0

0 h0h

h1 h1

h1 > h1

p = p0γh

p = p0φ2(h)

E∗3

B0

p0/ǫ

T (h, p, x∗3) > 0

p0φ0

p

p0

0 h0 h1 hh2 h1

h1 < h1

♥stt② r♦♥s ♥ t (h, p) ♣♥ ♦ t ♦①st♥ qr♠ stt E∗3(h, p)

2.5

Stability

propertie

sofcoexistenceequilib

rium

states

65

0 ≤ h ≤ h0 h0 < h < h1 h > h1

E0 st ♥♦ p > 0

Eǫ s 0 < p < p0/ǫ ♥st ♥♦ p > p0/ǫ

E1 st ♥♦ 0 < p < p0 s p > p0

E∗1(h, p)rx ≥ r0

st p0 < p < p1(h)st p0 < p < p1(h) ♥st p1(h) < p < p0φ1(h)

♥st p1(h) < p < p0/ǫrx < r0 s

st p0 < p < p0(βh+ 1/ξ0)♥ p0(βh+ 1/ξ0) < p < p0φ1(h)

E∗2(h, p) ∄ s p0φ2(h) ≤ p ≤ p0φ1(h) s p0/ǫ ≤ p ≤ p0φ1(h)

E∗3(h, p) ∄♥st maxp0φ2(h), p0γh ≤ p ≤ p0/ǫ, h0 < h < h1 ∄♥ p0φ2(h) ≤ p ≤ minp0γh, p0/ǫ, h2 < h < h1

h0 < h < hR hR < h < hS hS < h < h1

E∗1(h, p) st p0 < p < p1(h) st p0 < p < p0(βh+ 1/ξ0) st p0 < p < p0(βh+ 1/ξ0)rx < r0, h0 < h < h1 ♥st p1(h) < p < p0φ1(h) ♥ p0(βh+ 1/ξ0) < p < p0γh ♥ p0(βh+ 1/ξ0) < p < p0φ1(h)

♥st p0γh < p < p0φ1(h)

♠♠r② ♦ stt② ♣r♦♣rts ♦ qr♠ stts ♥=♥♦t tr♠♥ ♥st=♥st ts ♦♥ stt② ♦E∗1(h, p) ♦r h0 < h < h1 ♥ rx < r0

3

Numerical study

♥ t ♣r♦s ♣tr ♥stt t ②♥♠ ♦rs ♦ ♣rt♦r♣r② s②st♠ ♥ t ♠♦ ♥t♦♥s sr♥ t ♦♦ ♣r♦sss ♦rr♥♥ t ♦♥sr tr♦♣ ♥ r ♥♦t s♣ ② ♥②t ①♣rss♦♥s t ②s♦♠ rtrst ♣r♦♣rts tr♠♥♥ tr s♣s ❲ s♦♥ tt ts ss ♦r t ①st♥ ♥ stt② ♥②ss ♦ t qr♠ stts ♦ t s②st♠ ❯♥♦rt♥t② t stt② ♥②ss ♦ ♠t ②s ♥♥♦t s② ♣r♦r♠♦♦♥ ts ♥r ♣♣r♦ ♥ t ♠♦ ♥t♦♥s t♦ s♣ t♦ ♦rtr ♥ t qtt ♥②ss

♥ ts ♣tr rsts ♦ ♥♠r s♠t♦♥s ♦t♥ ♦r s♦♠ ♦♥rt r③t♦♥ ♦ t ♠♦ ♥t♦♥s r ♣rs♥t t♦ strt t ♦rs ♦ ts②st♠ rsts ♦♥r♠ ♥②t ♣rt♦♥s ♥ tr♦ t ♦♥ s♦♠ s♣ts♦ t ②♥♠s ♦ t s②st♠

3.1 Behaviours of the system

r ♦s ♦♥ s♦♠ ♣r ♦rs ♦t♥ ② s♥ t ♠♦ ♥t♦♥s ♥ t ♣r♠tr s s♣ ♥ ♣♣♥① ♦rs r♦ ♦rs ♥ r♠♥t t t ♥②t rsts ♦t♥ ♥ t ♣r♦s ♣tr♥ t ♠♦ ♥t♦♥s r ① ♥ s♦ ♥♠r② ♥stt t ♠t②s rs♥ r♦♠ E∗1 ② ♦♣ rt♦♥ ♠♥ trs ♦ t rt♦♥strtr s♦♥ ♥ t ♦♦♥ r♠s r ♦♥② qtt s t r rt♦♥ rs s♠t r ② s♥ ♠♦ ♥t♦♥s ♥ ♥ ♣r♠tr s s ♥ r ♠♦st ♥st♥s r♦♠ ♦♥ ♥♦trr♦r s♦♠ ♦ t ♣♥♦♠♥ sr ♦ ♥ ♦♥② t r②♥ rs♦t♦♥ ♥②♦ ♦t♥ qtt② t s♠ s♥r♦s t r♥t

68 3 Numerical study

♦♠♥t♦♥s ♦ ♠♦ ♥t♦♥s rtr ♥♦r♠t♦♥ ♦♥ t rt♦♥ strtr ♦s t♦ rt s ♦ t rt♦♥ ♣r♠trs tt ♥ t♦♦♥ ♥②ss s s ♥♠r ♦ rt♥ ♠t ②s r ♦♠♥s♦♥♣♦♥ts ♦ ♦t♥ s ② s t s♣ ♠♦ ♥t♦♥s ♥ t ♠ttr ♦ tr ♦r

• ❲♥ 0 < h < h0 ♥ tst ss ♥♣♥♥t② ♦ t ♦ rx t ♥♠r rsts s♦ t ①st♥ ♦ rt h s tt

♦r ♥② ① 0 ≤ h < h st ♠t ②s ♠r ♦r p st② ♦ t♦♣ p1(h) ♥ s♣♣r ② ♦ rt♦♥ t t tr♦♥ ② ♥♦♥ qr Eǫ ♥ E1 t rtr rt p2(h) > p1(h) sr ♥ ♥ ts r♥ ♦ h t s②st♠ s qtt② st s②st♠ t h = 0 ❬ r ❪

♥ h > h r♣♥ ♠t ②s ♠r ♦r p st② ♦ t ♦♣ p1(h) ❲ ♥♠r ♥ ♦ t ①st♥ ♦ rtr rt hC s tt ♦r h < h < hC r t r♣♥ ♠t ②s s♣♣r② s♥♦ rt♦♥ t st ♠t ② ♦♥ t r p = p3(h)♥ st ♠t ②s t♥ s♣♣r ② ♦ rt♦♥ ♥♦♥ t tr♦♥ ② t♥ qr Eǫ ♥ E1 r ♥ ♦♥ tr p = p2(h) r r ♦ tr♦♥ ②s p = p2(h) ♥trsts p = p3(h) ♦r h = hC t r♦sss t ♦♣ r p = p1(h) ♦r h ∈ (h, hC) ♦r hC < h < h0 t r♣♥ ♠t ②s s♣♣r ♥st② ♦ rt♦♥ t t tr♦♥ ② ♥♦♥ Eǫ ♥ E1 r ♦♥ t r p = p2(h) < p1(h)

♦♣ rt♦♥ s ts s♣rrt ♦r h < h ♥ srt ♦r h > h tr p = p3(h) ♦ s♥♦ rt♦♥ ♦ ♠t ②s ♠♥ts r♦♠ t ♣♦♥t(h, p1(h)) t s ♣♦ss t♦ tt ② s♥ t ♦♥t♥t♦♥ s♦tr ❬❪ tt t ♦♣ rt♦♥ r s ♥r③ ♦♣ ♦♠♥s♦♥t♦ ♣♦♥t t h = h ♣r ♥stt♦♥ ♦ t rt♦♥ ♦rr♥ t h = hC r p = p2(h) ♥ p = p3(h) ♥trst ♦tr ♦ ♣r♦r♠ s ②s ♦♦♥ ❬❪ t s♣ ♠♦ ♥t♦♥s ♥ ♦ ♦ t♦ tr♠♥♦ ♠♥② ②s rt ♥ ts r♦♥ ♦ t ♣r♠tr s♣ ♥ t s trstt② ts ♥②ss ♠ttr ♦ tr sts

3.1 Behaviours of the system 69

• ❲♥ h0 < h < h1 ♥ tr rx tt t ♦①st♥ r♦♥ ♦ tqr E∗1, E∗2, E∗3 r r② r♦♥ ♥ r s r♦ss ② t ♦♣rt♦♥ r p = p1(h) s r ♥ t ♦♣ rt♦♥ r♥rts ♥st ♠t ②s ♦r p st② ♦ t ♦♣ p1(h) s②s s♣♣r ② ♦ rt♦♥ ♦♥ t r p = p2(h) ♥ ♦rtr t t tr♦♥ ② ♦♥♥t♥ Eǫ ♥ E1 ♦r t ♦♠♦♥ ②t t s ♣♦♥t E∗2 ♥ t t hU t sss ♦ t ♥trst♦♥♦ t rs p = p2(h) ♥ p = p0φ2(h) r t♥ t tr♦♥② ♦rs ♦r h < hU ♥ t ♦♠♦♥ ② ♦r h > hU tr♥st♦♥ r♦♠♥ tr♦♥ t♦ ♦♠♦♥ ♦rts tr♥s ♦t t♦ s ② t ♦r♠t♦♥ th = hU ♦ tr♦♥ ② ♦♥♥t♥ t st ♣♣r qr♠ E∗2 = E∗3

♥ E1 s s t♦ t ♥trt♦♥ ♦ t ♠t ②s t t qr♠ E∗2

♥ t♥ t♦ tr s♣♣r♥ ② ♦ rt♦♥ t ♦♠♦♥ ②tr♦ E∗2 ♥st ♦ ② tr♦♥ ② ♦♥♥t♥ Eǫ ♥ E1 ♣♦♥th = hU s♠s t♦ ♦♠♥s♦♥tr ♦r ♥ r rt♦♥ ♣♦♥t ♥②② ♣r ♥②ss ♠♥ t tt♥ ♣♦ss ♥♦♥ q♥t ♣s ♣♦rtrtsr♦♥ ts ♣♦♥t ♥♥♦t rr ♦t ♥ ♥r ♥ t ♠ttr ♦ tr♥stt♦♥s t s♣ ♠♦ ♥t♦♥s♦♠ ①♠♣s ♦ t ♣r ②♥♠s ♦t♥ ♦r r♥t p ♥ h0 < h < h1r r♣♦rt ♥ t ♣s ♣♦rtrts ♥ r ❲ st ss ♥ ♦①st♥ qr r ♣rs♥t ♥ ♦s ♦♥ stt② ♦rr♥ ♥ ts②st♠ ♥ r r p < p2(h) t trt♦rs t♥ t♦ E0 ♦r E∗1 ♣♥♥ ♦♥ t ♥t ♦♥t♦♥s ♥ r r p2(h) < p < p1(h) ♥♥st ♠t ② s♣rts t s♥s ♦ ttrt♦♥ ♦ t st qr E∗1

♥ E0 st② ♥ r r p > p1(h) t ♦①st♥ qrr ♥st ♥ t s②st♠ ♦s t♦rs ♦ ①t♥t♦♥

• ♦r h > h1 ♥ rx ≥ r0 ♣r♦ ♥ ♦r♠ tt t ♦♣ rt♦♥r p = p1(h) s ②s ♦ t stt♦♥r② rt♦♥ r p = p0φ1(h)r ♥♠r ♥stt♦♥ s♦s tt ♥st ♠t ②s rs♦ t r p = p1(h) ♥ s♣♣r ② ♦ rt♦♥ t ♦♠♦♥② tr♦ t s ♣♦♥t E∗2 ♦♥ t r p = p2(h) < p1(h) r ♦r h > h1 ♥ rx < r0 ♥♥♦t stt ♥ ♥r t ♠t ♣♦st♦♥s ♦ trs p = p1(h), p = p0φ1(h) ♦r♦r t stt② ♣r♦♣rts ♦ E∗1 ♥♥♦t tr♠♥ ♥ ♥r ♥ t r♦♥ ♥ ♥ r r② r♦♥ ♥r ❲ ♥♠r② tt ♦♣ rt♦♥ s p1(h) ♦r

70 3 Numerical study

E∗1 s♦ ♥ h s ♦ t st♠t trs♦ hR s r ♥♥st ♠t ②s rs ♦ t r p = p1(h) ♥ s♣♣r ② ♦rt♦♥ t t ♦♠♦♥ ② tr♦ E∗2 ♦♥ t r p = p2(h) < p1(h) ♦♣ rt♦♥ r p = p1(h) s ♦ t ♥ p = p0(βh + 1/ξ0) ♦rh < hR ♥trsts t ♦r h = hR t ♣♦♥t R r γh = βh + 1/ξ0 ♥ st②s♥t② ♦ ♦r h > hR ♥ t s♥ ♦ t tr T (h, p, x∗1) ♦♥ tr p = p0φ1(h) ♥s r♦♠ ♣♦st t♦ ♥t s ♣♦♥t ♦t ♥ st♦♥ tr ①sts t rt hBT > h1 t t ♦♣ r p = p1(h)♥trsts t r p = p0φ1(h) s♦ t ♦ rt♦♥ r p = p2(h) ♥♦s t ♦♠♦♥ ② tr♦ t s ♣♦♥t E∗2 ♣sss tr♦ts ♥trst♦♥ ♥ t♥ ♦t♥ ♦♠♥s♦♥t♦ ♣♦♥t ♥ t (h, p) ♣♥♥ D(hBT , pBT , x∗1) = 0 ♥ T (hBT , pBT , x∗1) = 0 tt tr♥s ♦t t♦ ♦♥♦♥s rt♦♥ ♣♦♥t t s rtr③ ② t t②♣ ♣sr♠ ♦ r ts t②♣ ♦ rt♦♥ s ♥ tt ② ♦tr t♦rs♥ s♠r ♠♦s ❬ ❪

p

p0/ǫ

p0

0 hC h0hh

h1

E∗1,E∗2

E∗1 p = p0φ2(h)

E∗1,E∗2,E∗3

p = p3(h)

p = p2(h)

p = p1(h)

p = p0φ1(h)

E∗3

B0

GH

hǫ hU

tt r♣rs♥tt♦♥ ♦ ♦ ♥ ♦ rt♦♥ rs ♦r t qr♠ E∗1 ♥ ♠t②s ♥ r② t r♦♥s ♦ t (h, p) ♣♥ ♥ st ♦r ♥st ♠t ②s ①st ♥r③♦♣ ♣♦♥t

p

x∗

p2(h)p1(h)ε

1

H

LPC

0 < h < h

p

x∗

H

LPC

LPC

1

ǫ

p3(h) p1(h) p2(h)

h < h < hC ♦s t♦ h

p

xeq

LPCLPC

H

ǫ

1

x∗1

p3(h) p2(h) p1(h)

h < h < hC ♦s t♦ hC

p

xeq

LPC

H

1

εp1(h)p2(h)

x∗1

hC < h < hU

LPC

LP

1

ǫ

x∗1

x∗2

Hxeq

p2(h) p1(h) p0φ1(h)

h > hU

♠t ②s ♦rs rss p t ♠t ②s ♠r ♥ s♣♣r ② ♦ rt♦♥ ♣♥ ♠t ②s ♠r ♥ s♣♣r ② s♥♦ rt♦♥ t st ♠t ②♦♥ t r p = p3(h) ♥ st ♠t ②s t♥ s♣♣r ② ♦ rt♦♥ ♦♥ t r p = p2(h) ♣♦st♦♥ ♦ ts r t rs♣t t♦ p = p1(h) ♥s ♥rs♥ t ♦ h ♣♥ ♠t②s ♠r ♥ s♣♣r ♥st ② ♦ rt♦♥ rt ♥s r♣rs♥t t ♣r♦t♦♥s ♦t ♠t ②s ♦♥ t (p, x) ♣♥ ♦ ♥s st ②s s ♥s ♥st ②s P ♥♦tsrt♦♥ ♦ ♠t ②s tt ② ♥ts ♦♣ ♣♦♥t

x

y

E0 Eε E1

E∗2

E∗3

E∗1

x

y

E0 E1

E∗3

E∗2

E∗1

Eε

x

y

E0 Eε

E∗3

E∗1

E1

E∗2

Ps ♣♦rtrts ♦r h0 < h < h1 ♥ r♥t p p < p2(h) p2(h) < p < p1(h) p > p1(h)

hBT

BT

h1

p = p1(h)

p = p0γh

p = p0φ1(h)

R

S

T (p,h,x∗1) > 0

p = p2(h)

hS

h

hR

p p = p0(βh+ 1/ξ0)

tt r♣rs♥tt♦♥ ♦ ♦ ♥ ♦ rt♦♥ rs ♦r t qr♠ E∗1 ♥♠t ②s ♦r h > hR ♥ t s rx < r0 t r② ♥♦ts t r♦♥ ♥ T (h, p, x∗1) > 0 ♦♥♦♥s ♣♦♥t

x

y

Eε E1E0

E∗1, E∗2

Ps r♠ ♦s t♦ t ♦♥♦♥s rt♦♥ h = hBT , p = pBT

4

Concluding remarks

❲♥ ♠♦♥ ♣rt♦r♣r② s②st♠s t ♣r♦♣rts ♦ t ♣r② r♦t ♥t♦♥g(·) ♣♥ ♦♥ t ♥tr♦t♦♥ ♦ ♥trs♣ ♦♠♣tt♦♥ ♠♦♥ t ♣r② ♥ ♦♥t ss♠♣t♦♥ ♦ tr t s♥ ♦r t ♣rs♥ ♦ str♦♥ t♥ t ♦tr ♥ t tr♦♣ ♥t♦♥ f(·) ♠② ss♠ tr ♦♥ ♦r s♣ ♥ t ♥ ♣r②♣♥♥t rt♦♣♥♥t ♦r ♣rt♦r♣r② ♣♥♥t♦r♦r ♦♥② s♦♠ ♦ tr qtt ♣r♦♣rts r ♥♦♥ s ts ♥t♦♥ss♦ ♥♦t s♣ ② ♥② ♥②t ①♣rss♦♥ t ♦♥② ② ♥r ♣r♦♣rtstt ② ♦♦♦ ♦♥srt♦♥s ♠♦r♦r t② s♦ sts② s♦♠ t♥ss♠♣t♦♥s t♦ ♠ t ♥②ss trt

ss♠♣t♦♥s ♠ ♦♥ t ♠♦ ♥t♦♥s ♠t t ♥♠r ♦ qr♠♣♦♥ts ♥ ♦ t♦ ♣r♦r♠ s♥t② ♥r ①st♥ stt② ♥ rt♦♥♥②ss ♦ t qr♠ stts s s t ♠♥ r♥ t rs♣t t♦ t♥stt♦♥ ♦ t ②♥♠s ♥ ♥r③ ♠♦s rr ♦♥ ② ♦trs t♦rs❬ ❪ ♦ ss♠ tt t st ♦♥ st② stt ①sts t♦t ②♣♦tss ♦♥t ♥r ♥t♦♥s ♥ t♥ ♣r♦r♠ t ♦ s②♠♣t♦t stt② ♥②ss ♦t st② stt ♥r ♦♥srt♦♥ ♥ t ♦♥trr② ♥ ts ♦r ss♠♣t♦♥s ♦♥t ♣r② r♦t ♥t♦♥ ♥ ♦♥ t ♥t♦♥ rs♣♦♥s r rqr ♥ ♦rr t♦t ♥t♦ ♦♥t ♣r♦♣r ♦♦ ♣r♦♣rts ♦r♦r ts ss♠♣t♦♥s ♠tt ♥♠r ♦ qr♠ stts ♠tt ② t s②st♠ ♥ t stt② ♥rt♦♥ ♥②ss s st ♦r qr♠ stt

♦r ts ♣♣r♦ s t♦ s♦♠ rstrt♦♥s ♥ t ♥②ss ♦ t ②♥♠s t s ♦♦ t t ss♠♣t♦♥s ♦♥ g ♥ f s ♦♥t♦♥s ♠♣②rt♦♥s♣s t♥ f(s) f ′(s) ♥ g(s) g′(s) g′′(s) t ♥② rt ♥ ♦♣rt♦♥ s tt t stt② ♣r♦♣rts ♦ t ♠t ②s ♥♥♦t ♥ ♥r sts r♦♠ t s♥ ♦ t rst ②♣♥♦ ♥♠r s t ♣♥s ♦♥t rst s♦♥ ♥ tr rts ♦ t rt ♥ s ♦ t t

76 4 Concluding remarks

t qr♠ ♣♦♥t ♥ t r② r♥t ♣♣r ② ♠s♦♥ ♥ ♦r♦③♦ ❬❪ t s♣♦♥t ♦t tt t s ♦ t♦ r♥t ♥t♦♥s ♦♥♥ t♦ t s♠ ss ♥rst ♥ qtt② r♥t ②♥♠ ♦r ♥ t ♠♦ ♥ r♥tt②♣ ♦ rt♦♥ ♥ t trtr t ♦♥♥t♦♥ ② t♦ ♦ s ♠t②s t♦ ♥rr♦ t ss ♦ ♥♥♦♥ ♥t♦♥s ♥ t② ♦♥ tt ts ♣♣r♦♠② t♦ ♠rs♦♠ ①♣rss♦♥s ♦♦② ♠♥♥ ❲ ♦sr tt s♦♠♥rt♥ts ♦ r♠♦ ② rr②♥ ♦t ♠♥② ♥♠r s♠t♦♥s tr♥t ♠♦ ♥t♦♥s

♥ ♦r ♣♣r♦ t s ♦ r♥t ♥t♦♥s ♦♥♥ t♦ t s♠ ss s t♦s♦♠ ♦♠♠♦♥ ♦rs ♦r ♥st♥ ♥ ♦♥♥t♦♥ t t ♥♠r ♦ qr♥ tr stt② ♣r♦♣rts ♦r s♦♠ ♣r ②♥♠s s s t ♣sr♠ ♦s t♦ t ♦♥♦♥s rt♦♥ ♣♦♥t ♥ t s ♦ rx < r0♥ ♦♥ ♦♥② t s♣ ♥t♦♥s ♥ t ss ♠s♦♥ ♥ ♦r♦③♦ ❬❪♥②s ♥ t ts r ♣r♦♠ ♥ s♦ tt r t ♠♦ ♥t♦♥s r♥♦t s♣ ② ♥②t ①♣rss♦♥s t rt♦♥s ♥ sr ♦♥② t rt♥ ♣r♦t②

❲ s t♦ ♣♦♥t ♦t tt st② r♥t ♦r♠t♦♥s ♦ t ♠♦ qt♦♥st♦tr t rs♥ ♦ stt rs ♥ ♣r♠trs ♥ t s ♦ r♥trt♦♥ ♣r♠trs ♥ t stt② ♥②ss ♠ t ♦♠♣rs♦♥ t♥ tr♦s s♥r♦s r ♦r ♥ ♥② ♥t ♦rs ♦ ♣rt♦r♣r② s②st♠s♦ t②♣ rtr③ ② r♥t ♠♦ ♥t♦♥s g(·) ♥ f(·) ♥②s ♥t trtr ❬ ❪ sr s♦♠ tt♥t♦♥ ❯♥r tss♠♣t♦♥ ♦ str♦♥ t t ♥t♦♥s ♠♣♠♥t ♥ t ♦r♠♥t♦♥trtr ♥ ♦♠♥ ♥ r♥t ②s ♥ s②st♠ r

♥ t s ♦ ♣r②♣♥♥t tr♦♣ ♥t♦♥ r③ ② stt♥ h = 0 ♥ ♦♠♣rs♦♥s ♦ ♠♦s rtr③ ② ♦♠♥t♦♥s ♦ ♥ ♥ r♣♦rt ♥ ❬❪ ①st♥ ♥ stt② ♥②ss ♦ t qr♠stts ♥ ♣r♦r♠ ② t♥ p ♥ t rt♦ θ = m/(cb) s rt♦♥ ♣r♠trs t ♦♥sr ♠♦s ♠t t s♠ qr♠ stts ♥ s♦ ts♠ qtt ♦rs rr♥ t ♦ stt② ♣r♦♣rts t ♦rr♥♦ tr♦♥ ② ♥ t ♦♥sq♥t ♦ rt♦♥ ♦s② t stt②r♥s ♠② r② r♥t t t s♠ s ♦ t ♦♦♦ ♣r♠trsrx b m c

s♦ t ♠♦ ♥ t r♥t ♣♣r ❬❪ t ♥ t ♦ rt♦♥ t②♣g(s) = g0(s − ǫ)(1 − s)/(ǫ0 + s) t ǫ0 > 0 ♦ t ♥ ♥rtr♦♣ ♥t♦♥ f(s) = s ♦t❱♦trr ♥trt♦♥ ♥r♦s tr♦♥ ♦♦♣rt♦♥ ♥ ♠♦r♦r srt ♥ s♣rrt ♦♣ rt♦♥

4 Concluding remarks 77

♥ t st ♠♦s t t ♥ ♣r②♣♥♥t tr♦♣ ♥t♦♥♦♥② ♦♥ ♣♦ss ♦①st♥ stt t s ♥ ♦sr ♠② ♥r♦ srstt② ♥s ♦r ♥st♥ ss♠♥ ǫ0 ♥ t ♦ t s rt♦♥♣r♠tr t ♦①st♥ qr♠ stt ♥ st r♦♠ st t♦ ♥st ♥t♥ ♥ t♦ st ❬ ❪

rt♦♣♥♥t tr♦♣ ♥t♦♥ ①♣rss ♥ tr♠s ♦ t rt♦ x/(x + y) s s♥r ♥ t ♦r♥ s ♥ r♥t② ♥tr♦ ♥ ♠♦s t t ♦ qrt t②♣ ♣♥ ♠♦ ♥ ❬ ❪ ♥ ♦ rt♦♥ t②♣ ♦ t ♥ ❬❪ ♥ ts ♠♦s ♦♥② t♦ ♦①st♥ stts ♠② ♦♥♥ ♦♥ s ②s s ♣♦♥t ♥ t r♥t ♣r♠tr s♣s ♦♥sr ②ts t♦rs ♦♥♦♥s rt♦♥ ♣♦♥t ♦ ♦♠♥s♦♥t♦ s ♥tt ♥ t s ♦♥sr s ♥ ♦r♥s♥ ♥tr ♦ t ♦ ②♥♠s ❬❪t♦ ts ♠♦s t s♠ strtr t ♥②ss ♥ t ♥♠r s♠t♦♥s ♣r♦r♠ ♥ ❬ ❪ s♦ ♥ sst t s♥ ♦ st ♠t ②s ♥ ❬❪ ♦r ① st ♦ ♣r♠trs t ♦♦♥ ♠② ♣♣♥ t ①t♥t♦♥♦ ♦t ♣♦♣t♦♥s ♦①st♥ ♦r tr♠♥ ♣♦♣t♦♥ s③s ♦r t ♦st♦♥♦ ♦t ♣♦♣t♦♥ s♠ rsts r ♦t♥ ♥ ❬❪ ♣♦ss ①♣♥t♦♥ st ♦♦♥ ♥ ❬ ❪ ♣r♠tr ♥tr♦ ♥ t ♣r② r♦t ♥t♦♥ ss♦t t ǫ ♥ ♦r ♦r♠t♦♥ s s s rt♦♥ ♣r♠tr ♥ r ♥ t♥♠r s♠t♦♥s t s ♠♥t♥ ① ♥ ❬❪ rtr♠♦r ♥ ❬ ❪♠♦r ♦ ♣♥♦♠♥ r sr tr♦♥ ♦♦♣ ♥ rt♦♥ ♦ ♠t ②s rsts ♦t♥ ♥ ❬ ❪ sst tt t s ♦ ♣r♠trs♥tr♦ ♥ t ♣r② r♦t ♥t♦♥ s rt♦♥ ♣r♠tr ♣t ♥ ♥♣♥♦♠♥ r ♥♦t tt ② ♣r♦r♠♥ t ♥②ss t ♦tr rt♦♥♣r♠trs

♥ ♦r ♥②ss ① t ♠♥s♦♥ ♦♦♦ rts rx, b, m tr♠♥♥ t t♠ s ♦ t ②♥♠s ♥ t ♦♥rs♦♥ t♦r c r♦♠ ♣r② t♦♣rt♦r ♦♠ss r ♠t♣t t♦rs ♥ qt♦♥ ♦r♦r s♦t trs♦ ε s ♠♥t♥ ① ♣r♠trs h ♥ p ♣♣r♥ ♥ tr♠♥t ♦ t tr♦♣ ♥t♦♥ srs t ♣rt♦r♣r② ♥trt♦♥ ♥ ♦s♥ s rt♦♥ ♣r♠trs ♥ ♦r ♥②t st② ♦♥ tt ts②st♠ ♠ts t ♠♦st tr ♦①st♥ qr E∗1, E∗2, E∗3 ♣♥♥ ♦♥ ts ♦ ♣r♠trs h ♥ p s ♥ ♦①st♥ qr♠ E∗2 ♥①sts s ②s s ♣♦♥t qr♠ E∗3 s ②s ♥st ♥ tstrt ♥ p = p0γh ♦s ♥♦t ♥trst ts ①st♥ r♦♥ ♥ t (h, p) ♣♥

st② ♦ t stt② ♣r♦♣rts ♦ E∗1 ♥ t (h, p) ♣♥ s ♠♦r ♥trt ❲♦♥ ♦♣ rt♦♥ r p = p1(h) ♥ ♠t ②s ♠r st ♦r ♥st♣♥♥ ♦♥ t ♦ h ♥ s♣♣r ② ♦ rt♦♥ ♥♦♥ t

78 4 Concluding remarks

tr♦♥ ② t♥ t ♥♦♥ ♦①st♥ qr Eǫ, E1 ♦r t ♦♠♦♥② tr♦ t ♦①st♥ qr♠ E∗2 rs♣t② rtr♠♦r s♦ ♥♠r② tt r♦♥ ♥ t (h, p) ♣♥ r st ♥ ♥st ♠t②s ♦①st ♥ s♣♣r ② s♥♦ rt♦♥ ♦ ②s ♥② ♣r♦t ①st♥ ♦ ♦♥♦♥s rt♦♥ ♣♦♥t ♦r rx < r0 ❲ ♣♦♥t ♦ttt ♥ s♦♠ r♦♥s ♦ t ♣r♠tr s♣ t ♠♦ ♣rs♥ts ♠t♣ ttrt♦rs♦r♦r t ①t♥t♦♥ ♦ ♦t ♣♦♣t♦♥s s ②s ♣♦ss s♥ t ♦①t♥t♦♥ E0 s ②s ♦② s②♠♣t♦t② st ♥ ♦② st ♦r s♦♠♣r♠tr s

t s ♦rt ♥♦t♥ tt r♦♠ ♠t♠t ♣♦♥t t s②st♠ ♠ts r♥t ♣s ♣♦rtrts ♣♥♥ ♦♥ t ♣r♠tr s t ts ♥♦t ♥ssr②♠♣s s♥♥t r♥ ♥ t ♦♦ ♥tr♣rtt♦♥ ♦ t rsts ♦r ♥st♥ t r♥ t♥ st st② stt ♥ s♠ st ♠t ②r♦♥ t s♠ t ♥st st② stt s ♦ ♠♥♦r ♠♣♦rt♥ ♥ t♦ t♠t♠t st② ssrts tt t ②♥♠s ♥ ts t♦ ss r ♥♦t q♥t

st r r♠r ♦♥r♥s t ♠①♠ s♣ ♣r♦t♦♥ rts rx ♥ ry rt r0 ♥ ♥ ♦ rx s ♦♥ t ♣♥s ♦♥ t ♦♦♦♣r♠trs b m c ♥ ♦♥ t t②♣ ♦ t tr♦♣ ♥t♦♥ ♦r ♦ ts②st♠ s♦s r♥t trs ♣♥♥ ♦♥ t s♥ ♦ t r♥ rx − r0 st s ♥♣♥♥t ♦ tr t ♣rs♥ ♦r t s♥ ♦ ♥ t ♥ t♠♦ qt♦♥s s st♦♥s ♥ ❬❪ t s ♦rt ♥♦t♥ ttt r♥ ♥ t s②st♠ ♦r t♦ t s♥ ♦ rx−r0 ♦♠s r② ♠r♥ tr♦♣ ♥t♦♥ s♥r ♥ t ♦r♥ s s ❬❪ rtr♠♦r t ♥ ♦sr s tt t ♦♥t②♣ tr♦♣ ♥t♦♥ r0 = ry t ♥ t②♣ r0 > ry t t s♠ ♦♦♦ ♣r♠trs m, c, b s rx < ry t♥ rx < r0 t ♦♥t②♣ tr♦♣ ♥t♦♥ ♦♥trr② t t tr♦♣ ♥t♦♥ ♥ rx > ry ♠t tr rx < r0 ♦r rx > r0 ❲r tt ♥ ♥r ♥ ♦♦ t t♠ ♥ ♦r r♣r♦t♦♥ ♥ r♦t♦ t ♥s ♦ ♣♦♣t♦♥ s ♥rs♥ t t tr♦♣ ❬ ❪ t ♥②♥t ♥ s♦♠ ♣rt♦r♣r② s②st♠s s s s♦♠ r♥ s②st♠s ♥② stt♦♥♠② ♦r

♠♠r③♥ t ♥r ♣♣r♦ t♦ ♣rt♦r♣r② s②st♠s s ♥ ts ♦r♥ t ♠t♠t ♦r♠t♦♥ ♦ ♠♦ ♥t♦♥s s ♥s♣ ①♣t ♦rs♦♠ ♥r qtt ♣r♦♣rts s ♣t ♥ ♥ t ♦r ♦♠♣①t② ♦t rt♦♥ strtr ♦ t ♠♦ ♦r♥ s♦ t♦ r♥t ♦rs ❬ ❪ ♦stt♦♥r② ♥ ♦♣ rt♦♥s ♥ tr♠♥ ♥②t② ♥ ts ♥rr♠♦r s♦ t ♣rs♥ ♦ t ♦♥♦♥s ♥ t s♣ ♦♠♥s♦♥t♦

4 Concluding remarks 79

rt♦♥ ♣♦♥ts s ♥♣♥♥t ♦ t ♣rtr ①♣rss♦♥ ♦ t ♠♦ ♥t♦♥s ♥①t st♣ ♦♥ssts ♥ ♣r♦r♠♥ ♥ ♥♦♦s ♥stt♦♥ ♦r ♥♦♥♦ ♥♦♠♥s♦♥t♦ rt♦♥s s s t ♥r③ ♦♣ ♥ t rt♦♥s♦ ②s ♠tt ② ts ♥r ♠♦ t ♦t ♠♦r ♥♥ ♥♦♣② ♦ t♦♥ ♠♥ts t♦ tr② t♦ ①♣♥ ♣♥♦♠♥ st ♥r ♥r ♦s②st♠s

Appendix

A Details on some parameters

t

µ(s) =sf ′(s)

f(s), s > 0.

r♦♠ ♣r♦♣rts ♥ t ♦♦s tt

µ(0) = 1, 0 < µ(s) < 1, lims→+∞

µ(s) = 0.

rst t♦ ♣r♦♣rts ♦ µ(s) r s② r ♦ ♣r♦ t ♠t ♥ ts ♦♥sr t ♥tt②

∫ s

s0

µ(a)

ada =

∫ s

s0

f ′(a)

f(a)da,

r s0 > 0 s ① ♥ s > s0 r♦♠ tt tr ①sts a(s)s0 < a(s) < s s tt

µ (a(s)) =log f(s)/f(s0)

log s/s0.

r♦♠ t♥ ♥t♦ ♦♥t t ♦♦s tt

lims→+∞

µ (a(s)) = 0, lims→+∞

∂µ (a(s))

∂s= lim

s→+∞

µ(s)− µ (a(s))

s log s/s0= 0,

♠♣② t ♠t ♥ ♦r r♦♠ ♣r♦♣rts r ♥♦t t♦s♦ t ♠♦♥♦t♦♥t② ♦♥t♦♥ µ′(s) < 0 ♦s ♦r t ♥t♦♥s f(s)♥r② s ♥ t ♣♣t♦♥s

82 4 Concluding remarks

②♥♠ ♦rs ♦ ♣rt♦r♣r② s②st♠s ♠② r♥t ♣♥♥♦♥ t s♥ ♦ t r♥ rx − r0 ♥ rx − ry r r0 s ♥ ♥ ♥ry ♥ ❬ ❪ ❲t rs♣t t♦ ts ss ♣♦♥t ♦t tt t ♦♥t②♣ tr♦♣ ♥t♦♥ t ♣r♠trs p0 ♥ µ0 r ♥ ②

p0 =α

1− α, µ0 = 1− α,

s♦ tt

r0 =m(1− α)

α= ry.

trs t ♥ tr♦♣ ♥t♦♥ ♦t♥

p0 = − ln(1− α), µ0 =−(1− α) ln(1− α)

α,

s♦ tt

r0 = ry−α ln(1− α)

α + (1− α) ln(1− α)> ry.

s s st ♦ ♦♦♦ ♣r♠trs t♦ s ♥ ♥♠r s♠t♦♥s t♥ t s r♦♠ ♦♥ t ❬❪ r♣♦rt ♥ t ♦♦♥ t

Pr♠tr rx b m c ǫ(d−1) (d−1) (d−1)

❱ 0.11 0.88 0.19 0.39

s t ♥ st♠t ♥ ❬❪ ② rr t♦ ♥ r♥ s②st♠ sr②♥ ♦♦ ♦♥tr♦ ①♣r♠♥ts t ♣②t♦♣♦s ♠t tr♥②s rt♥ ts ♦♦ ♦♥tr♦ ♥t t ♣rt♦r ♠t P②t♦ss ♣rs♠s ❬❪❲ s t♦ ♣♦♥t ♦t tt ② s♥ ts ♦ ♦ ♣r♠trs s♦♠ rt♦♥s rtr③♥ t ♦♦ s②st♠ r sts ♥qt② α = m/(cb) = 0.55 < 1 s ♥ t♥ ♥♦♥ tr ②♥♠s ♥ ♦♥ ♠①♠♠ r♦t rt rx ♦ t ♣r② s ss t♥ t ♦♥ ♦ t ♣rt♦r

rx = 0.11 d−1 < ry = cb−m = 0.15 d−1.

❲t ♦♥t②♣ tr♦♣ ♥t♦♥ tt r0 = ry t ♦♦s tt ♥t (h, p) ♣♥ t t♦ strt ♥s p/p0 = γh ♥ p/p0 = βh+ 1/ξ0 ♥trst❲ t♥ rx = 0.16 d−1 ♥ s♦♠ ♥♠r ①♣r♠♥ts t♦ s♠t t srx > ry

B The cusp bifurcation point 83

B The cusp bifurcation point

♥ t♦♥ ♦ ♣tr ♦♥ s♣ rt♦♥ ♣♦♥t ♥ B0 ♥ t (h, p)♣♥ rr♥ t ♦①st♥ qr♠ stts r ♥t t♦ r② t ♦♥t♦♥s ♦r s♣ rt♦♥ ♣♦♥t

rst ♦ t♦ r t ♠♥s♦♥ s②st♠ ♦♥ t ♥tr ♠♥♦ ♥ t♥ r② t ♦♥t♦♥s ♦r s♣ rt♦♥ ♣♦♥t ♥ ♦r ♥ ♦♥sr t ♦♦♥

x = F (x, h, p) =1

x+ hβg(x)− p

p0.

ts qr♠ ♣♦♥ts r t x−♦♦r♥ts ♦ t ♦①st♥ qr ♦ t y−♦♦r♥ts ♥ ♦♥ t♥s t♦

y∗ = βx∗g(x∗).

♦r♦r ts qr♠ sr s t s♠ ♦ t ♥r s②st♠

• ❲ tt F (η0, h0, p0Φ0) = 0.• ❲ ♥t t♦ ♣r♦ tt Fx(η0, h0, p0Φ0) = 0.

Fx = − 1

x2+ hβg′(x) =

1

x2(−1 + hβx2g′(x)

)

=1

x2(−1 + hψ(x)) .

Fx(η0, h0, p0Φ0) =1

η20(−1 + h0ψ(η0)) = 0.

• ❲ ♥t t♦ ♣r♦ tt Fxx(η0, h0, p0Φ0) = 0.

Fxx = − 2

x3(−1 + hψ(x)) +

1

x2hψ′(x).

Fxx(η0, h0, p0Φ0) = − 2

η30(−1 + h0ψ(η0)) +

1

η20h0ψ

′(η0) = 0.

• ❲ ♥t t♦ ♣r♦ tt Fxxx(η0, h0, p0Φ0) 6= 0.

84 4 Concluding remarks

Fxxx =6

x4(−1 + hψ(x))− 4

x3hψ′(x) +

1

x2hψ′′(x).

Fxxx(η0, h0, p0Φ0) =6

η40(−1 + h0ψ(η0))−

4

η30h0ψ

′(η0) +1

η20h0ψ

′′(η0)

=1

η0h0ψ

′′(η0) < 0,

s♥ ψ s ♦♥② ♦♥ ♠①♠♠ ♥ η0• ❲ ♥t t♦ ♣r♦ tt (FhFxp − FpFxh)(η0, h0, p0Φ0) 6= 0.

Fh = βg(x),

Fxp = 0,

Fp = − 1

p0,

Fxh = βg′(x).

(FhFxp − FpFxh)(η0, h0, p0Φ0) = βg′(η0)1

p0> 0.

s t ♦♥t♦♥s r sts

C List of main symbols

t t♠ dx rt♦ t♥ ♣r② ♦♠ss ♥ ♠①♠♠ ♦ ♣r② ♣♦♣t♦♥ s③y rt♦ t♥ ♣rt♦r ♦♠ss ♥ ♠①♠♠ ♦ ♣r② ♣♦♣t♦♥ s③rx ♠①♠♠ s♣ rt ♦ ♣r② r♦t d−1b ♠①♠♠ s♣ rt ♦ ♣r② ♦♥s♠♣t♦♥ d−1m s♣ rt ♦ ♣rt♦r ♠♦rtt② d−1c ♦♥rs♦♥ ♥② r♦♠ ♣r② t♦ ♣rt♦r ♦♠ssǫ rt♦ t♥ ♠♥♠♠ ♥ ♠①♠♠ ♦ ♣r② ♣♦♣t♦♥ s③

g(x) ♠♥s♦♥ ♥t♦♥ s♣②♥ t t②♣ ♦ ♣r② r♦t

f

(px

1 + hy

)

♠♥s♦♥ ♥t♦♥ rs♣♦♥s ♦ ♣rt♦r t♦ ♣r② ♥♥

p ♣rt♦♥ ♥②h ♠sr ♦ t ♣rt♦r ♥trr♥ r♥ t ♣rt♦♥ ♣r♦ss

α m/cb

C List of main symbols 85

β rxc/m

ry cb−m (d−1)

p0 f−1(α)

µ(s) sf ′(s)/f(s)

µ0 µ(p0)

γ cµ0/(1− µ0)

r0 mµ0/(1− µ0) (d−1)

Λ(h, p) 1− µ0 − cµ0hp0/p

φ(x, h) 1/x+ hβg(x)

ψ(x) βx2g′(x)

ξ0 ♥q s♦t♦♥ ♦ g′(x) = 0

φ(ξ0, h) βh+ 1/ξ0

η0 ♥q s♦t♦♥ ♦ ψ′(x) = 0

h0 1/ψ(η0)

h1 1/ψ(ǫ)

hǫ ♥q s♦t♦♥ ♦ φ1(h) = 1/ǫ

η1 ♥q s♦t♦♥ ♦ ψ(x) = 1/h1 ♥ t r♥ (η0, ξ0)

ξ1(h), ξ2(h) s♦t♦♥s ♦ ψ(x) = 1/h ξ1(h) ≥ ξ2(h)

φ0 φ(η0, h0)

B0 s♣ ♣♦♥t (h0, p0φ0)

φj(h) φ(ξj(h), h), j = 1, 2

p1(h) ♦♣ rt♦♥ ♦r p

p2(h) ♦ rt♦♥ ♦r p

hR 1/(ξ0(γ − β)) ♥q s♦t♦♥ t♦ γh = βh+ 1/ξ0

hS ♥q s♦t♦♥ ♦ φ1(h) = γhhU ♥q s♦t♦♥ ♦ p2(h) = p0φ2(h)hBT ♥q s♦t♦♥ ♦ p1(h) = p0φ1(h)

Prt

♦t rt♦♥s♦♥ ♣rt♦r♣r②

s②st♠s ♥♦♥ t ♦♥t②♣ ♥

t ♥t♦♥♥s ♥t♦♥

rs♣♦♥ss

5

Introduction

s Prt ♦ss ♦♥ t st② ♦ t♦ ♣rt♦r♣r② ♠♦s t s♦♥ ttst② ♥ st ♠t t♦ ss t②♣s ♦ ♥t♦♥ rs♣♦♥ss ♥ t rt♦♥♣rt ♥ ♣rs♥t r♦sss♦♥ tr♠ ♥ t ♣rt♦r qt♦♥ ❲ s♦ ♦♦ ♦r♦♥t♦♥s ♦♥ t ♣r♠trs s t♦ r♥ ♥stt②

♦♠♣① ♥t♦♥ rs♣♦♥ss r qt s ♥ ♣rt♦r♣r② ♠♦s ❬ ❪ ♦r ①♠♣ t ♦♥t②♣ ♥t♦♥ rs♣♦♥s ❬❪ s s ♦♥ t tt ♣rt♦rs t ♠t ♠♦♥t ♦ ♣r② ♥ t s ♥ ♣r② r♥♥t ♥♦t♥ t N := N(t) t ♣r② ♦♠ss ♥ t P := P (t) t♣rt♦r ♦♠ss ts t②♣ ♦ ♥t♦♥ rs♣♦♥s s t♦ t ♦♦♥ s②st♠ ♦t♦ s

N = r0g(N)− bNP

1 + kN,

P =cbNP

1 + kN− µP,

t r0, b, c, k, µ > 0 t ♥t♦♥ g srs t ♣r② r♦t ♥ ♥ tr♥r tt s g(N) = N ♦r ♥♦ ♦st ♣rt s♦ tt g(N) = (1− ηN)N tη ≥ 0 ❬❪ ♦t tt ♥ g(N) = N ♥ k = 0 ♦♥ r♦rs t ♦t❱♦trr♣rt♦r♣r② ♠♦

♦♥ s♦ ss t♦ t ♥t♦ ♦♥t t ♦♠♣tt♦♥ t♥ ♣rt♦rs ♥t② tr② t♦ t ♣r② ♦♥ ♥ s t st② ♠♦r ♦♠♣① ♥t♦♥♥s♥t♦♥ rs♣♦♥s ❬ ❪

1 ♦st ♦ t ♦♥t♥ts ♦ ts Prt ♥ ♥ t ♣♣r ♥ ♣r♣rt♦♥ ♦♥♦rt♦ stts ♦rs♥ ♦t rt♦♥s♦♥ s②st♠s ♥♦♥ t ♦♥t②♣ ♥ t ♥t♦♥♥s♥t♦♥ rs♣♦♥ss ♦r ♣rt♦r♣r② ♠♦s

90 5 Introduction

N = r0g(N)− bNP

1 + kN + hP,

P =cbNP

1 + kN + hP− µP,

t r0, b, c, k, µ > 0 ♥ s♦ h > 0♥ ♠♣♦rt♥t ♣♦♥t ♥ t sq t ♦srt♦♥ tt ♣rt♦r♣r②

rt♦♥s♦♥ ♠♦s t t ♥t♦♥♥s ♥t♦♥ rs♣♦♥s ♥ trt♦♥ ♣rt r ♥♦♥ t♦ ♣r♦ ♣ttr♥s ♦♠♥ ♦t ♦ r♥ ♥stt② ♥s♦♥ tr♠s t st rts ♥♦t ② dN , dP r t♦ t rt♦♥tr♠s ❬ ❪ ♥ ts s t ♥sts N, P s♦ ♣♥s ♦♥ s♣ ♦♦r♥ts♥ t s②st♠ rts

∂tN − dN∆xN = r0g(N)− bNP

1 + kN + hP,

∂tP − dP∆xP =cbNP

1 + kN + hP− µP.

♥ t ♦♣♣♦st ♥♦ r♥♣ttr♥s r ♥♦♥ t♦ ♣♣r ♥ t s ♦ rt♦♥s♦♥ ♣rt♦r♣r② ♠♦ t ♦♥t②♣ ♥t♦♥ rs♣♦♥s ♥ st♥r s♦♥ tr♠s ❬❪ ①♣t ♥ rr ②♥♠s r ♦♥sr ♦r ①♠♣♥ ♦♥ s qrt ♥tr♣rt♦r ♥trt♦♥ ♦r t♥ tr♠ ❬ ❪ ♦r ♥ ♥st②♣♥♥t ♣rt♦r ♠♦rtt② ❬❪ t s ♦ ♥♦♥r♥ ♣ttr♥s tt♠♥s dN = dP s st ♥ ❬ ❪

♥ ♥trst♥ ♠♦♥ ss ♦♥ssts ♥ ♥♥ s♠♣ ♥ s♦♠t rst ♠r♦s♦♣ ♠♦ ♥ tr♠s ♦ t♠ ss ♥ s♦♠ ♠t s tst ♦r♠② t♦ s②st♠s ♦r ♠r♦s♦♣ ♠♦ s s♥ ②t③ ♥ ♠♥♥ ❬❪ ♦r t ♦♥t②♣ ♥t♦♥ rs♣♦♥s ♥ ② rt③♥ ②♥r ❬❪ s♠♥ ♥ ♦r ❬❪ ♦r t ♥t♦♥♥s ♦♥t③ ♥ ♠♥♥ ♣r♦♣♦s s②st♠ ♦ tr s ♥ t ♣rt♦rs r ♥ t♦ sss sr♥ ♥ ♥♥ ♣rt♦rs t ♥trt♦♥ t♥ ♣rt♦rs ♥ ♣r② s trt ♥ qt s♠♣ ② ♦t❱♦trr tr♠sr ♦♥sr Prt♦rs r sr♥ ♦r ♣r② ♦♠ ♥♥ t rt♣r♦♣♦rt♦♥ t♦ t ♥♠r ♦ ♣r② ♥ ♦♠ t♦ t sr♥ stt t ♦♥st♥t rt ♥② ♥♥ ♣rt♦rs ♦♥trt t♦ t r♣r♦t♦♥ ♥ rst♦ sr♥ ♣rt♦r t ♠♦rtt② rt ♥ s♥ ♦ ♣r② s ♦♥st♥t ♥q ♦r t t♦ sss sr♥♥♥ st s s♣♣♦s t♦ ♣♣♥ ♦♥ ♠ str s t♥ t r♣r♦t♦♥ ♥ ♠♦rtt② ♣r♦sss ♦rrs♣♦♥♥

5 Introduction 91

♣r♠tr ♥ t s②st♠ ♦ s s tr♦r 1/ε ♥ t s②st♠ rts ♦rs♦♠ r0 α γ Γ µ ε > 0

N ε = r0Nε − αN εpεs,

pεs =1

ε(−αN εpεs + γpεh) + Γpεh − µpεs,

pεh =1

ε(αN εpεs − γpεh)− µpεh.

t s s♦♥ tt ♥ t ♦r♠ ♠t ε → 0 N ε → N ♥ pεs + pεh → P r N, Psts② t b = α, k = α/γ, c = Γ/γ ♥ g(N) = N ❬❪

s♠ ♣r♦r s ♣♣ tr ② rt③ ♥ ②♥r ❬❪ t② ♥♦t ♦♥② t ♣rt♦r ♣♦♣t♦♥ ♥t♦ srrs ♥ ♥rs t s♦ strtrt ♣r② ♣♦♣t♦♥ ♥t♦ t♦ sss t ss ♦ t ♣r② t②♣② ♦r♥♥ ♣r♦♥ t♦ ♣rt♦♥ ♥ t ss ♦ t♦s ♣r② ♥s ♦ ♦♥ r ♥ ♥♥♦t t ② ♣rt♦rs ♥ ts ② t② r t ♥t♦♥♥s ♥t♦♥ rs♣♦♥s ♥ tr♠s ♦ ♠♥s♠s t t ♥ ♦♥t s ♥trr♥ t♥ ♣rt♦rs s♦ s♠♥ ♥ ♦r ❬❪ strt♥r♦♠ r♥t ♦r♠♥s♦♥ ♠♦ s rt♦♥ s♠s ♥ qsst②stt ss♠♣t♦♥s ♦t♥♥ s②st♠ ♦ t♦ ♦r♥r② r♥t qt♦♥s ♦rt② t♦ s♠♣② ♦♠♣t qrt ①♣rss♦♥ t Pé ♣♣r♦①♠t♦♥t♦ r♦r t st♥r ♦r♠ ♦ t ♥t♦♥♥s ♥t♦♥ rs♣♦♥s ♥♦t ss t♦ r♥t t♠ ss r ①♣♦t

♥ ts Prt r ♥trst ♥ t ♥tr♦t♦♥ ♦ s♦♥ ♣r♦sss ♥ ts②♠♣t♦t ♣r♦♠ ♥♦t♥ ② d1, d2, d3 t s♦♥ rts ♦ ♣r② sr♥♣rt♦rs ♥ ♥♥ ♣rt♦rs rs♣t② ♥ t♥ ♥t♦ ♦♥t ♣♦ss♦st tr♠ ♥ t ♦t♦♥ ♦ ♣r②s ♦t♥ t ♦♦♥ rt♦♥s♦♥s②st♠

∂tNε − d1∆xN

ε = r0 (1− ηN ε)N ε − αN εpεs,

∂tpεs − d2∆xp

εs =

1

ε(−αN εpεs + γ pεh) + Γpεh − µpεs,

∂tpεh − d3∆xp

εh =

1

ε(αN εpεs − γ pεh)− µpεh.

♦t tt s②st♠t② ①♣t t s♦♥ rt d3 ♦ ♥♥ ♣rt♦rs t♦ s♠r t♥ t t ♦♥ ♦ sr♥ ♣rt♦rs d2 ♦r♠ ♠t ♦ ts s②st♠♥ ε→ 0 s t st ♦ t♦ r♦ss rt♦♥s♦♥ qt♦♥s

92 5 Introduction

∂tN − d1∆xN = r0(1− ηN)N − γαN

αN + γP,

∂tP −∆x

(d2γ + d3αN

αN + γP

)

= ΓαN

αN + γP − µP,

♥ t rt♦♥ tr♠s r ♥t ♣ t♦ t ♥ ♦ ♥♠ ♦ t ♦♥st♥t♣r♠trs t♦ t♦s ♦ t ♥ t s♦♥ tr♠ rt t♦ ♣rt♦rss ♠ ♠♦r ♦♠♣t t♥ ♦♥st♥t t♠s ♣♥ ♦ P tr♠s d∆xP s②st♠t② ♥r s tr♠s ♥ t sq r♦ss s♦♥ rr t♦ tr♠s ∆x(f(N,P )P ) r f s s♠♦♦t ♥♦♥♦♥st♥t ♥t♦♥s♦ N, P s ♥ t s♦♥ qt♦♥ ♦ ♥ st♦♥ stt r♦r♦st♦r♠ s♦♥ tt ♦♥r♥ ♦ s♦t♦♥s t♦ s②st♠ t♦rs s♦t♦♥s t♦s②st♠ ♥ ♦s ♥ st ♥t♦♥ s♣s r ♥tr♦

s♠ ♣r♦r ♥ ♣♣ ♥ t s ♦ ♥t♦♥♥s ♥t♦♥ rs♣♦♥s tt s s②st♠s ♦ s ♦s t♦ rst ♥tr♦ s②st♠ ♦ tr s ♠♦♥ t ♥trt♦♥ t♥ ♣r②s ♥♥ ♥ sr♥♣rt♦rs s ♥ t ♥ s♦ t ♥t♦ ♦♥t t ♦♠♣tt♦♥ ♠♦♥♣rt♦rs ♥ t② ♦♦ ♦r ♣r② s s ♦♥ t♥s t♦ t ♥tr♦t♦♥ ♦ t♥♦♠♥t♦r 1 + ξ P ♦r s♦♠ ξ > 0 ♥ t ♥trt♦♥ tr♠ t♥ ♣rt♦rs ♥♣r② s②st♠ rts t♥ s ♦♦s

N ε = r0(1− ηN ε)N ε − αN εpεs1 + ξpεs

,

pεs =1

ε

(

− αN εpεs1 + ξpεs

+ γpεh

)

+ Γpεh − µpεs,

pεh =1

ε

(αN εpεs1 + ξpεs

− γpεh

)

− µpεh.

ts ♦r♠ ♠t ♥ ε→ 0 s t♥ s②st♠ ♦s t♦ ♥ s♦ ♦t♥ ♥ ❬❪strt♥ r♦♠ ♠♦r ♦♠♣① s②st♠ ♦ ♦r s

rt♦♥s♦♥ s②st♠ ♥ d1, d2, d3 r st s rts ♦r ♣r② sr♥ ♥ ♥♥ ♣rt♦rs rts

5 Introduction 93

∂tNε − d1∆xN

ε = r0 (1− ηN ε)N ε − αN εpεs1 + ξpεs

,

∂tpεs − d2∆xp

εs =

1

ε

(

− αN εpεs1 + ξpεs

+ γpεh

)

+ Γpεh − µpεs,

∂tpεh − d3∆xp

εh = −1

ε

(

− αN εpεs1 + ξpεs

+ γpεh

)

− µpεh.

❲ ♣rs♥t ♥ t♦♥ r♦r♦s rst ♦ ♦♥r♥ ♦ t s♦t♦♥s ♦ tss②st♠ t♦rs t s♦t♦♥ ♦ rt♦♥r♦ss s♦♥ s②st♠ r t rt♦♥♣rt s ♦s t♦ s s②st♠ rts

∂N

∂t− d1∆xN = r0 (1− ηN)N − γ

2αNP

γ + αN + γξP +√

(γ + αN − γξP )2 + 4γ2ξP,

∂P

∂t−∆x (f(N,P )P ) = Γ

2αNP

γ + αN + γξP +√

(γ + αN − γξP )2 + 4γ2ξP− µP,

r

f(N,P ) = d22γ

γ + αN − γξP +√

(γ + αN − γξP )2 + 4γ2ξP

+ d32αN

γ + αN + γξP +√

(γ + αN − γξP )2 + 4γ2ξP.

♣r♦♦ ♦ t♦s t♦r♠s r s ♦♥ st♠ts ♦♠♥ ♦t ♦ t♦ sss♦ ♠t♦s ♥ ♦♥ ♥ s t t② ♠♠s s ♦r rt♦♥s♦♥s②st♠s ② Prr ♥ ♠tt ❬❪ ♦r ♣rs② s rs♦♥ ♦ t♦s♠♠s ♦♥ t♦ r♦r Lp rrt② ♦r p > 2 ♦r t s♦t♦♥s ♦ s s②st♠s❬ ❪ ♥ t ♦tr ♥ s♦ s ♥tr♦♣② ♥t♦♥s r str♦♥②r♠♥s♥t ♦ t♦s s ♥ ♦rs ♥ ♠r♦s♦♣ ♠♦s ♦r t ssr♠♦t♦ s②st♠ ❬❪ r st ❬❪

s ♣r♦♦s ♥ s♦ ♦♠♣r t♦ r♥t rsts ♥ r♦ss rt♦♥s♦♥ s②st♠s r ♦t♥ s ♠ts ♦ st♥r rt♦♥s♦♥ s②st♠s t♠♦r qt♦♥s ♥ t ♦♥t①t ♦ ♠str② ♦r ♦♦② ❬ ❪

s r② ♠♥t♦♥ ♥t♦♥♥s ♥t♦♥ rs♣♦♥s s ♣rtr② ♥trst♥ s♥ t s ♥♦♥ tt ♣rt♦r♣♥♥t ♥t♦♥ rs♣♦♥ss

94 5 Introduction

t♦ ♣ttr♥s ♥ ♥rs♦♥ tr♠s r t♦ t rt♦♥ ♣rt ♦r①♠♣ ♥ t ♦♦♥ s②st♠

∂N

∂t− d1∆xN = r0 (1− ηN)N − γ

2αNP

γ + αN + γξP +√

(γ + αN − γξP )2 + 4γ2ξP,

∂P

∂t−D∆xP = Γ

2αNP

γ + αN + γξP +√

(γ + αN − γξP )2 + 4γ2ξP− µP.

♦r ♦♥ ♦♥sr tt t ♥t♦♥♥s ♥t♦♥ rs♣♦♥s s♦♠♥ ♦t ♦ ♥ s②♠♣t♦ts ♥ ε→ 0 ♦ ♦♥ s♦ rtr st② t ♣♦ss ♣♣r♥ ♦ ♣ttr♥s strt♥ r♦♠ s②st♠ t r♦ss s♦♥ tr♠s ♦t tt ♦r ♦♥t②♣ ♥t♦♥ rs♣♦♥s ♥♦ ♣ttr♥s s♠ t♦ ♣♣r ♥t t r♦ss s♦♥ ♠♦ ♥r t ♦♦② rs♦♥ ss♠♣t♦♥d3 < d2

♥ ♣tr st② t r♥ ♥stt② r♦♥s ss♦t t♦ s②st♠ ♥ ♥ ♦rr t♦ ♦ s♦ rst ♣r♦r♠ ♥ ♠♥s♦♥③t♦♥ ♥st♦ ♣ ♦♥② s♠ ♥♠r ♦ ♣r♠trs ♥ t qt♦♥s

♥ ♠ ①♣t t ♦♥t♦♥ ♦♥ t ♣r♠trs t♦ t ①st♥♦ ♥ ♦♠♦♥♦s ♦①st♥ qr♠ ♦r ♥ ❲ s♦ ♣r♦r♠ stt② ♥②ss ♦ ts qr♠ ♥ t ①sts t t s s s♦ tt t r♥ ♥stt② r♦♥ ♥ tr♠s ♦ ♣r♠trs s ♥♦♥♠♣t② s①♣t ♦r ♦t s②st♠s ♥ ♥ ♥② ♦♠♣r t s③ ♦ tsr♦♥s ♠♥ ♣♦♥t s t t tt t r♥ ♥stt② r♦♥ ss♦t t♦s②st♠ s ②s strt② ♥ ♥ t r♥ ♥stt② r♦♥ ♦ s②st♠

s ♦♥sq♥ t s ♦ rt♦♥s♦♥ s②st♠s ♦r ♣rt♦r♣r② ♥trt♦♥s ♦ ♥t♦♥♥s ♥ st♥r s♦♥ s s♠♣② t♦t rt♦♥ tr♠s ❬❪ ♠② t♦ ♥ ♦rst♠t♦♥ ♦ t ♣♦sst② ♦ ♣♣r♥ ♦ ♣ttr♥s ♥ t s ♥ t ♥t♦♥♥s ♥t♦♥ rs♣♦♥s s ♦♥sq♥ ♦ t ♥trt♦♥s t♥ sr♥ ♥ ♥♥ ♣rt♦rs

t s ♦rt t♦ ♠♥t♦♥ tt ♥ ♠♥② ♥st♥s t ♥tr♦t♦♥ ♦ r♦sss♦♥tr♠s ♥st ♦ st♥r ♥r s♦♥ tr♠s s ①t② t♦ t ♦♣♣♦st rsttt s t ♥rs ♦ t st ♦ ♣r♠tr s ♥ ♣ttr♥s ♦♣ ❬ ❪

6

A justification of classical functional responses

♥ ts ♣tr ♦♥sr ♣rt♦r♣r② s②st♠ ♦ tr rt♦♥s♦♥ qt♦♥s ♥♦r♣♦rt♥ t ②♥♠s ♦ ♥♥ ♥ sr♥ ♣rt♦rs ♥ s♦ ttts s♦t♦♥s ♦♥r ♥ s♠ ♣r♠tr t♥s t♦ 0 t♦rs t s♦t♦♥s ♦ rt♦♥r♦ss s♦♥ ♣rt♦r♣r② s②st♠ ♥♦♥ r♦sss♦♥ tr♠ ♥ ♦♥t②♣ ♦r ♥t♦♥♥s ♥t♦♥ rs♣♦♥s ❲t rs♣t t♦t s②st♠s ♣rs♥t ♥ t ♥tr♦t♦♥ s②st♠t② s ♥ t sq ①♣t t♥ ♦r t♦♥ t ♣r♠tr k := 1/η ♥ ♦rr t♦ st t♦ t trt♦♥♦♥♣t ♦ rr②♥ ♣t② s ♠♥s ♦ ♦rs tt ♦r ♦♥② ♥ t s♥ η > 0 ♦rrs♣♦♥s t♦ ♦st ♣r② r♦t

6.1 The derivation of the Holling-type II functional

response

❲ ♦♥sr rt♦♥s♦♥ ♣rt♦r♣r② ♠♦ ♥ st♥s t♦t②♣s ♦ ♣rt♦rs t♦s sr♥ ♦r ♣r② ♥ t♦s s② ♥♥ ♣r② trr t ♥trt♦♥ t♥ ♣rt♦rs ♥ ♣r② s trt ♥ qt s♠♣② ♦t❱♦trr t②♣ ♥trt♦♥ s ♦♥sr ❲ s♣♣♦s tt t ♣rt♦♥♣r♦ss ♣♣♥s ♥ r♦♥ Ω ⊂ RN , (N = 1, 2, 3) ss♠ t♦ s♠♦♦t C2♦♥ ♥ ♦♥♥t r t ♥s r ♠♦♥ ❲ ♥♦t t nε :=nε(x, t) ≥ 0 t ♥st② ♦ ♣r② t pεs := ps(x, t) ≥ 0 ♥ pεh := ph(x, t) ≥ 0t ♥st② ♦ sr♥ ♥ ♥♥ ♣rt♦rs rs♣t② r t ∈ R, x ∈ Ω❲ ss♠ tt s♣s s ts ♦♥ s♦♥ ♦t② ♥ s t♥ ♥♦♥rt r♦♥ s♣♣♦s t♦ t rst ❲ s s♦ ss♠ tt ♥s r♦♥♥ ♥ t r♦♥ Ω s♦ tt t ① ♦ ♥st② ♦ s♣s t t ♦♥r②∂Ω s ③r♦ s s t ♦♠♦♥♦s ♠♥♥ ♦♥r② ♦♥t♦♥s ♥♦ ① ♦

96 6 A justification of classical functional responses

♥st② ♦ s♣s t t ♦♥r②

n(x) · ∇xnε = 0, n(x) · ∇xp

εs = 0, n(x) · ∇xp

εh = 0,

r n(x) s t ①tr♦r ♥t ♥♦r♠ t♦r t♦ ∂Ω t ♣♦♥t xPrt♦rs r sr♥ ♦r ♣r②s ♦♠ ♥♥ t rt ♣r♦♣♦rt♦♥

t♦ t ♥♠r ♦ ♣r②s ♥ ♦♠ t♦ t sr♥ stt t ♦♥st♥t rt♥② ♥♥ ♣rt♦rs ♦♥trt t♦ t r♣r♦t♦♥ ♥ rs t♦ sr♥♣rt♦r t ♠♦rtt② rt s ♦♥st♥t ♥ q ♦r t t♦ sss sr♥♥♥ st s s♣♣♦s t♦ ♣♣♥ ♦♥ ♠ str s t♥ tr♣r♦t♦♥ ♥ ♠♦rtt② ♣r♦sss ♥ t♥ tr r r ss ♣rt♦rs t♥♣r② ♦rrs♣♦♥♥ ♣r♠tr ♥ t s②st♠ ♦ s s tr♦r 1/ε♥ t s②st♠ rts ♦r s♦♠ r0 α γ Γ µ ε > 0 d1, d2, d3

∂tNε − d1∆xN

ε = r0Nε

(

1− N ε

k

)

− αN εpεs

∂tpεs − d2∆xp

εs =

1

ε

(− αN εpεs + γpεh

)+ Γpεh − µpεs,

∂tpεh − d3∆xp

εh =

1

ε

(αN εpεs − γpεh

)− µpεh,

t♦tr t ♥t ♦♥t♦♥s ♥ ♦♠♦♥♦s ♠♥♥ ♦♥r② ♦♥t♦♥s♥ t♦♥ r♦r♦s② ♣r♦ tt ♥ t ♠t ♥ ε → 0 t s♦t♦♥N ε, pεs, p

εh ♦ ts s②st♠ ♦♥rs ♥ st t♦♣♦♦② t♦rs N ≥ 0, ps ≥

0, ph ≥ 0 s tt

ph =αNpsγ

.

❲ ♥♦ s t♦ rt t ♠t♥ s②st♠

∂tN − d1∆xN = r0N

(

1− N

k

)

− αNps

∂t(ps + ph)−∆x (d2ps + d3ph) = Γph − µ(ps + ph),

♥ tr♠s ♦ t ♥st② ♦ t ♣r② N ♥ t♦tt② ♦ ♣rt♦rs P := ps + ph ♦♥②t qt♦♥ ♦r t t♦tt② ♦ ♣rt♦rs s ♦t♥ ♥ ♥ t ♦r♠ ①♣t ♥ ε → 0 t♦ ♦t♥ αNps = γph tt s rrtt♥ ♥tr♠s ♦ P

ps =γP

αN + γ, ph =

γNP

αN + γ,

6.1 The derivation of the Holling-type II functional response 97

♥

P := ps + ph = ps +αNpsγ

=γps + αNps

γ.

♥ t ♠t♥ s②st♠ ♥ rtt♥ t N, P s ♥♥♦♥s

∂tN − d1∆xN = rN

(

1− N

k

)

− γαNP

αN + γ

∂tP −∆x

(d2γ + d3αN

αN + γP

)

=ΓαNP

αN + γ− µP.

s s②st♠ st sts② ♠♥♥ ♦♥r② ♦♥t♦♥s ♥ s ss♦t t t♥t ♦♥t♦♥s N(0, x) = Nin(x), P (0, x) = ps,in(x) + ph,in(x) ❲ ♥t t♦ ♥♦ttt t ♠t♥ s②st♠ ♣rs♥ts ♦♥t②♣ tr♦♣ ♥t♦♥ ♥ t rt♦♥♣rt ♥ ts s♥s r ts t②♣ ♦ ♥t♦♥ rs♣♦♥s ② t♠sr♠♥t r♦♠ ♠♦r ♦♠♣t ♠♦ s t s r② ♥♦♥ ♥ t ♦♥t①t♦ ♣rt♦r♣r② ♠♦s sr ② ♦r♥r② r♥t qt♦♥ ❬❪ ♦r♦rt ♠t♥ s②st♠ ♣rs♥ts r♦sss♦♥ ♥ t ♣rt♦r qt♦♥ t s♦♥rt ♣♥s ♦♥ t ♣r② ♦♠ss t ♣r② s♦♥ s st ♥r s ♠♥stt t s♦♥ tr♠ rt t♦ ♣rt♦rs s ♠ ♠♦r ♦♠♣t t♥ ♦♥st♥t t♠s ♣♥ ♦ P ♥r s tr♠ ♥ s♠♣② t♦ trt♦♥ ♣rt ❬❪ ♥ ♣rtr t s♦♥ tr♠ ♦t♥ ② t t♠s r♠♥t ♣♥s ♦♥ ♦t t s♦♥ ♦♥t ♦ sr♥ ♥ ♥♥ ♣rt♦rsd2 ♥ d3 ♥ ts s ♥ rt t t r♦sss♦♥ tr♠ s

∆x (f(N)P ) , f(N) =d2γ + d3αN

αN + γ,

s ♠♦♥♦t♦♥② rs♥ ♥t♦♥ t f(0) = d2 ♥ f(N) → d3 ♥N → +∞ ❲ s♦ ♥♦t tt tr s ♥♦ ♣r② ♥♦ ♥♥ ♣rt♦rs r ♣rs♥t♥ t s♦♥ ♦♥t rs t♦ ♦♥st♥t ♦♥ q t♦ t s♦♥ ♦♥t♦ sr♥ ♣rt♦rs ♥ t ♦♥trr② ♥ tr s ♣r② ♥♥ ①♣t tt ♥♥ ♣rt♦rs r ♠♦r ♥♠r♦s t♥ sr♥ ♣rt♦rs ♥ ts♦♥ ♦♥t ♥ rs t♦ ♦♥st♥t q t♦ t s♦♥ ♦♥t ♦♥♥ ♣rt♦rs ❲t rs♣t t♦ t ♥r s ♥ ♦t♥ t ♦♥st♥t f t r♦sss♦♥ tr♠ rs t ♣rt♦r s♦♥ ♥ t ♣r②♥st② ♥rss

6.1.1 Adimensionalization

♥ ♦rr t♦ s♠♣② t ♥♦tt♦♥s ♥ t♦ ♣ ♦♥② ♠♥♥ ♣r♠trs ♥♦♣r♦♣♦s ♥ ♠♥s♦♥③t♦♥ ♣r♦r s♥ t ♥ rs T, n, p ♥st

98 6 A justification of classical functional responses

♦ t, N, P ♥ t ♦♦♥ ②

t = ΘT, N = Σn, P = Πp.

tr s♠♣t♦♥s t s②st♠ ♦♠s

∂n

∂T− d1Θ∆xn = r0Θ

(

1− n

k/Σ

)

n− αΠΘnpαΣ

γn+ 1

,

∂p

∂T−∆x

d2Θ + d3ΘαΣ

γn

αΣ

γn+ 1

p

= ΘΓ

αΣ

γnp

αΣ

γn+ 1

−Θµp.

♦♦s♥ Θ, Σ, Π ♥ s ② tt αΠΘ = 1, αΣ/γ = 1, r0Θ = 1 ♥ ♣t t s②st♠

∂n

∂T− d1Θ∆xn =

(

1− n

k/Σ

)

n− np

n+ 1,

∂p

∂T−∆x

(d2Θ + d3Θn

n+ 1p

)

= ΘΓnp

n+ 1−Θµp.

❲ st D1 := d1Θ, D2 := d2Θ, D3 := d3Θ, ν := k/Σ, c = ΘΓ, m = Θµ ♥ ♥ ♣ t

∂tn−D1∆xn = n(

1− n

ν

)

− np

1 + n

∂tp−∆x

(D2 +D3n

1 + np

)

=cnp

1 + n−mp.

6.2 The derivation of the Beddington-DeAngelis-like

functional response

s ♥ t ♣r♦s st♦♥ ♥tr♦ s②st♠ ♦ tr s ♠♦♥ t ♥trt♦♥ t♥ ♣r② ♥♥ ♥ sr♥ ♣rt♦rs t ♥ s♦ t♥t♦ ♦♥t t ♦♠♣tt♦♥ ♠♦♥ ♣rt♦rs ♥ t② ♦♦ ♦r ♣r② s s ♦♥t♥s t♦ t ♥tr♦t♦♥ ♦ t ♥♦♠♥t♦r 1 + ξpεs ♦r s♦♠ ξ > 0 ♥ t ♥trt♦♥ tr♠ t♥ ♣rt♦rs ♥ ♣r② ❬❪ s②st♠ ♦♠s

6.2 The derivation of the Beddington-DeAngelis-like functional response 99

∂N ε

∂t− d1∆xN

ε = r0

(

1− N ε

k

)

N ε − αN εpεs1 + ξpεs

,

∂pεs∂t

− d2∆xpεs =

1

ǫ

(

− αN εpεs1 + ξpεs

+ γpεh

)

+ Γpεh − µpεs,

∂pεh∂t

− d3∆xpεh = −1

ǫ

(

− αN εpεs1 + ξpεs

+ γpεh

)

− µpεh.

♥ t♦♥ r♦r♦s② ♣r♦ tt ♥ t ♠t ♥ ε → 0 t s♦t♦♥N ε, pεs, p

εh ♦ ts s②st♠ ♦♥rs ♥ st t♦♣♦♦② t♦rs N ≥ 0, ps ≥

0, ph ≥ 0 s tt

ph =1

γ

αNps1 + ξps

,

♥

P := ps + ph = ps +1

γ

αNps1 + ξps

=γ(1 + ξps)ps + αNps

γ(1 + ξps).

❲ ♥♦ s t♦ rt t ♠t♥ s②st♠

∂tN − d1∆xN = r0

(

1− N

k

)

N − αNps1 + ξps

,

∂t(ps + ph)−∆x(d2ps + d3ph) = Γph − µ(ph + ps),

♥ tr♠s ♦ N ♥ P ♦♥② r t s♦♥ qt♦♥ s ♦t♥ ♥ ♥ ❲ ♥♦t tt ps stss s♦♥ r qt♦♥ ♥ P s ♥

γξp2s + (γ + αN − γξP )ps − γP = 0.

♥ ♦♥sr ♦♥② t ♣♦st r♦♦t ♦ ts qt♦♥

ps =−A+

√∆

2γξ=

2γP

A+√∆,

r ♥♦t

A := γ + αN − γξP, ∆ = A2 + 4γ2ξP.

♦t tt ∆ > 0 s♥ P > 0 ♥♦t♥ ②

B := γ + αN + γξP,

100 6 A justification of classical functional responses

s♦ ♦t♥ r♦♠ t ♦r♠

ph =2αNP

B +√∆,

r ∆ ♥ ♦♠♣t ♥ tr♠s ♦ B

∆ = A2 + 4γ2ξP = (γ + αN − γξP )2 + 4γ2ξP

= γ2 + (αN)2 + (γξP )2 + 2γαN − 2γ2ξP − 2αNγξP + 4γ2ξP

= γ2 + (αN)2 + (γξP )2 + 2γαN + 2γ2ξP + 2αNγξP − 4αNγξP

= (γ + αN + γξP )2 − 4αNγξP = B2 − 4αNγξP.

♥ t ♠t♥ s②st♠ ♥ rtt♥ t N, P s ♥♥♦♥s

∂N

∂t− d1∆xN = r0

(

1− N

k

)

N − γ2αNP

B +√∆,

∂P

∂t−∆x

(

d22γP

A+√∆

+ d32αNP

B +√∆

)

= Γ2αNP

B +√∆

− µP,

r♠♠r♥ t ♥t♦♥ ♦ A, B ♥ ∆ ♥ ♥ ♠t♥ s②st♠♣rs♥ts r♦sss♦♥ tr♠ ♥ t ♣rt♦r qt♦♥ t s♦♥ rt ♣♥s♦♥ t ♣r② ♦♠ss ♥ tr♦♣ ♥t♦♥ ♦s t♦ t ♥t♦♥♥s ♦♥

s s②st♠ st sts② ♠♥♥ ♦♥r② ♦♥t♦♥s ♥ s ss♦t t t♥t ♦♥t♦♥s N(0, x) = Nin(x), P (0, x) = ps,in(x) + ph,in(x) ❲ ♥♦t tt t♠t♥ s②st♠ ♣rs♥ts ♥t♦♥ rs♣♦♥s ♦s t♦ t ♥t♦♥♥s ♥t rt♦♥ ♣rt ♥ ts s♥s r ts t②♣ ♦ ♥t♦♥ rs♣♦♥s ② t♠s r♠♥t r♦♠ ♠♦r ♦♠♣t ♠♦ s t②♣ ♦ ♥t♦♥ rs♣♦♥s s r s♦ ② s♠♥ t ❬❪ ♥ t ♦♥t①t ♦ ♦r♥r② r♥tqt♦♥s strt♥ r♦♠ s②st♠ ♦ ♦r s ② qsst②stt ♣♣r♦①♠t♦♥ ❲t tr ♣♣r♦ s♦ t ♦st r♦t ♥ t ♣rt♦r tr♥♦r tr♠sst ♣♥ ♦♥ ts ♦♠♣① ①♣rss♦♥ rtr t② s♠♣ ts ♦♠♣t qrt ①♣rss♦♥ t Pé ♣♣r♦①♠t♦♥ ♥ t② r♦r tst♥r ♦r♠ ♦ t ♥t♦♥♥s ♥t♦♥ rs♣♦♥s

♦r♦r ♥ s♦ ♥♦t tt s ♥ t ♣r♦s t♦♥ r rt ♦♥t②♣ ♥t♦♥ rs♣♦♥s t ♠t♥ s②st♠ ♣rs♥ts r♦sss♦♥tr♠ ♥ t ♣rt♦r qt♦♥ t s♦♥ rt ♣♥s ♦♥ t ♣r② ♦♠ss t ♣r② s♦♥ s st ♥r s ♠♥s tt t s♦♥ tr♠ rt t♦

6.2 The derivation of the Beddington-DeAngelis-like functional response 101

♣rt♦rs s ♠ ♠♦r ♦♠♣t t♥ ♦♥st♥t t♠s ♣♥ ♦ P ♥rs tr♠ ♥ s♠♣② t♦ t rt♦♥ ♣rt ❬❪ ♥ ♣rtrt s♦♥ tr♠ ♦t♥ ② t t♠s r♠♥t ♣♥s ♦♥ ♦t t s♦♥♦♥t ♦ sr♥ ♥ ♥♥ ♣rt♦rs d2 ♥ d3 trs r t r♦sss♦♥ tr♠ ♣♥s s♦ ♦♥ P ♥ rt t t r♦sss♦♥ tr♠ s

∆x (f(N,P )P ) , f(N,P ) = d22γP

A+√∆

+ d32αNP

B +√∆,

r♠♠r♥ tt ∆ := ∆(N,P ) s ♥ t ♣r♦s s tt f(0, P ) = d2♥ f(N,P ) → d3 ♥ N → +∞

6.2.1 Adimensionalization

♥ ♦rr t♦ s♠♣② t ♥♦tt♦♥s ♥ t♦ ♣ ♦♥② ♠♥♥ ♣r♠trs ♥♦♣r♦♣♦s ♥ ♠♥s♦♥③t♦♥ ♣r♦r s♥ t ♥ rs T, n, p ♥st♦ t, N, P ♥ t ♦♦♥ ②

t = ΘT, N = Σn, P = Πp.

tr s♠♣t♦♥s t s②st♠ ♦♠s

∂n

∂T− d1Θ∆xn = r0Θ

(

1− n

k/Σ

)

n− 2γαΠΘnp

B +√∆

,

∂p

∂T−∆x

(

d2Θ2γp

A+√∆

+ d3Θ2αΣnp

B +√∆

)

=2ΓαΣΘnp

B +√∆

− µΘp,

r

A+√∆ = γ + αΣn− γξΠp+

√

(γ + αΣn+ γξΠp)2 − 4γαξΣΠnp,

B +√∆ = γ + αΣn+ γξΠp+

√

(γ + αΣn+ γξΠp)2 − 4γαξΣΠnp.

♦♦s♥ Θ, Σ, Π ♥ s ② tt 2αΠΘ = 1, αΣ = 1, γξΠ = 1 ♥ ♣t t s②st♠

∂n

∂T− d1Θ∆xn = r0Θ

(

1− n

k/Σ

)

n− γnp

B +√∆,

∂p

∂T−∆x

(

d2Θ2γp

A+√∆

+ d3Θ2np

B +√∆

)

=Γ γΣξnp

B +√∆

− µΘp,

102 6 A justification of classical functional responses

r ♥♦

A+√∆ = γ + n− p+

√

(γ + n+ p)2 − 4np,

B +√∆ = γ + n+ p+

√

(γ + n+ p)2 − 4np.

❲ st D1 := d1Θ, D2 := d2Θ, D3 := d3Θ, r = r0Θ, ν := k/Σ rtr♠♦r ♥♦t ♥ n ② N p ② P γ ② γ ♥ Γ := Γ γΣξ, µ := µΘ ❲ ♥ ♣ t

∂N

∂T−D1∆xN = r

(

1− N

ν

)

N − γNP

B +√∆,

∂P

∂T−∆x

(

D22γP

A+√∆

+D32NP

B +√∆

)

=ΓNP

B +√∆

− µP,

r ♥♦

A = γ +N − P, B = γ +N + P, ∆ = (γ +N + P )2 − 4NP.

t♦♥③♥ t ♥♦♠♥t♦rs ♥ ♦t♥ ♥ q♥t ①♣rss♦♥ s ♦r t stt② ♥②ss ♦ qr♠ stts ♦ t ♥♦♥ s ♣rt

∂N

∂T−D1∆xN = r

(

1− N

ν

)

N − γ

4

(

B −√B2 − 4NP

)

,

∂P

∂T−∆x

(D2

2(√∆− A) +

D3

2(B −

√∆)

)

=Γ

4

(

B −√B2 − 4NP

)

− µP,

r A, B ♥ ∆ r ♥ ②

6.3 Rigorous results of convergence

❲ ♦♥sr ♥ ts st♦♥ t ♠② ♥ ε > 0 ♦ s②st♠s sr ♥ tt s

∂tNε − d1∆xN

ε = r0 (1− ηN ε)N ε − αN εpεs1 + ξpεs

,

∂tpεs − d2∆xp

εs =

1

ε

(

− αN εpεs1 + ξpεs

+ γpεh

)

+ Γpεh − µpεs,

∂tpεh − d3∆xp

εh = −1

ε

(

− αN εpεs1 + ξpεs

+ γpεh

)

− µpεh,

6.3 Rigorous results of convergence 103

t♦tr t ♦♠♦♥♦s ♠♥♥ ♦♥r② ♦♥t♦♥s

n(x) · ∇xNε = 0, n(x) · ∇xp

εs = 0, n(x) · ∇xp

εh = 0,

r η ≥ 0 ♥ ξ ≥ 0

❲♥ ♦t η ♥ ξ r q t♦ ③r♦ t rt♦♥ ♣rt ♦ t s②st♠ rs t♦ ♥ ts sst♦♥ ss♠ tt η ≥ 0 tt s trts♠t♥♦s② ss t ♦r t♦t ♦st ts ♦r t ♣r②s ❲ ♦r trts♣rt② t s ♥ ξ = 0 ♥ t♦ ♦♥t②♣ ♥t♦♥ rs♣♦♥s ♥t s ♥ ξ > 0 ♥ t♦ ♥t♦♥♥s ♥t♦♥ rs♣♦♥s

❲ ♥t t♦ r♦r♦s② ♣r♦ tt ♥ t ♠t ♥ ε→ 0 t s♦t♦♥ N ε, pεs, pεh

♦ ts s②st♠ ♦♥rs ♥ st t♦♣♦♦② t♦rs N ≥ 0, ps ≥ 0, ph ≥ 0 ♠♥ rsts r s ♦♥ t ♥♦t♦♥ ♦ r② s♦t♦♥s s ♣♣♥①

♦♦♥ t♦r♠ rrs t♦ t s ξ = 0

♦r♠ t Ω s♠♦♦t ♦♠♥ ♦ Rd ♦r s♦♠ ♠♥s♦♥ d ∈ N−0d1, d2, d3 > 0 s♦♥ rts r0, α, γ , Γ, µ > 0 ♥ η ≥ 0 ♣r♠trs ♥Nin := Nin(x) ≥ 0 ph,in := ph,in(x) ≥ 0 ps,in := ps,in(x) ≥ 0 ♥♦♥♥t ♥tt rs♣t② ♥ L∞(Ω) L2(Ω) ♥ L2(Ω)

♥ ♦r ε > 0 tr ①sts ♥q ♦ ss ♦r t > 0 s♦t♦♥N ε pεh p

εs ♦ s②st♠ t ξ = 0 sts②♥ ♦♠♦♥♦s ♠♥♥

♦♥r② ♦♥t♦♥s ♥ t t ♥t t ♥ ♦♦r♦r ♥ ε → 0 ♦♥ ♥ ①trt r♦♠ N ε ssq♥ s ♦♥

♥ L∞([0, T ]× Ω) ♦r T > 0 ♥ ♦♥rs t♦rs ♥t♦♥ N ≥ 0 ②♥♥ L∞([0, T ]×Ω) ♥ ♥ ①trt r♦♠ pεs ssq♥ ♦♥rs ② ♥L2+δ([0, T ]×Ω) t♦rs ♥t♦♥ ps ≥ 0 ②♥ ♥ L2+δ([0, T ]×Ω) ♦r T > 0 ♥s♦♠ δ > 0 ♥② ♦♥ ♥ ①trt r♦♠ pεh ssq♥ ♦♥rs t♦rs ♥t♦♥ ph ≥ 0 ♦♥♥ t♦ L2+δ([0, T ] × Ω) ② ♥ L2+δ([0, T ] × Ω) ♦r T > 0 ♥ s♦♠ δ > 0

♦r♦r N ps ♥ ph r r② s♦t♦♥s ♦ t r♦sss♦♥ s②st♠

∂tN − d1∆xN = r0(1− ηN)N − αNps,

∂t(ps + ph)−∆x(d2ps + d3ph) = Γph − µ(ph + ps),

αNps = γph,

t ♦♠♦♥♦s ♠♥♥ ♦♥r② ♦♥t♦♥s

n(x) · ∇xN = 0, n(x) · ∇xps = 0, n(x) · ∇xph = 0,

104 6 A justification of classical functional responses

♥ t ♥t t N(0, x) = Nin(x) ♥ (ps + ph)(0, x) = ps,in(x) + ph,in(x) ss②st♠ ♥ rrtt♥ ♥ t s♠♣r ♦r♠ t P = ph + ps

∂tN − d1∆xN = r0(1− ηN)N − γαN

αN + γP,

∂tP −∆x

(d2γ + d3αN

αN + γP

)

= ΓαN

αN + γ− µP,

t ♠♥♥ ♦♥r② ♦♥t♦♥s n(x) · ∇xN = 0, n(x) · ∇xP = 0, ♥ t♥t t N(0, x) = Nin(x) ♥ P (0, x) = ps,in(x) + ph,in(x)

♥② N s ♥ W 1,2+δ([0, T ];L2+δ(Ω)) ∩ L2+δ([0, T ];W 2,2+δ(Ω)) ♦r T > 0♥ s♦♠ δ > 0 ♥ P s ♥ L2+δ([0, T ]×Ω])

♦♥s♦♥ ♦ ts t♦r♠ ♥ s♦♠t ①t♥ ♥ t s ♦ ♦♠♥s♦♥ ♥ strt② ♣♦st ♥t t ❲ ♥ ♥ t s♦ t ♦♦♥♣r♦♣♦st♦♥

Pr♦♣♦st♦♥ ❯♥r t s♠ ss♠♣t♦♥s s ♥ ♦r♠ ♥ ♥r t①tr ss♠♣t♦♥s tt t ♠♥s♦♥ s d = 1 ♦r d = 2 ♥ tt inf ss Nin(x) > 0t sq♥s pεh ♥ pεs ♦♥r t♦rs ph ♥ ps ♦r♦r t q♥ttsph, ps ♥ P ♥ L1([0, T ];W 1,1(Ω)) ♦r T > 0

❲ ♥♦ tr♥ t♦ t s ♥ ξ 6= 0

♦r♠ t Ω s♠♦♦t ♦♠♥ ♦ Rd ♦r s♦♠ ♠♥s♦♥ d ∈ N−0d1, d2, d3 > 0 s♦♥ rts r0, α, ξ, γ, Γ, µ > 0 ♥ η ≥ 0 ♣r♠trs ♥Nin := Nin(x) ≥ 0 ph,in := ph,in(x) ≥ 0 ps,in := ps,in(x) ≥ 0 ♥♦♥♥t♥t t rs♣t② ♥ L∞(Ω) L2(Ω) ♥ L2(Ω) ❲ ss♠ ♠♦r♦r ttinf ssNin(x) > 0

♥ ♦r ε > 0 tr ①sts ♥q ♦ ss ♦r t > 0 s♦t♦♥N ε pεh p

εs ♦ s②st♠ sts②♥ ♦♠♦♥♦s ♠♥♥ ♦♥r②

♦♥t♦♥s ♥ t t ♥t t ♥ ♦♦r♦r ♥ ε → 0 ♦♥ ♥ ①trt r♦♠ N ε ssq♥ s ♦♥

♥ L∞([0, T ]×Ω) ♦r T > 0 ♥ ♦♥rs t♦rs ♥t♦♥ N ≥ 0 ②♥ ♥L∞([0, T ]×Ω) r♦♠ pεs ssq♥ ♦♥rs str♦♥② ♥ L2+δ([0, T ]×Ω)t♦rs ♥t♦♥ ps ≥ 0 ②♥ ♥ L2+δ([0, T ]×Ω) ♦r T > 0 ♥ s♦♠ δ > 0 ♥r♦♠ pεh ssq♥ ♦♥rs str♦♥② t♦rs ♥t♦♥ ph ≥ 0 ②♥ ♥L2+δ([0, T ]×Ω) ♥ L2+δ([0, T ]×Ω) ♦r T > 0 ♥ s♦♠ δ > 0

♦r♦r N ps ♥ ph r r② s♦t♦♥s ♦ t r♦ss s♦♥ s②st♠

6.3 Rigorous results of convergence 105

∂tN − d1∆xN = r0 (1− ηN)N − αNps1 + ξps

,

∂t(ps + ph)−∆x(d2ps + d3ph) = Γph − µ(ph + ps),

αNps1 + ξps

= γph,

t ♦♠♦♥♦s ♠♥♥ ♦♥r② ♦♥t♦♥s

n(x) · ∇xN = 0, n(x) · ∇xps = 0, n(x) · ∇xph = 0,

♥ t ♥t t N(0, x) = Nin(x) ♥ (ps + ph)(0, x) = ps,in(x) + ph,in(x)♥② N s ♥ W 1,p([0, T ];Lp(Ω)) ♥ Lp([0, T ];W 2,p(Ω)) ♦r T > 0 ♥

p ∈ [1,+∞[ ph s ♥ L2([0, T ];H1(Ω)) ♥ ps s ♥ L1([0, T ];W 1,1(Ω))

❲ ♥♦t r♦♠ ♥♦ ♦♥ ② CT > 0 ♦♥st♥t ♠② ♣♥ ♦♥ T t♣r♠trs ♥ t ♥t t ♦ t ♦♥sr s②st♠s

6.3.1 Proof of Theorem. 6.3.1

❲ ♦♥sr s②st♠ t ξ = 0 ①st♥ ♦ ♦ ♥ t♠ s♦t♦♥s ♦r N ε, pεs, p

εh r ♥♦♥♥t

♦r ♥ ε > 0 t♦ ts s②st♠ s ss ❬❪

rs ♦ qt♦♥ s ♦♥ ♦ ② r0Nε ♥ ♦r T > 0

tr ①sts CT > 0 s tt

supε>0

||N ε||L∞([0,T ]×Ω) ≤ CT ,

t♥s t♦ t ♠①♠♠ ♣r♥♣ ♦r ♦♠♣rs♦♥ ♣r♥♣ ♥ t CT ♥ ♦♥ ♦ ② ||Nin||L∞er0T s ♦♥sq♥ tr ①sts N ∈ L∞([0, T ]×Ω)♥ ssq♥ st ♥♦t ② N ε s tt N ε N ♥ L∞([0, T ]×Ω) ∗ t♦♣♦♦②

♥ t qt♦♥s ♦r pεh ♥ pεs ♥ ♣ t

∂tPε −∆x(A

εP ε) = Γpεh − µP ε ≤ ΓP ε,

t

P ε = pεh + pεs, Aε =d2p

εh + d3p

εs

pεh + pεs.

♥ t♥s t♦ t② ♠♠ ❬❪ ♥ t ♦r rr♥ ❬❪ s tt

106 6 A justification of classical functional responses

supε>0

||P ε||L2([0,T ]×Ω) ≤ CT .

r♥ rs♦♥ ♦ t s♠ ♠♠ ♦r ①♠♣ ❬❪ ♦r ❬❪ ②s ♥ t tttr st♠t

supε>0

||P ε||L2+δ([0,T ]×Ω) ≤ CT ,

♦r s♦♠ δ > 0s ♦♥sq♥ tr ①st ph, ps ∈ L2+δ([0, T ] × Ω) ♥ ssq♥s st

♥♦t ② pεh, pεs s tt pεs ps p

εh ph ♥ L2+δ([0, T ]×Ω) t♦♣♦♦② ♦r

s♦♠ δ > 0

sr♥ tt ∂tNε − d1∆xN

ε s ♦♥ ♥ L2+δ([0, T ]× Ω) ♦r T > 0 ♥s♦♠ δ > 0 t t♥s t♦ t ♠①♠ rrt② st♠ts ♦r t t r♥

supε>0

||∂tN ε||L2+δ([0,T ]×Ω) ≤ CT ,

supε>0

||∂xixjN ε||L2+δ([0,T ]×Ω) ≤ CT .

s ♦♥sq♥ s tt t sq♥ N ε s str♦♥② ♦♠♣t ♥ L2([0, T ]×Ω) s♦ tt ♣ t♦ ♥ ①tr ①trt♦♥ N ε ♦♥rs t♦rs N ❲ s♦ stt N s ♥ W 1,2+δ([0, T ];L2+δ(Ω)) ∩ L2+δ([0, T ];W 2,2+δ(Ω))

❯s♥ t ♦♥ ♥ L2+δ ♦ pεs ♥ ♣ t t ♦♥r♥ N εpεs Nps ♥L2+δ([0, T ]×Ω) ♦r T > 0 ♥ s♦♠ δ > 0

Pss♥ t♦ t ♠t ♥ t qt♦♥s ♥ ♥ t s♥s ♦ strt♦♥s ♥ ♣ t t qt♦♥s ♥ ♦r ♣rs② ♣ss♥t♦ t ♠t ♥ t r② ♦r♠t♦♥ ♦ qt♦♥s ♥ tt r② ♦r♠t♦♥ ♦ t ♦ s②st♠ t♦tr t t ♦♠♦♥♦s ♠♥♥ ♦♥r② ♦♥t♦♥ ♦♥ N ♥ P ♥ t ♥t t ♦r N ♥ P tt sP (0, x) = ps,in(x) + ph,in(x)

sr♥ tt

pεh −N εpεs = ε(∂tpεs − d2∆xp

εs)− ε(pεh − µpεs),

♥ ♣ss♥ t♦ t ♠t ♥ t s♥s ♦ strt♦♥s ♥ ts stt♠♥t t ♥tt② ♦♥s t ♣r♦♦ ♦ ♦r♠

6.3 Rigorous results of convergence 107

6.3.2 Proof of Proposition 6.3.1

❲ ♦♠♣t ♦r q ∈]0, 1[ t t♠ rt ♦ st ♥♦♥♥t q♥tt②

d

dt

∫ [(pεh)

q+1

q + 1+

∣∣∣∣

α

γN ε

∣∣∣∣

q(pεs)

q+1

q + 1

]

= − d3q

∫

(pεh)q−1|∇xp

εh|2

︸ ︷︷ ︸

1

− d2q

(α

γ

)q ∫

(N ε)q(pεs)q−1|∇xp

εs|2

︸ ︷︷ ︸

2

− γ

ε

∫ (

pεh −(α

γ

)

N εpεs

) (

(pεh)q −

(α

γN εpεs

)q)

︸ ︷︷ ︸

3

− µ

∫ [

(pεh)q+1 +

(α

γN ε

)q

(pεs)q+1

]

︸ ︷︷ ︸

4

+Γ

(α

γ

)q ∫

(N ε)q(pεs)qpεh

︸ ︷︷ ︸

5

+q

q + 1

(α

γ

)q ∫

(pεs)q+1(N ε)q−1(∂tN

ε + d2∆xNε)

︸ ︷︷ ︸

6

− d2q(1− q)

1 + q

(α

γ

)q ∫

(pεs)q+1(N ε)q−2|∇xN

ε|2︸ ︷︷ ︸

7

.

❲ ♦sr tt t tr♠s 1 2 3 4 7 r ♥♦♥♣♦st ♠♠r♥ ttN ε s ♦♥ ♥ L∞ ♥ tt pεs, p

εh r ♦♥ ♥ L2+δ ♦r s♦♠ δ > 0 s

tt ∫ T

0

∫

(N ε)q(pεs)qpεh ≤ CT ♦r q ∈]0, 1[.

♠♠r♥ t♥ tt ∂tNε + d2∆xN

ε s ♦♥ ♥ L2+δ ♦r s♦♠ δ > 0 stt

∫ T

0

∫

|pεs|q+1|∂tN ε + d2∆xNε| ≤ CT ♥ q ∈]0, 1[ s s♠ ♥♦.

s ♦♥sq♥ ♥trt♥ ♦♥ [0, T ] s tt ♦r q ∈]0, 1[ s♠ ♥♦

108 6 A justification of classical functional responses

∫ (

pεh − (α/γ)N εpεs

)(

(pεh)q − ((α/γ)N εpεs)

q

)

≤ CT ε,

∫ T

0

∫

Ω

(pεh)q−1|∇xp

εh|2 ≤ CT ,

∫ T

0

∫

Ω

(N ε)q(pεs)q−1|∇xp

εs|2 ≤ CT .

sr♥ tt

(∂t − d1∆x) lnNε =

1

N ε(∂t − d1∆x)N

ε + d1|∇xN

ε|2(N ε)2

≥ r0(1− ηN ε)− αpεs ≥ (−ηr0 − αpεs)CT ,

s tt s♥ pεs s ♦♥ ♥ L2+δ([0, T ]×Ω) ♦r s♦♠ δ > 0 ♥ ♠♥s♦♥ d = 1♦r d = 2 ♦t♥ ttN ε s ♦♥ ♦ ② strt② ♣♦st ♦♥st♥t ♥ r tt (∂t−d1∆x)

−1 ts s ♦♥♦t♦♥ t ♥t♦♥ ②♥ ♥ L3−ε ♥d = 1 ♦r L2−ε ♥ d = 2 ♦r ♥② ε > 0 s♦ tt t♥s t♦ ❨♦♥s ♥qt②♥ t ss♠♣t♦♥ tt t ♥t t♠ N ε s ss♥t② strt② ♣♦st lnN ε s♦♥ ♦ ② strt② ♣♦st ♦♥st♥t s ♦♥sq♥ st ♦r q ∈]0, 1[s♠ ♥♦

∫ T

0

∫

Ω

(pεs)q−1|∇xp

εs|2 ≤ CT .

♥ ②r③ ♥qt② ♥srs tt

(∫ T

0

∫

Ω

|∇xpεs,h|)2

≤∫ T

0

∫

Ω

(pεs,h)q−1|∇xp

εs,h|2 ×

∫ T

0

∫

Ω

(pεs,h)1−q ≤ CT .

❯s♥ s tt ∂tPε s ♦♥ ♥ L2([0, T ];H−2(Ω)) s♦ tt t♥s

t♦ ♥s ♠♠ ❬❪ P ε ♦♥rs t♦ P ♥ pεh − N εpεs → 0 ♣ t♦①trt♦♥ ♦ ssq♥ s ♦ ♥ N ε ♦♥rs t♦ N stt pεh ♦♥rs t♦ ph ♥ p

εs ♦♥rs t♦ ps ♥② ♠♣s tt

ph, ps ∈ L1([0, T ],W 1,1(Ω)) ♦r T > 0

6.3.3 Proof of Theorem. 6.3.2

s ♥ ♦r♠ ①st♥ ♦ s♦t♦♥s ♦r N ε, pεs, pεh r ♥♦♥♥t

♦r ♥ ε s ss ❬❪

6.3 Rigorous results of convergence 109

s ♥ ♦r♠ ♥ ♦r T > 0 tr ①sts CT > 0 s tt

supε>0

||N ε||L∞([0,T ]×Ω) ≤ CT ,

♥ s ♦♥sq♥ tr ①sts N ∈ L∞([0, T ] × Ω) ♥ ssq♥ st♥♦t ② N ε s tt N ε N ♥ L∞([0, T ]×Ω) ∗ ♥ s tt

supε>0

||P ε||L2+δ([0,T ]×Ω) ≤ CT ,

♦r s♦♠ δ > 0 ♥ s ♦♥sq♥ tr ①st ph, ps ∈ L2+δ([0, T ] × Ω) ♥ssq♥s st ♥♦t ② pεh, p

εs s tt pεs ps p

εh ph ♥ L2+δ([0, T ]×Ω)

♦r s♦♠ δ > 0

♦ ♦sr♥ tt t rs ♦ s ♦♥ ♥ L∞([0, T ] × Ω) ts ♦♥② ♥ L2+δ([0, T ] × Ω) ♦r s♦♠ δ > 0 ♥ ♦r♠ t ♠①♠ rrt②st♠ts ♦r t t r♥ ② t ♦♥s

supε>0

||∂tN ε||Lp([0,T ]×Ω) ≤ CT ,

supε>0

||∂xixjN ε||Lp([0,T ]×Ω) ≤ CT ,

♦r p ∈ [1,+∞[ i, j = 1, .., d s♦ tt t sq♥ N ε s str♦♥② ♦♠♣t ♥Lp([0, T ]×Ω) ♦r p ∈ [1,+∞[ ♥ N ε ♦♥rs t♦rs N

❲ ♥♦ ♦♠♣t t rt ♦ st ♥♦♥♥t q♥tt②

1

2

d

dt

∫ [γ

α(pεh)

2 +N εψ(pεs)

]

,

t ψ(x) = 2/ξ(x − ln(1 + ξx)/ξ) s♦ tt ψ′(x) = 2x/(1 + ξx) ♥ ψ′′(x) =2/(1 + ξx)2 ❲ ♦t♥

1

2

d

dt

∫ [γ

α(pεh)

2 +N εψ(pεs)

]

=

∫γ

αpεh∂tp

εh +

∫ψ(pεs)

2∂tN

ε +1

2

∫

N εψ′(pεs)∂tpεs

=

∫γ

αpεh

(

d3∆xpεh −

1

ε

(

γpεh −αN εpεs1 + ξpεs

)

− µpεh

)

+

∫ψ(pεs)

2∂tN

ε

+

∫

N ε pεs1 + ξpεs

(

d2∆xpεs +

1

ε

(

γpεh −αN εpεs1 + ξpεs

)

− µpεs + Γpεh

)

110 6 A justification of classical functional responses

= d3γ

α

∫

pεh∆xpεh − 1

εα

∫ (

γpεh −αN εpεs1 + ξpεs

)2

− µγ

α

∫

(pεh)2

+d12

∫

ψ(pεs)∆xNε +

r02

∫

ψ(pεs)(1− ηN ε)N ε − α

2

∫

ψ(pεs)pεs

1 + ξpεsN ε

+ d2

∫

N ε pεs1 + ξpεs

∆xpεs − µ

∫N ε(pεs)

2

1 + ξpεs+ Γ

∫N εpεsp

εh

1 + ξpεs.

♦① tr♠s ♥trt♥ ② ♣rts ♥ s♥ t ♦♥r② ♦♥t♦♥s ♥ rrtt♥ s

d3γ

α

∫

pεh∆xpεh = −d3

γ

α

∫

|∇pεh|2

d2

∫

N ε pεs1 + ξpεs

∆xpεs = −d2

∫

∇pεs∇(

N ε pεs1 + ξpεs

)

= −d2∫

∇pεs∇(

pεs1 + ξpεs

)

N ε − d2

∫

∇pεs∇N ε pεs1 + ξpεs

= −d22

∫

∇pεs∇ψ′(pεs)Nε − d2

2

∫

∇ψ(pεs)∇N ε

= −d22

∫

|∇pεs|2∇ψ′′(pεs)Nε +

d22

∫

ψ(pεs)∆xNε

s ♥ rrtt♥ s

= − 1

εα

∫ (

γpεh −αN εpεs1 + ξpεs

)2

︸ ︷︷ ︸

1

− d3γ

α

∫

|∇pεh|2︸ ︷︷ ︸

2

−µγ

α

∫

(pεh)2

︸ ︷︷ ︸

3

− d22

∫

N εψ′′(pεs)|∇pεs|2︸ ︷︷ ︸

4

+1

2

∫

ψ(pεs) (∂tNε + d2∆N

ε)︸ ︷︷ ︸

5

− µ

∫N ε(pεs)

2

1 + ξpεs︸ ︷︷ ︸

6

+Γ

∫N εpεsp

εh

1 + ξpεs︸ ︷︷ ︸

7

.

6.3 Rigorous results of convergence 111

tr♠s 1 2 3 4 ♥ 6 r ♥♦♥♣♦st ♥ r♠♠r♥ tt 0 ≤ ψ(x) ≤2x/ξ pεs ♥ p

εh r ♦♥ ♥ L2+δ ♦r s♦♠ δ > 0 N ε s ♦♥ ♥ L∞ ♥ ♥②

∂tNε ∆xNε r ♦♥ ♥ Lp ♦r p < +∞ s tt tr♠s 5 ♥ 7 ♦♥

♥trt ♦♥ [0, T ] r ♦♥ ② s♦♠ ♦♥st♥t CT > 0 s ♦♥sq♥ ♥ ♣ t t st♠ts

∫ T

0

∫

Ω

(

γpεh −αN εpεs1 + ξpεs

)2

≤ CT ε,

∫ T

0

∫

Ω

|∇xpεh|2 ≤ CT ,

∫ T

0

∫

Ω

N εψ′′(pεs)|∇xpεs|2 ≤ CT .

❲ s tt t CT := α + r0η supε>0 ||N ε||L∞([0,T ]×Ω)

(∂t − d1∆x)Nε ≥ −CTN

ε,

s♦ tt ♥♦t♥ ② inf t ss♥t ♥♠ ♥ t ♦r♠ ♦

infε>0,x∈Ω

N ε(t, x) ≥ e−CT infε>0,x∈Ω

N ε(0, x).

s ♦♥sq♥ t♥s t♦

∫ T

0

∫

Ω

ψ′′(pεs)|∇xpεs|2 ≤ CT .

♥ t ②r③ ♥qt② ♥srs tt

(∫ T

0

∫

Ω

|∇xpεs|)2

≤(∫ T

0

∫

Ω

ψ′′(pεs)|∇xpεs|2)

×(∫ T

0

∫

Ω

(1 + ξpεs)2

)

≤ CT .

❯s♥ s tt ∂tPε s ♦♥ ♥ L2([0, T ];H−2(Ω)) s♦ tt t♥s t♦

st♠ts ♥ ♥s ♠♠ ❬❪ P ε ♦♥rs t♦ P ♦t ttγpεh−αN εpεs/(1+ ξp

εs) → 0 ♣ t♦ ①trt♦♥ ♦ ssq♥ t♥s t♦

♥ tt N ε ♦♥rs t♦ N ♥

γP ε −(

γpεh −αN εpεs1 + ξpεs

)

= γpεs +αN εpεs1 + ξpεs

♦♥rs t♦rs γP sr♥ tt

112 6 A justification of classical functional responses

∣∣∣∣

(

γpεs +αN εpεs1 + ξpεs

)

−(

γpεs +αNpεs1 + ξpεs

)∣∣∣∣≤ α

ξ|N ε −N |,

s tt

γpεs +αNpεs1 + ξpεs

→ γP.

❯s♥ t ♦♥t♥t② ♥ t strt ♠♦♥♦t♦♥t② ♦ y → γy + αNy/(1 + ξy) ♦r N ≥ 0 s tt pεs ♦♥rs t♦rs ♥♦♥♥t ♥t♦♥ ♥♦t ②ps ♥ p

εh s♦ ♦♥rs t♦rs ♥♦♥♥t ♥t♦♥ ♥♦t ② ph s

♦♥sq♥ t② s♦ ♦♥r ♥ L2+δ str♦♥ ♥ δ > 0 s s♠ ♥♦ ♥②t s r tt

ps + ph = P, γph =αNps1 + ξps

,

s♦ tt ♦s

❲ ♥♦ ♣ss t♦ t ♠t ♥ qt♦♥ ♥ ♥ t s♥s ♦ strt♦♥s♠♦r ♣rs② ♥ t s♥s ♦ r② s♦t♦♥s ♥ t ♠♥♥♦♥r② ♦♥t♦♥s ♥ t ♥t t N(0, x) = Nin(x) ♥ (ps + ph)(0, x) =ps,in(x) + ph,in(x) s♦ tt ♥ ♦

♥② t♥s t♦ ♥ s tt N s ♥ Lp([0, T ],W 2,p(Ω)) ∩W 1,p(]0, T [, Lp(Ω)) ♦r p ∈]1,+∞[ ♥ t♥s t♦ ♥ s ttps ♥ ph rs♣t② ♦♥ t♦ L1([0, T ],W 1,1(Ω)) ♥ L2([0, T ], H1(Ω))

7

The Turing Instability analysis

♥ ts ♣tr st② t r♥ ♥stt② r♦♥s ss♦t t♦ s②st♠ ♥ ❲ ♠ ①♣t t ♦♥t♦♥ ♦♥ t ♣r♠trs t♦ t ①st♥♦ ♥ ♦♠♦♥♦s ♦①st♥ qr♠ ♥ ♣r♦r♠ stt② ♥②ss ♦ tsqr♠ ♥ t ①sts t t s s st② r♥ ♥stt②♦r ♦t r♦sss♦♥ s②st♠s ♥ ♦♠♣r t rsts t rs♣t t♦ t s ♥ ♥r s♦♥ tr♠ s s♠♣② t♦ t ♣rt♦r qt♦♥ ♥st ♦t r♦sss♦♥ tr♠

7.1 The Holling-type II

♥ ts t♦♥ ♥t t♦ st② t r♥ ♥stt② ♦ t r♦sss♦♥ s②st♠t t ♦♥t②♣ ♥t♦♥ rs♣♦♥s ❲ r s♦ ♥trst ♥ t st② ♦t ss♦t ♥rs♦♥ s②st♠ tt s ♥ ♥rs♦♥ tr♠s r s♠♣② t♦ t rt♦♥ ♣rt

7.1.1 Homogeneous equilibrium states

rst ♦♦ ♦r t ♦♠♦♥♦s qr♠ stts tt s t s s②st♠ ♦t rt♦♥ ♣rt ♦

n = n(

1− n

ν

)

− np

1 + n

p =cnp

1 + n−mp.

r♦♠ t rst qt♦♥ p = 0 ♦t♥ n = 0 ♦r N = ν ♦rrs♣♦♥♥ t♦ tt♦t ①t♥t♦♥ E0(0, 0) ♥ t ♥♦♥♦①st♥ E1(ν, 0) qr trs

114 7 The Turing Instability analysis

♦♦ ♦r ♦①st♥ qr♠ E∗(n∗, p∗) ② ♦①st♥ qr♠ ♠♥tt n∗, p∗ > 0 r♦♠ t s♦♥ qt♦♥ t

n∗ =m

c−m, p∗ = n∗

(

1− n∗

ν

) n∗ + 1

n∗

,

t ①st♥ ♦♥t♦♥s ♦♥ t ♣r♠trs

c−m > 0, ν >m

c−m.

♦ st② t stt② ♣r♦♣rts ♦ ts qr♠ stts ❲ ♥♦t ②Ji,j, i, j = 1, 2 t ♠♥ts ♦ t ♦♥ ♠tr① ♦ t s②st♠

J11 =∂

∂nn = 1− 2

νn− p

(1 + n)2,

J12 =∂

∂pn = − n

1 + n,

J21 =∂

∂np =

cp

1 + n,

J22 =∂

∂pp =

cp

1 + n−m.

t♥ t ♦♥ ♠tr① ♥ t qr♠ stts ♦t♥

J(E0) =

(1 00 −m

)

, J(E1) =

−1 − n∗

1 + n∗

0cp∗

1 + n∗

−m

,

♥ ttcp∗

1 + n∗

−m < 0 ⇔ ν <m

c−m.

♥ t qr♠ E0 s ♥st ♠♦r ♣rs② s ♣♦♥t t qr♠E1 s ♦② s②♠♣t♦t② st ♠♦r ♣rs② ♥♦ ♥ E∗ ♦s ♥♦t ①st♥ t qr♠ E1 s ♥st ♠♦r ♣rs② s ♣♦♥t ♦trs

♠♥ts ♦ t ♦♥ ♠tr① t t t qr♠ E∗ r

7.1 The Holling-type II 115

J∗11 = 1− 2

νn∗ −

p∗(1 + n∗)2

= 1− 2

νn∗ − n∗

(

1− n∗

ν

) n∗ + 1

n∗

1

(1 + n∗)2

= 1− 2

νn∗ −

(

1− n∗

ν

)(

1− m

c

)

= −n∗

ν

(

1 +m

c

)

− m

c,

J∗12 = − n∗

1 + n∗

< 0,

J∗21 =

cp∗1 + n∗

> 0,

J∗22 =

cp∗1 + n∗

−m = 0.

♥ tt t tr♠♥♥t ♦ t ♦♥ ♠tr① s ♣♦st ♥ t trs t s♠ s♥ ♦ J∗

11 ♥ ♣rtr

J∗11 > 0 ⇔ m

c−m< ν < 2

m

c−m,

ts E∗ ①sts st ♥ ♥st ♦r rtr s ♦ ν

7.1.2 Turing Instability analysis

❲ ♦♦ ♥♦ t♦ t rt♦♥s♦♥ s②st♠ t ♣r♠tr s ♦r t♦①st♥ qr♠ s st ♦♥sr ♥rs♦♥ tr♠s t s②st♠rts

∂tn−D1∆xn = n(

1− n

ν

)

− np

1 + n

∂tp−D∆xp =cnp

1 + n−mp,

r D s ♦♥st♥t s♦♥ ♦♥t ♦ ♣rt♦rs D 6= D1 ♥ ts s t s♥♦♥ tt r♥ ♥stt② ♥ ♥♦t ♣♣r ❬❪ ♥ t

M ∗k =

(J∗11 −D1k

2 J∗12

J∗21 −Dk2

)

,

116 7 The Turing Instability analysis

♥ t s s② t♦ r② tt tr M ∗k < 0 ♥ detM ∗

k > 0 ∀ k.♦ st② t r♦sss♦♥ s②st♠ t ♥r③t♦♥ ♦ t s♦♥

tr♠s r♦♥ E∗ s t ♠tr①

D1 0

(D3 −D2)

(1 + n∗)2p∗

D2 +D3n∗

1 + n∗

,

s ♣♦st ♦♥ ♠♥ts ♥♦♥③r♦ ♦♦♥ ♠♥t s t r♦sss♦♥ ♦♥t ♦ ♣rt♦rs ♥ t ♥r③ s②st♠ ❲ ♥♦t tt t t♥s t♦③r♦ ♥ p∗ → 0 s ♥ ts s tr ♥ ♥♦ ① ♦ ♣rt♦rs s♥ ♦♥♦♥③r♦ ♦♦♥ ♠♥t s ♣rsr ② t s ♦ t s♦♥ ♦♥ts♦ sr♥ ♥ ♥♥ ♣rt♦rs ♥ ♣rtr ♥ ss♠ tt D3 < D2 ♠♥s tt sr♥ ♣rt♦rs r ♠♦r s t♥ ♥♥ ♣rt♦rs♦♦② rs♦♥ ❲t ts ss♠♣t♦♥ D3 − D2 < 0 ♥ t r♦sss♦♥ ♦♥t s ♥t s ♠♣s tt t ① ♦ ♣rt♦rs s rtt♦r ♥rs♥ s ♦ t ♦♥♥trt♦♥ ♦ ♣r②s rtrst ♠tr①k ∈ R s t ♥♠r s t ♦r♠

M ∗k =

J∗11 −D1k

2 J∗12

J∗21 −

(D3 −D2)p∗(1 + n∗)2

k2 −D2 +D3n∗

1 + n∗k2

.

s tr M ∗k < 0 ♥ detM ∗

k > 0, ∀k ❲ ♥ ♦♥ tt ♥♦ r♥ ♥stt② ♣♣rs s♦ ♥ t r♦sss♦♥ s②st♠ ♥r t ♦♦② rs♦♥ss♠♣t♦♥ D3 < D2

♥tr♦t♦♥ ♦ r♦sss♦♥ tr♠s r ② t♠s r♠♥t ♣rs♥t ♥ t ♣r♦s ♣tr ♥st ♦ st♥r ♥r s♦♥ tr♠s ♦s ♥♦t t♦ t ♥rs ♦ t st ♦ ♣r♠tr s ♥ ♣ttr♥s ♦♣ ts s①t② t ♦♣♣♦st rst ♦t♥ ♥ ❬ ❪ t ♥ ♥♦tr t②♣ ♦ r♦sss♦♥

7.2 The Beddington-DeAngelis

♥ ts t♦♥ ♥t t♦ st② t r♥ ♥stt② ♦ t r♦sss♦♥ s②st♠t t ♥t♦♥♥s ♥t♦♥ rs♣♦♥s ❲ r s♦ ♥trst ♥ tst② ♦ t ss♦t ♥rs♦♥ s②st♠ tt s ♥ ♥rs♦♥ tr♠sr s♠♣② t♦ t rt♦♥ ♣rt rst ♦ ♦♦ ♦r t qr♠ stts♦ t s s②st♠ ♥ t♥ ♣r♦r♠ t r♥ ♥stt② ♥②ss ♥ ♦tss ♥ ♦♠♣r t rsts

7.2 The Beddington-DeAngelis 117

7.2.1 Homogeneous equilibrium states

♥ ts sst♦♥ ♦♦ ♦r t qr♠ stts ♦ t s s②st♠ tt st rt♦♥ ♣rt ♦ t ♦ s②st♠ ♦r q♥t②

N = r

(

1− N

ν

)

N − γNP

B +√∆,

P =ΓNP

B +√∆

− µP,

r A, B ♥ ∆ r ♥ ♥

r♦♠ t rst qt♦♥ P = 0 ♦t♥ N = 0 ♦r N = ν ♦rrs♣♦♥♥ t♦t t♦t ①t♥t♦♥ E0(0, 0) ♥ t ♥♦♥♦①st♥ E1(ν, 0) qr

trs ♦♦ ♦r ♦①st♥ qr♠ E∗(N∗, P∗) ② ♦①st♥ qr♠ ♠♥ tt N∗, P∗ > 0 r♦♠ t s♦♥ qt♦♥ t t ♥tt②

ΓN∗

γ +N∗ + P∗ +√

(γ +N∗ + P∗)2 − 4N∗P∗

− µ = 0

r♦♠ rt♦♥③♥ t ♥♦♠♥t♦r ♥ ♦t♥

√

(γ +N∗ + P∗)2 − 4N∗P∗ = γ +N∗ + P∗ −4µ

ΓP∗.

r♦♠ ♥ s♦ t ♥ ①♣rss♦♥ ♦ P∗ ♥ tr♠s ♦ N∗ ♥ t

ΓN∗ − µ(γ +N∗ + P∗) = µ√

(γ +N∗ + P∗)2 − 4N∗P∗.

❲ s tr♦r tt t ♦①st♥ qr♠ ♥ ①st ♦♥②

ΓN∗ − µ(γ +N∗ + P∗) > 0,

♥ t♥ t♥ t sqr ♦ ♦t tr♠s ♥ ♣ t

P∗ =Γ

2µ

(Γ − 2µ)N∗ − 2γµ

(Γ − 2µ)=

Γ

2µN∗ −

Γγ

Γ − 2µ.

sttt♥ ♥ s tt s q♥t t♦

Γ − 2µ

2N∗ + γµ

(2µ

Γ − 2µ

)

> 0.

118 7 The Turing Instability analysis

♥ r ♦♦♥ ♦r qr t N∗ > 0 s tt ♦r ♥ s♦ rrtt♥ s

Γ − 2µ > 0.

sttt♥ t ①♣rss♦♥ ♥ t qt♦♥ N = 0 r♦♠ t rst qt♦♥ ♦ rtt♥ s

N = r

(

1− N

ν

)

N − γ

4(B −

√∆),

♦t♥

r

(

1− N∗

ν

)

N∗ −γµ

ΓP∗ = 0,

r♦♠ ♥♦tr ①♣rss♦♥ ♦ P∗ ♥ tr♠s ♦ N∗

P∗ =Γ

γµr

(

1− N∗

ν

)

N∗.

sttt♥ t ①♣rss♦♥ ♥ ♦t♥ s♦♥ ♦rr qt♦♥ ♥ t♥♥♦♥ N∗

r

νN2

∗ −(

r − γ

2

)

N∗ −µγ2

Γ − 2µ= 0.

❲ s tt t♥s t♦

∆N :=(

r − γ

2

)2

+ 4r

ν

µγ2

Γ − 2µ> 0,

s♦ tt qt♦♥ s ♦♥ ♥ ♦♥② ♦♥ strt② ♣♦st s♦t♦♥ ♥ ②

N∗ =ν

2r

(

r − γ

2+√

∆N

)

.

♥ t ♦♥t♦♥ P∗ > 0 s q♥t t♦

Γ

2µN∗ −

Γγ

Γ − 2µ> 0 ⇔ 0 < N∗ < ν,

♣♥♥ ♦♥ t ♦s♥ ①♣rss♦♥ ♦r P∗ s ♦♥t♦♥ ♥ rrtt♥ s

Γ − 2µ >2γµ

ν,

② ssttt♥ ♥ t st tr♠ ♦ ♦t tt ts st ♥ssr② ♦♥t♦♥ ♦r t ①st♥ ♦ t ♦①st♥ qr♠ E∗ ♠♣s ♦♥t♦♥ ❲

7.2 The Beddington-DeAngelis 119

♥♦ r② ①♣♥ ② ♦♥t♦♥ s s♥t ♥ ts ♦t ♥ssr② ♥s♥t ♦r t ①st♥ ♦ E∗ ♥ ♥ rrtt♥ s

√

∆N < r +γ

2,

s♦ tt N∗ ♦♠♣t r♦♠ ♦r♠ ♥ s s tt 0 < N∗ < ν♠♠r♥ tt ts st ♦♥t♦♥ s q♥t t♦ P∗ > 0 ♥ P∗ s ♥ ② ♦r ①♠♣ s tt ♦t N∗ ♥ P∗ ♥ ♥ ts ② r strt②♣♦st ♥ ♥ s② tt t② sts② N = 0, P = 0 ♥

♦ st② t stt② ♣r♦♣rts ♦ ts qr♠ stts ❲ ♥♦t ②Ji,j, i, j = 1, 2 t ♠♥ts ♦ t ♦♥ ♠tr① ♦ t s②st♠

J11 =∂

∂NN = r − 2r

νN − γ

4

(

1− γ +N − P√

(γ +N + P )2 − 4NP

)

,

J12 =∂

∂PN = −γ

4

(

1− γ −N + P√

(γ +N + P )2 − 4NP

)

,

J21 =∂

∂NP =

Γ

4

(

1− γ +N − P√

(γ +N + P )2 − 4NP

)

,

J22 =∂

∂PP =

Γ

4

(

1− γ −N + P√

(γ +N + P )2 − 4NP

)

− µ.

t♥ t ♦♥ ♠tr① ♥ t qr♠ stts ♦t♥

J(E0) =

[r 00 −µ

]

, J(E1) =

[−r ∗0 J22(E1)

]

,

J22(E1) =Γν

2(γ + ν)− µ > 0 ⇔ Γ − 2µ >

2γµ

ν.

♥ t qr♠ E0 s ♥st ♠♦r ♣rs② s ♣♦♥t t qr♠E1 s ♦② s②♠♣t♦t② st ♠♦r ♣rs② ♥♦ ♥ E∗ ♦s ♥♦t ①st♥ t qr♠ E1 s ♥st ♠♦r ♣rs② s ♣♦♥t ♦trs

s ♦ E∗ s ♠♦r ♥trt rst ♦♠♣t t q♥tts

Q∗ := γ +N∗ + P∗ −4µ

ΓP∗ =

Γ − 2µ

2µN∗ +

2µγ

Γ − 2µ> 0

120 7 The Turing Instability analysis

♥ t♥ t♦ ♥

P∗

Q∗

=Γ

Γ − 2µ

(

1− 2µγΓ

(Γ − 2µ)2N∗ + 4µ2γ

)

.

♥s t♦ ♥ ♦t♥ r♠♠r♥ t ♥t♦♥ ♦ ∆N ♥ ①♣t ①♣rss♦♥s ♦r t ♠♥ts ♦ t ♦♥ ♠tr① t t t qr♠ E∗ ♥♦t ② J

J∗11 := J11(E∗) = r

(

1− 2

νN∗

)

− γ(Γ − 2µ)

2Γ

P∗

Q∗

= − r

Q∗

2µγ

Γ − 2µ

[(Γ − 2µ)2

4νµ2γN2

∗ +2

νN∗ − 1

]

=µγ2Γ

(Γ − 2µ)2N∗ + 4µ2γ−√

∆N ,

J∗12 := J12(E∗) = −γ

4

(√

(γ +N∗ + P∗)2 − 4N∗P∗ − (γ +N∗ + P∗) + 2N∗√

(γ +N∗ + P∗)2 − 4N∗P∗

)

= − γ

2ΓQ∗

(−2µP∗ + ΓN∗) = − γ2µ

Q∗(Γ − 2µ)< 0,

J∗21 := J21(E∗) =

Γ

4

(√

(γ +N∗ + P∗)2 − 4N∗P∗ − (γ +N∗ + P∗) + 2P∗√

(γ +N∗ + P∗)2 − 4N∗P∗

)

=1

2Q∗

(Γ − 2µ)P∗ > 0,

J∗22 := J22(E∗) = −µ+

Γ

4

(√

(γ +N∗ + P∗)2 − 4N∗P∗ − (γ +N∗ + P∗) + 2N∗√

(γ +N∗ + P∗)2 − 4N∗P∗

)

= −µ+Γ

2Q∗

(

N∗ −2µ

ΓP∗

)

=1

2Q

(

−2µ

(

γ +N∗ + P∗ −4µ

ΓP∗

)

+ ΓN∗ − 2µP∗

)

=1

2Q∗

(

(−2µγ + (Γ − 2µ)N∗)− 4µP∗ +8µ2

ΓP∗

)

=1

2Q∗

(2µ

Γ(Γ − 2µ)P∗ − 4µP∗ +

8µ2

ΓP∗

)

= − µ

ΓQ∗

(Γ − 2µ)P∗ < 0.

7.2 The Beddington-DeAngelis 121

♦t tt t s♥ ♦ J∗11 s ♥♦t ♣rsr r t♦ tr♠♥ t

s♥ ♦ t ♦trs ♠♥ts J∗12 < 0, J∗

21 > 0, J∗22 < 0 ♦r ♥ ♣r♦ tt

detJ = J∗11J

∗22 − J∗

12J∗21 > 0,

ts ♦s tr s t s♥ ♦ J∗11 ♥ t ssttt♥ ♥ t ①♣rss♦♥ ♦

detJ(E∗) t ♦r♠s ♦r J∗i,j, i, j = 1, 2

detJ =r

Q2∗

(

1− N∗

ν

)

N∗

[

r

((Γ − 2µ)2

2νµγN2

∗ +4µ

νN∗ − 2µ

)

+Γγ

2

]

♥ ssttt♥ ♥ t ♥r tr♠ ♥s t rts ♦t♥

detJ =r

Q2∗

(

1− N∗

ν

)

N∗

[

r(Γ − 2µ)2

2νµγN2

∗ + 2µ√

∆N +γ

2(Γ − 2µ)

]

> 0.

♥ t ♦♥trr② t s♥ ♦ t tr ♦ t ♦♥ ♠tr① t t E∗ s ♥♦t♣rsr ❲♥ J∗

11 < 0

tr J = J∗11 + J∗

22 < 0,

s♦ tt E∗ s ♦② s②♠♣t♦t② st ❲♥ J∗11 > 0 t tr ♥ ♥♦♥♣♦st

♦r ♥♦♥♥t ❲ ♥♠r ♥ tt ♦t ss ♥ ♦ ♦r r♥ts ♦ t ♣r♠trs ♦ t ♠♦

7.2.2 Turing Instability with linear diffusion terms

❲ ♦♥sr t rt♦♥s♦♥ s②st♠ ♥ ♥rs♦♥ tr♠s r s♠♣② t♦ t rt♦♥ ♣rt ♦ ♦r ♥ D1, DP > 0

∂N

∂t−D1∆xN = r

(

1− N

ν

)

N − γNP

B +√∆,

∂P

∂t−DP∆xP =

ΓNP

B +√∆

− µP,

r A, B ♥ ∆ r ♥ ♥ ♥ sts ♦ ♣r♠tr s s tttr J(E∗) < 0 r♠♠r tt s ♣r♠trs ♥ ①st

♦r ♥② ♥♠r k ∈ R t rtrst ♠tr① t t t qr♠E∗ s ♥ ②

M =

[J∗11 −D1k

2 J∗12

J∗21 J∗

22 −DPk2

]

,

122 7 The Turing Instability analysis

♥ s ♥t tr ♥ t

tr M = J∗11 −D1k

2 + J∗22 −DPk

2 = tr J −D1k2 −D2k

2 < 0.

ts tr♠♥♥t s

detM = detJ − (DPJ∗11 +D1J

∗22)k

2 +D1DPk4,

s♦ tt ♥ssr② ♦♥t♦♥ ♦r t r♥ ♥stt② s

D1J∗22 +DPJ

∗11 > 0.

J∗11 < 0 r♠♠r♥ tt J∗

22 < 0 s tt DPJ∗11 +D1J

∗22 < 0 ♥ ts s

t s②st♠ ♦s ♥♦t s♦ ♥② r♥ ♥stt②♥ t ♦♣♣♦st J∗

11 > 0 ♦r ♥② k 6= 0 ♥ st DP r ♥♦ s♦ ttdetJ − DPJ

∗11k

2 < 0 ♥ ♥ D1 > 0 s s♠ ♥♦ detM < 0 ♥ r♥♥stt② ♣♣rs

7.2.3 Turing Instability with cross-diffusion

♥♦ ♦♥sr s②st♠ ♦r t q♥t ♦r♠ tt s

∂N

∂T−D1∆xN = r

(

1− N

ν

)

N − γ

4

(

B −√B2 − 4NP

)

,

∂P

∂T−∆x

(D2

2(√∆− A) +

D3

2(B −

√∆)

)

=Γ

4

(

B −√B2 − 4NP

)

− µP,

r A, B ♥ ∆ r ♥ ② r♦r t rtrst ♠tr① st ♦r♠

M =

[

J∗11 − J∗

∆11k2 J∗

12 − J∗∆12k

2

J∗21 − J∗

∆21k2 J∗

22 − J∗∆22k

2

]

,

r k ∈ R s t ♥♠r ♥ t tr♠s J∆ij, i, j = 1, 2 r ♦t♥ t♥st♦ ♥r③t♦♥ ♦ t s♦♥ tr♠s r♦♥ E∗ ② r ♥ ② t ♦♦♥①♣t ♦r♠s

7.2 The Beddington-DeAngelis 123

J∗∆11 = D1 > 0,

J∗∆12 = 0,

J∗∆21 =

D2 −D3

2

((γ +N∗ − P∗)−

√∆∗√

∆∗

)

= −D2 −D3

Q∗

Γ − 2µ

ΓP∗,

J∗∆22 =

D2

2

((γ −N∗ + P∗) +

√∆∗√

∆∗

)

+D3

2

(√∆∗ − (γ −N∗ + P∗)√

∆∗

)

=D2

Q∗

(

γ + P∗(Γ − 2µ)

Γ

)

+D3

Q∗

2µγ

Γ − 2µ> 0.

❲ ♥♦t tt ♦♥② t s♥ ♦ J2∆21 ♣♥s ♦♥D2, D3 t♦ t ♦♦ ♠♥♥

♦ ts ♣r♠trs r tt s②st♠t② ss♠ tt D2 > D3 s♥sr♥ ♣rt♦rs r ①♣t t♦ ♠♦r s t♥ ♥♥ ♣rt♦rs❲t ts ss♠♣t♦♥ J∗

∆21 < 0

❲ st ♦♥sr sts ♦ ♣r♠tr s s tt tr J(E∗) < 0 ♥ trtrst ♠tr① M s ♥t tr s

tr M = J∗11 −D1k

2 + J∗22 − J∗

∆22k2 = tr J

︸︷︷︸

−

−D1k2 − J∗

∆22︸︷︷︸

+

k2 < 0.

ts tr♠♥♥t s

detM = detJ︸ ︷︷ ︸

+

−(J∗11J

∗∆22 + J∗

22J∗∆11 − J∗

12J∗∆21)k

2 + (D1J∗∆22)

︸ ︷︷ ︸

+

k4,

s♦ tt ♥ssr② ♦♥t♦♥ ♦r t r♥ ♥stt② s

J∗11J

∗∆22 + J∗

22J∗∆11 − J∗

12J∗∆21 > 0.

J∗11 < 0

J∗11

︸︷︷︸

−

J∗∆22︸︷︷︸

+

+ J∗22

︸︷︷︸

−

J∗∆11︸︷︷︸

+

− J∗12

︸︷︷︸

−

J∗∆21︸︷︷︸

−

,

s♦ tt ♦♥t♦♥ ♦s ♥♦t ♦ ♥ s ♥ t s ♦ ♥r s♦♥ ts②st♠ ♦s ♥♦t s♦ r♥ ♥stt② ♥ t ♦♣♣♦st J∗

11 > 0

J∗11

︸︷︷︸

+

J∗∆22︸︷︷︸

+

+ J∗22

︸︷︷︸

−

J∗∆11︸︷︷︸

+

− J∗12

︸︷︷︸

−

J∗∆21︸︷︷︸

−

,

s♦ tt ♦r ♥② k 6= 0 ♥ st D2 r ♥♦ ♥ D3 ∼ D2 s♦ tt detJ −(J∗

11J∗∆22 − J∗

12J∗∆21)k

2 < 0 ♥ ♥ D1 > 0 s s♠ ♥♦ detM < 0 ♥r♥ ♥stt② ♥ ♣♣r

124 7 The Turing Instability analysis

7.2.4 Turing instability regions: linear vs cross diffusion

❲ r tt t rt♦♥ ♦ qt♦♥s s♥ t ♠♥st ♣♣r♦ ♥t ♠r♦s♦♣ ♠♦ ♣r♦s r♦ss s♦♥ tr♠ ♥ t ♣rt♦r qt♦♥rs t ♣r② s♦♥ rt s st ♦♥st♥t ❲ r ♦r rrs ♦♥♥♥t ♦♥sr qt♦♥s

∂N

∂T−D1∆xN = r

(

1− N

ν

)

N − γNP

B +√∆,

∂P

∂T−∆x (f(P,N)P ) =

ΓNP

B +√∆

− µP,

r

A = γ +N − P, B = γ +N + P, ∆ = (γ +N + P )2 − 4NP,

f(P,N) := D22γ

A+√∆

+D32N

B +√∆.

❲ ♥t t♦ ♦♠♣r tr ♥tr strts t♦ ♠♦ t s♦♥ ♥ s②st♠s ♦s ♠♦♥ t ♣rt♦r♣r② ♥trt♦♥s

rst t t rt♦♥ ♣rt ♦ ♥ s♦♥ tr♠s ♥ s ② tt D2 s ♦♥st♥t rt ♦r ♣rt♦rs s ♠♥s tt ①t② tt s♦♥ ♦♥t ♦ sr♥ ♣rt♦rs ♦r ♣rt♦rs ♣♣r♥ ♥ t♠t♥ ♠♦

♦♥② t t rt♦♥ ♣rt ♦ ♥ s♦♥ tr♠s ♥ s ② tt DP = f(P∗, N∗) r f s ♥ ♥ s ♦♥st♥t rt ♦s♦♥ ♥ t qt♦♥ ♦r ♣rt♦rs s ♠♥s tt ♥♦ t ♥t♦ ♦♥tt r♥ ♥ tr♠ ♦ st② ♠♦♥ sr♥ ♥ ♥♥ ♣rt♦rss♥ ♦t s♦♥ rts D2 ♥ D3 r ♣rs♥t ♥ qt♦♥

❲ ♥② ♦♥sr t s♦♥ ♠♦ ♦♠♥ ♦t ♦ t rt♦♥ ②s♥r ♣rtrt♦♥ ♦ ♠r♦s♦♣ s②st♠ s ①♣♥ ♥ t♦♥

♦t rst tt t♥s t♦ t s ♣♦ss t♦ ♦t♥ s♠♣ ①♣rss♦♥ ♦r DP ♥

7.2 The Beddington-DeAngelis 125

DP = f(P∗, N∗) = D22γ

A∗ +√∆∗

+D32N∗

B∗ +√∆∗

= D2γ

γ +N∗ −2µ

ΓP∗

+D3N∗

γ +N∗ + P∗ −2µ

ΓP∗

= D2γ

γ +2µγ

Γ − 2µ

+D3N∗

γ +Γ

2µN∗ −

Γγ

Γ − 2µ+

2µγ

Γ − 2µ

= D2

(

1− 2µ

Γ

)

+D32µ

Γ.

❲ ♥♦t tr♦r tt DP s ♦♥① ♦♠♥t♦♥ ♦ t s♦♥ ♦♥tsD2, D3 ♦rrs♣♦♥♥ t♦ sr♥ ♥ ♥♥ ♣rt♦rs rtr♠♦r ss♠♥tt D3 < D2 r♠♠r tt r♦♠ t ♠♦♥ ♣♦♥t ♦ ♥♥ ♣rt♦rs ♦r s♦♥ rt t♥ sr♥ ♣rt♦rs

DP = D2

(

1− 2µ

Γ

)

+D32µ

Γ= D2 −

D2 −D3

Γ2µ < D2.

rtrst ♠trs ♦ t ss tt ♦♥sr r ♥② ♥ ②

♥r s♦♥ t D2

ML2 =

[

J∗11 −D1k

2 J∗12

J∗21 J∗

22 −D2k2

]

;

♥r s♦♥ t DP ♥ ♥

MLP =

[

J∗11 −D1k

2 J∗12

J∗21 J∗

22 −DPk2

]

;

r♦ss s♦♥

MC =

[

J∗11 −D1k

2 J∗12

J∗21 − J∗

∆21k2 J∗

22 − J∗∆22k

2

]

;

t J∗∆21, J

∗∆22 ♥ ♥ ♥

❲ rst ♥t t♦ ♦♠♣r t r♥ ♥stt② r♦♥s ♦ ss ♥ ♥r s♦♥ tr♠s ♦rrs♣♦♥♥ t♦ ♦♥st♥t s♦♥ ♦♥ts D2 ♥ DP ♦r♣rt♦rs ♥♠② t r♥ ♦r k2 tt s detM < 0 rtrst ♠trsr ♥ r ♦♥ s♦ r♠♠r tt D2 > DP s ♦ t

126 7 The Turing Instability analysis

♠♦♥ ss♠♣t♦♥ ♦♥sr♥ t ♥r ♠tr① ♦ ♥r s♦♥ ♣♥♥♦♥ t ♣r♠tr D > 0

M (D) =

[J∗11 −D1k

2 J∗12

J∗21 J∗

22 −Dk2

]

,

♥ st② t r♥ ♥stt② r♦♥ t rs♣t t♦ D ② ♦♠♣t♥

detM (D) =(J∗11 −D1k

2) (J∗22 −Dk2

)− J∗

12J∗21 =

= DD1k4 − (D1J

∗22 +DJ∗

11) k2 + detJ .

♥trst♥ s r♥ ♥stt② ♣♣r♥ s ♦t♥ ♥r t ♥ssr②♦♥t♦♥ D1J

∗22 +DJ∗

11 > 0 tt s

D > D := −D1J∗22

J∗11

,

r D s trs♦ s♦t♦♥s t♦ t qt♦♥ detM (D) = 0 ♦♥rs ♦ t r♥ ♥stt② r♦♥ ♥ ts r♦♥ ①sts ♥ rtt♥s

sol1,2 =(D1J

∗22 +DJ∗

11)±√

(D1J∗22 +DJ∗

11)2 − 4D1D(J∗

11J∗22 − J∗

12J∗21)

2D1D.

s♦t♦♥s ①st t sr♠♥♥t ♥ s ♥♦♥♥t tt s

(D1J∗22 +DJ∗

11)2 − 4D1D detJ ≥ 0,

s t♦ ♥ ♥qt② ♥ D

(J∗11D)2 − 2D1(J

∗11J

∗22 − 2J∗

12J∗21)D + (D1J

∗22)

2 ≥ 0.

ss♦t qt♦♥ s ♥♦♥♥t sr♠♥♥t tt s

(D1(J∗11J

∗22 − 2J∗

12J∗21))

2 − (D1J∗11J

∗22)

2 = −4D21J

∗12J

∗21 detJE∗

≥ 0,

♥ t qt♦♥ ♠ts t♦ r r♦♦ts

D1,2 =D1(J

∗11J

∗22 − 2J∗

12J∗21)± 2D1

√−J∗

12J∗21 detJE∗

J∗211

=D1

J∗211

[√

detJE∗±√

−J∗12J

∗21

]2

≥ 0,

7.2 The Beddington-DeAngelis 127

r ♦t ♥♦♥♥tt s s♦ ♣♦ss t♦ ♣r♦ tt D1 < D < D2 ♥ t t ♥ s② s♥ tt

D2 > D s q♥t t♦√

−J∗12J

∗21 detJ > − detJ ,

D1 < D s q♥t t♦√

−J∗12J

∗21 detJ > detJ

♦ ts ♦♥t♦♥s r ②s sts ♥ t s ♦ D t♦ r♥♥stt② r D > D2 s ♥ D < D < D2 t q♥tts sol1,2 r ♥♦t r

♥ ♦r D < D ♥♦ r♥ ♥stt② ♥ ♣♣r❲ ♥♦ ♥ ♣r♦r♠ qtt st② ♦ t ♦r ♦ t r♦♦ts sol1,2(D)

♥ r② t ♣r♠tr D > D2 ♥ ♣rtr s♥ r♦♠ s tt

sol1,2(D) > 0 ∀D > D2.

♦r♦r ♥ r♦♠ ♦r♠ t ♥ s② s♥ tt

sol1(D2) = sol2(D2) =D1J

∗22 + D2J

∗11

2D1D2

> 0,

♥ s♦ tt

limD→+∞

sol2(D) =J∗11

D1

> sol2(D2),

♥lim

D→+∞sol1(D) = 0.

rtr♠♦r t♥ t rt ♦ t rs♣t t♦ D ♦t♥

∂

∂Dsol2(D) = − J∗

22

2D2+

1

2

(

J∗

22

D+

J∗

11

D1

)(

− J∗

22

D2

)

+ 2 tJD1 D2

√(

D1

DJ∗22 + J∗

11

)2

− 4D1

DtJ

,

♥

∂

∂Dsol1(D) = − J∗

22

2D2

√(

D1

DJ∗22 + J∗

11

)2

− 4D1

DtJ−

(

D1

DJ∗22 + J∗

11

)

√(

D1

DJ∗22 + J∗

11

)2

− 4D1

DtJ

128 7 The Turing Instability analysis

− tJ

D2

√(

D1

DJ∗22 + J∗

11

)2

− 4D1

DtJ

.

s♦ tt∂

∂Dsol1(D) < 0

∂

∂Dsol2(D) > 0.

s ♠♥s tt t ♦ sol1 s strt② rs♥ t rs♣t t♦ D sol2s strt② ♥rs♥

♥ s tt t r♥ ♥stt② r♦♥ r♦s ♥ D ♥rss ②♦♥D2 s ♦ ts ♦♦s♥ s♦♥ rt s ♦♥② ♦♥ t ♦r ♦ sr♥ ♣rt♦rs ♦ t♦ ♥rt ♦♥s♦♥s ♦t t ♣♦sst② ♦ ♣ttr♥♦r♠t♦♥

❲ ♥♦ s t♦ ♦♠♣r t r♥ ♥stt② r♦♥s ♥ t s ♦ ♥r s♦♥ t ♦♥st♥t s♦♥ rt DP s ♥ ♥ s t rtrst♠tr①

MLP =

[

J∗11 −D1k

2 J∗12

J∗21 J∗

22 −DPk2

]

,

♥ t s ♦ r♦ss s♦♥ tt s ♥ t rtrst ♠tr① s

MC =

[

J∗11 −D1k

2 J∗12

J∗21 − J∗

∆21k2 J∗

22 − J∗∆22k

2

]

.

❲ ♦sr tt

detMLP = D1DPk4 − (DPJ

∗11 +D1J

∗22) k

2 + detJ ,

detMC = D1J∗∆22k

4 − (J∗∆22J

∗11 +D1J

∗22 − J∗

12J∗∆21) k

2 + detJ ,

♥ t r♥ ♥stt② r♦♥s ♥ st♠t t♥s t♦ t st② ♦ t♦str♠♥♥ts ♦t r s♦♥ ♦rr ♣♦②♥♦♠s ♥ k2 t ♣♦st ♥ ♦♥ts rtr♠♦r ♦r k = 0 s tt detMLP = detMC = detJ ❲ ♥tt♦ ♦♠♣r t ♥ ♦♥ts AL, AC ♦♥ ♦♥ ♥ ♥ t ♦♥ts ♦ k2♥♠② BL, BC ♦♥ t ♦tr ♥ ♦s ♦♥ts r ♥ ♥ ts ②

AL := D1DP , BL := DPJ∗11 +D1J

∗22,

AC := D1J∗∆22, BC := J∗

∆22J∗11 +D1J

∗22 − J∗

12J∗∆21.

7.2 The Beddington-DeAngelis 129

sttt♥ t ①♣rss♦♥s ♦ DP ♥ ♥ ♥ J∗∆22 ♥ ♥ tr♠s ♦ D2

♥ D3 ♥ AL, AC , BL, BC s tt

AL = D1

[

D2

(

1− 2µ

Γ

)

+D32µ

Γ

]

,

AC = D1

[

D2

(γ

Q∗

+P∗

Q∗

(

1− 2µ

Γ

))

+D32µγ

Q∗(Γ − 2µ)

]

,

BL =

[

D2

(

1− 2µ

Γ

)

+D32µ

Γ

]

J∗11 +D1J

∗22,

BC =

[

D2

(γ

Q∗

+P∗

Q∗

(

1− 2µ

Γ

))

+D32µγ

Q∗(Γ − 2µ)

]

J∗11 +D1J

∗22 + J∗

12

D2 −D3

Q∗

(

1− 2µ

Γ

)

P∗.

❲ rst ♦♦ t AL ♥ AC ♦♥ts ♦ t ♥ tr♠ ♦ t tr♠♥♥ts♦t AL ♥ AC r ♦♥① ♦♠♥t♦♥s ♦ D2 ♥ D3 s♥

(

1− 2µ

Γ

)

+2µ

Γ= 1,

♥(γ

Q∗

+P∗

Q∗

(

1− 2µ

Γ

))

+2µγ

Q∗(Γ − 2µ)=

1

Q∗

[

γ + P∗

(

1− 2µ

Γ

)

+2µγ

(Γ − 2µ)

]

=1

Q∗

[Γ − 2µ

2µN∗ +

2µγ

(Γ − 2µ)

]

=Q∗

Q∗

= 1.

❲ ♥♦ ♦♠♣r t ♦♥ts ♦ D2 ♥ D3 ♥ t ♦♥① ♦♠♥t♦♥s ♦r D2 t (

γ

Q∗

+P∗

Q∗

(

1− 2µ

Γ

))

> 1− 2µ

Γ⇔ N∗ >

2µγ

Γ − 2µ,

♥ ♦r D3 t

2µ

Γ>

2µγ

Q∗(Γ − 2µ)⇔ N∗ >

2µγ

Γ − 2µ,

♦s t♥s t♦ s ♦♥sq♥ r t♦ ♣r♦ tt DP < J∗∆22

♥

D2

(

1− 2µ

Γ

)

+D32µ

Γ< D2

(γ

Q∗

+P∗

Q∗

(

1− 2µ

Γ

))

+D32µγ

Q∗(Γ − 2µ).

❲ ♥♦ ♣r♦ tt BL > BC ♥ t ♥ rt

130 7 The Turing Instability analysis

BL =

[

D2 − (D2 −D3)2µ

Γ

]

J∗11 +D1J

∗22,

BC =

[

D2 − (D2 −D3)2µγ

Q∗(Γ − 2µ)

]

J∗11 +D1J

∗22 + J∗

12

D2 −D3

Q∗

(

1− 2µ

Γ

)

P∗.

trt♥ r♦♠ ts ①♣rss♦♥s s tt BL > BC ♥ ♦♥②

[

D2 − (D2 −D3)2µ

Γ

]

J∗11 +D1J

∗22 >

[

D2 − (D2 −D3)2µγ

Q∗(Γ − 2µ)

]

J∗11 +D1J

∗22 + J∗

12

D2 −D3

Q∗

(

1− 2µ

Γ

)

P∗.

♥ ♥ ts ♦r♠ ② t ♦♠♠♦♥ strt② ♣♦st t♦r D2−D3 ♥♠t♣② ♦t ss ② Q∗ ❲ ♦t♥

[

−(D2 −D3)

2µ

Γ

]

J∗11Q∗ >

[

−(D2 −D3)

2µγ

Q∗(Γ − 2µ)

]

J∗11Q∗ + J∗

12(D2 −D3)

Q∗

(

1− 2µ

Γ

)

P∗Q∗,

♥ s♥ t ①♣rss♦♥s ♦ J∗11 ♥ J∗

12 ♥ ♣ t

− 2µ

Γ

[

r

(

1− 2

νN∗

)

Q∗ − γΓ − 2µ

2ΓP∗

]

>

− 2µγ

(Γ − 2µ)Q∗

[

r

(

1− 2

νN∗

)

Q∗ − γΓ − 2µ

2ΓP∗

]

− γ2µ

Q∗ΓP∗.

❲ ♥ ①♣♥ t ♣r♦t ♥ t rs ♦t♥

−2µ

Γ

[

r

(

1− 2

νN∗

)

Q∗ − γΓ − 2µ

2ΓP∗

]

> − 2µγ

(Γ − 2µ)r

(

1− 2

νN∗

)

.

♦ ♥ ♦t ss ② 2µ r♥ tr♠s ♥ t s ♥ t

[γ

Γ − 2µ− Q∗

Γ

](

r − 2r

νN∗

)

+γ

Γ

Γ − 2µ

2ΓP∗ > 0.

❯s♥ ♦r♠ ♥ Q∗ ♥ tr♠s ♦ N∗ ♥ ♠♥t♥ t ♦♠♠♦♥ t♦rΓ ♦t♥

7.2 The Beddington-DeAngelis 131

[

γ − Γ − 2µ

2µN∗

](

r − 2r

νN∗

)

+ γΓ − 2µ

2ΓP∗ > 0.

❯s♥ ♥♦ t ①♣rss♦♥ ♦ P∗ ♥ tr♠s ♦ N∗ ♥ ♦r♠ ♦t♥

[

γ − Γ − 2µ

2µN∗

](

r − 2r

νN∗

)

+ γΓ − 2µ

2ΓΓ

2µ

(Γ − 2µ)N∗ − 2γµ

(Γ − 2µ)

> 0.

❯s♥ ♥② t ①♣rss♦♥ ♦ N∗ ♥ ♦r♠ ♦♥② ♥ t s♦♥ N∗ ♥ tqt♦♥ ♦ ♥ ♣ t t ♥qt②

[

γ − Γ − 2µ

2µN∗

](

r − r +γ

2−√

∆N

)

+γ

4µ

(

(Γ − 2µ)N∗ − 2γµ

)

> 0,

s q♥t t♦

γ2

2−

γ

2

Γ − 2µ

2µN∗ −

√

∆N

[

γ − Γ − 2µ

2µN∗

]

+γ

4µ(Γ − 2µ)N∗ −

γ2

2> 0,

♥ ♥ r t♦

−√

∆N

[

γ − Γ − 2µ

2µN∗

]

︸ ︷︷ ︸

−

> 0,

s ②s tr s♥ P∗ > 0

♥② s tt t tr♠♥♥ts ♦ t rtrst ♠trs t ♥r♥ r♦ss s♦♥ rs♣t②

detMLP = ALk4 − BLk

2 + detJ , ♥ detMC = ACk4 − BCk

2 + detJ ,

r s tt AC > AL ♥ BL > BC ♦♦♥ t t r♥ ♥stt② r♦♥s ttr r♦♥s ♥ t tr♠♥♥t ♦ t rtrst ♠tr① s strt② ♥t s tt tr ss ♥tr② ♣♣r

tr r ♥♦ r♦♥s ♦ ♥t tr♠♥♥t ♦r ♦t ♥r ♥ r♦ss s♦♥r t♥ ♥♦ r♥♣ttr♥s ♣♣r ♥ ♦t s②st♠s

t ♥r s♦♥ s s r♥ ♥stt② r♦♥ t t tr♠♥♥t ♦ tr♦ss s♦♥ s s ♣♦st ♦r k r s♦ tt t r♦ss s♦♥s ♦s ♥♦t t♦ r♥ ♥stt②

♦t ss t♦ ♥♦♥♠♣t② r♥ ♥stt② r♦♥s r t ♥ ♣r♦ tt

132 7 The Turing Instability analysis

√

B2L − 4A1 detJ

2AL

>

√

B2C − 4AC detJ

2AC

,

♠♥s tt t r♥ ♥stt② r♦♥ ♦r t r♦ss s♦♥ s sstrt② ♥ ♥ t r♥ ♥stt② r♦♥ ♦ t ♥r s♦♥ s

♥ ss s tt t r♦sss♦♥ ♠♦ s t♦ ♣♦sst② ♦ ♦t♥♥♥♦♥tr ♣ttr♥s s ss ② t♥ ♥ t ♥r s♦♥ ♠♦ r♦rt s ♦ ♠♦ ♥ st♥r s♦♥ tr♠s r rt② t♦ t rt♦♥♣rt ♠② t♦ ♥ ♦rst♠t♦♥ ♦ t st ♦ t ♣r♠trs ♦r ♣ttr♥s♣♣r t rs♣t t♦ t r♦sss♦♥ ♠♦

7.2 The Beddington-DeAngelis 133

det J∗

k2

0

det(MC

k)

det(ML

k)

det J∗

TIRL

k20

det(MC

k)

det(ML

k)

TIRC

TIRL

k20

det(ML

k)

det(MC

k)

det J∗

r♥ ♥stt② r♦♥s ♦r ♥rs♦♥ ♥ r♦sss♦♥ ss r r ♥♦ r♦♥s♦ ♥t tr♠♥♥t ♦r ♦t ♥r ♥ r♦sss♦♥ s♦ tt ♥ ♦t ss r♥ ♥stt② ♥♥♦t ♣♣r ♥rs♦♥ s s r♥ ♥stt② r♦♥ TIRL t t tr♠♥♥t♦ t r♦sss♦♥ s s ♣♦st ♦r k s♦ tt t r♦sss♦♥ s ♦s ♥♦t t♦ r♥♥stt② ♦t ss t♦ ♥♦♥♠♣t② r♥ ♥stt② r♦♥s t t r♥ ♥stt② r♦♥ ♦rt r♦sss♦♥ TIRC s s strt② ♥ ♥ t r♥ ♥stt② r♦♥ ♦ t ♥rs♦♥s TIRL

8

Concluding remarks

s Prt ♦ss ♦♥ t st② ♦ t♦ ♠r♦s♦♣ ♥ tr♠s ♦ t♠ ss ♣rt♦r♣r② ♠♦s t s♦♥ tt st② ♥ st ♠t t♦ ss t②♣s ♦♥t♦♥ rs♣♦♥ss ♥ t rt♦♥ ♣rt ♥ ♣rs♥t r♦sss♦♥ tr♠ ❲ s♦ ♣rs♥t r♦r♦s rst ♦ ♦♥r♥ ♦ t s♦t♦♥s ♦ ts s②st♠ t♦rst s♦t♦♥ ♦ t rt♦♥r♦ss s♦♥ s②st♠

♥ t t♦ tr♦♣ s r ♦♥sr ♣r② ♥ ♣rt♦rs r rtr ♥t♦ sr♥ ♣rt♦rs ♥ ♥♥ ♣rt♦rs ♦r♠r r ♣rt♦rst ♥ t ♣rt♦♥ ♣r♦ss t ttr r rst♥ ♥s ♥ strtr♦♠ s②st♠ ♦ tr ♣rt r♥t qt♦♥s t st♥r ♥r s♦♥♥ tr♠s ♦ ♣♥ ♥ t ♦t❱♦trr rt♦♥ tr♠ r♦ qsst②stt ♣♣r♦①♠t♦♥ ♥ ♣ t s②st♠ ♦ t♦ Ps t ♣r② ♥t♦t ♣rt♦r ♥sts s ♥♥♦♥s ♥ ♦♥t②♣ ♥t♦♥ rs♣♦♥s♣♣rs t♦tr t r♦sss♦♥ tr♠ ♥ t ♣rt♦r qt♦♥ s ♠♥stt t s♦♥ tr♠ rt t♦ ♣rt♦rs s ♠ ♠♦r ♦♠♣t t♥ ♦♥st♥t t♠s ♣♥ ♦ P ♥r s tr♠ ♥ s♠♣② t♦ trt♦♥ ♣rt ❬❪ ♥ ♣rtr t s♦♥ tr♠ ♦t♥ ② t t♠s r♠♥t ♣♥s ♦♥ t ♣r② ♦♠ss ♥ ♦♥ ♦t t s♦♥ ♦♥t ♦ sr♥♥ ♥♥ ♣rt♦rs d2 ♥ d3 ♦♦♥ t ts ①♣rss♦♥ t r♦sss♦♥ tr♠rs t ♣rt♦r s♦♥ ♥ t ♣r② ♥st② ♥rss

♥ ♠♦② t strt♥ ♠♦ ♥srt♥ ♦♠♣tt♦♥ ♠♦♥ ♣rt♦rs❲t ts ♥ ♥ ♣ tr qs st②stt ♣♣r♦①♠t♦♥ t s②st♠♦ t♦ Ps ♦r ♣r② ♥ t♦t ♣rt♦r ♥sts rtr③ ② ♥t♦♥♥s ♥t♦♥ rs♣♦♥s ♥ r♦sss♦♥ tr♠ ♥ t ♣rt♦r qt♦♥

❲ ♥♦t tt t ♠t♥ s②st♠ ♣rs♥ts ♥t♦♥ rs♣♦♥s ♦s t♦ t♥t♦♥♥s ♥ t rt♦♥ ♣rt ♥ ts s♥s r ts t②♣

136 8 Concluding remarks

♦ ♥t♦♥ rs♣♦♥s ② t♠s r♠♥t r♦♠ ♠♦r ♦♠♣t ♠♦s t②♣ ♦ ♥t♦♥ rs♣♦♥s s r s♦ ② s♠♥ t ❬❪ ♥ t ♦♥t①t ♦ ♦r♥r② r♥t qt♦♥s strt♥ r♦♠ s②st♠ ♦ ♦r s ② qsst②stt ♣♣r♦①♠t♦♥ ❲t tr ♣♣r♦ s♦ t ♦st r♦t ♥t ♣rt♦r tr♥♦r tr♠s st ♣♥ ♦♥ ts ♦♠♣① ①♣rss♦♥ rtr t② s♠♣ ts ♦♠♣t qrt ①♣rss♦♥ t Pé ♣♣r♦①♠t♦♥ ♥t② r♦r t st♥r ♦r♠ ♦ t ♥t♦♥♥s ♥t♦♥ rs♣♦♥s

s♦ ♥ ts s t ♠t♥ s②st♠ ♣rs♥ts r♦sss♦♥ ♥ t ♣rt♦rqt♦♥ ♣♥s ♦♥ ♦t t s♦♥ ♦♥ts ♦ sr♥ ♥ ♥♥♣rt♦rs d2 ♥ d3 t ♣r② s♦♥ s st ♥r

r♥ ♥stt② ♥②ss ♦ t r ♠♦s s st ♥ ♣tr ♦rt rst ♦♥ t s ♥♦♥ tt ♣rt♦r♣r② ♠♦s t ♣r②♣♥♥t tr♦♣♥t♦♥ ♥ t rt♦♥ ♣rt ♥ st♥r ♥rs♦♥ ♥ ♥♦t rs t♦r♥ ♥stt② ❬❪ ♥ t t r♦sss♦♥ ♠♦ ♥♦ ♣ttr♥s s♠ t♦♣♣r ♥r ♦♦② rs♦♥ ss♠♣t♦♥ ♦♥ t s♦♥ ♦♥ts♦r t s♦♥ s②st♠ ♥ t ♥t♦♥ rs♣♦♥s s ♥t♦♥♥s ♦♦ ♦r ♦♥t♦♥s ♦♥ t ♣r♠tr s t♦ r♥ ♥stt②♥ ♦♠♣r ts r♥ ♥stt② r♦♥s t t ♦♥s ♦t♥ ♥ tr♦sss♦♥ tr♠ s ssttt ② ♥r s♦♥ ♠♥ ♣♦♥t s t ttt t r♥ ♥stt② r♦♥ ss♦t t♦ t r♦sss♦♥ s②st♠ s ②sstrt② ♥ ♥ t r♥ ♥stt② r♦♥ ♦ t ♥rs♦♥ s②st♠ s ♦♥sq♥ t s ♦ rt♦♥s♦♥ s②st♠s ♦r ♣rt♦r♣r② ♥trt♦♥s ♦♥t♦♥♥s t②♣ ♥ st♥r s♦♥ s s♠♣② t♦ t rt♦♥ tr♠s ♠② t♦ ♥ ♦rst♠t♦♥ ♦ t ♣♦sst② ♦ ♣♣r♥ ♦ ♣ttr♥s♥ t s ♥ t ♥t♦♥♥s ♥t♦♥ rs♣♦♥s s ♦♥sq♥ ♦t ♥trt♦♥s t♥ sr♥ ♥ ♥♥ ♣rt♦rs

t s ♦rt t♦ ♠♥t♦♥ tt ♥ ♠♥② ♥st♥s t ♥tr♦t♦♥ ♦ r♦sss♦♥tr♠s ♥st ♦ st♥r ♥rs♦♥ tr♠s s ①t② t♦ t ♦♣♣♦st rsttt s t ♥rs ♦ t st ♦ ♣r♠tr s ♥ ♣ttr♥s ♦♣ ❬ ❪ r st② s t♥ t♦ rtr ♥trst♥ ♦♥s♦♥ ♣ttr♥ ♦r♠t♦♥♦r♥t♥ r♦♠ r♥ ♥stt② s ♦♥trt ② t r♦sss♦♥ tr♠ r② t ♥ t♦ t ①st♥ trtr s♠s t♦ sst tt r♦sss♦♥♥♥s ♣ttr♥ ♦r♠t♦♥ s rss t qst♦♥ ♦ t ssst♦♥ ♦ r♦sss♦♥ t rs♣t t♦ tr ♣r♦♣rts ♦ ♥♥♥ s♦♥r♥ ♥stt② ♦ s ♥ ♥trst♥ t ts ♣♦♥t ♥ ♦r ♣♣r♦ ♣r♦s ♠♥st rt♦♥ ♦ t r♦sss♦♥ tr♠ t ♦t♥ ♠♦s r s ♦♥♣♥♦♠♥♦♦ ss♠♣t♦♥s ♥ t♥ ♣r♦r ♥♦ ♦t t ts ♥ s s ss ♥s r② rs♦♥ ♦ t trtr ♦♥ ts t♦♣ s rst

8 Concluding remarks 137

tt♠♣t t ①♣rss♦♥ ♦♥ t ♣♥ ts s t ② ♦ ts r♥ r♦sss♦♥ tr♠ ♣r♦ ② ♠♥st rt♦♥ rs ♥ rt♥ s♥st s♦♥ ♦♥t ♦ ♣rt♦rs ♦r ♦♦♥ t t ♦t❱♦trr ♠♦st ♥ ❬❪ r♥t t②♣ ♦ r♦sss♦♥ s ♦♥sr ♥rss ts♦♥ ♦ ♣r② ♥ ♣rt♦rs t♦rs ♣r♦ tt ts t②♣ ♦ r♦sss♦♥♥♥s t r♥ ♥stt②

t s ♦rt ♥♦t♥ tt ♥ t q♦t ♠♦ s♠r r♦sss♦♥ tr♠ s♦♥sr s♦ ♥ t ♣r② qt♦♥ ❲t ♦r ♣♣r♦ t s ♣♦ss t♦ ♦t♥ ♣rt♦r♣r② rt♦♥r♦ss s♦♥ t r♦sss♦♥ s♦ ♥ t ♣r② qt♦♥ s♥ t ♠♥st rt♦♥ ♣r♦♣♦s ② rt③ ♥ ②♥r ❬❪ ♥ t♦♥t①t ♦ ♦r♥r② r♥t qt♦♥s ♥ t ♥ t♦♥ t♦ t s♦♥ ♦ t♣rt♦r ♣♦♣t♦♥ ♥t♦ srrs ♥ ♥rs t ♣r② ♣♦♣t♦♥ s ♥t ♣r② t②♣② ♦r♥ ♥ ♣r♦♥ t♦ ♣rt♦♥ ♥ ♥♥r ♣r② ♥s ♦ ♦♥ r t♥ t♦ r♥t t♠ ss r ①♣♦t t♦r t ♦r♠♥s♦♥ s②st♠ t♦ s②st♠ ♦ t♦ qt♦♥s t r♦sss♦♥tr♠s ♥ ♥t♦♥♥s ♥t♦♥ rs♣♦♥s st② ♦ t ♥♥ ♦ts ♥ r♦sss♦♥ tr♠ ♦ s t♦ ♣r♦r♠ t sst sst♦♥♥ t s ♣♥♥ s tr ♦r

Appendix

A Very-weak formulation

❲ r② r t ♥♦t♦♥ ♦ r② s♦t♦♥s ♥ ♦ t ♦♥r② ♦♥t♦♥s rrt ② t ♦ ♦ tst ♥t♦♥s ♦r ts ♣r♣♦s ♦♥sr t ♦♦♥s②st♠ ♥ t str♦♥ ♦r♠t♦♥

∂tN − d1∆xN = f(N,P ) := rN

(

1− N

K

)

− γαNP

αN + γ, in R+ ×Ω

∂tP −∆x (µ(N)P ) = g(N,P ) :=ΓαNP

αN + γ− µP, in R+ ×Ω

∇xN · n|∂Ω = 0, ∇xP · n|∂Ω = 0, in R+ × ∂Ω

N(0, ·) = Nin, P (0, ·) = Pin, in Ω.

r N = N(t, x) ≥ 0, P = P (t, x) ≥ 0 r t ♥♥♦♥s t r (t, x) ∈R+ × Ω t Ω ♦♥ ♦♠♥ ♦ Rd, (d = 1, 2, 3) t ♥t♦♥ µ(N) ♥ ts♦♥ qt♦♥ s

µ(N) :=d2γ + d3αN

αN + γ,

n = n(x) st♥s ♦r t ♦tr ♥♦r♠ t ♣♦♥t x ♦ t ♦♥r② ∂Ω Nin ♥Pin r ♥♦♥♥t ♥t t ♥ t r♠♥♥ tr♠s r ♥♦♥♥t ♦♥st♥t♣r♠trs

♠r ♦♠♦♥♦s ♠♥♥ ♦♥r② ♦♥t♦♥s ♥ t ① ♦r♠t♦♥ rq♥t t♦ ♥ t

140 8 Concluding remarks

∇xN · n|∂Ω = 0,

∇xP · n|∂Ω = 0,⇔

∇xN · n|∂Ω = 0,

∇x (µ(N)P ) · n|∂Ω = 0.

♥

∇x (µ(N)P ) · n|∂Ω = [P∇xµ(N) + µ(N)∇xP ] · n|∂Ω

= [Pµ′(N)∇xN + µ(N)∇xP ] · n|∂Ω

= Pµ′(N)∇xN · n|∂Ω + µ(N)∇xP · n|∂Ω .

♣♣♦s N, P s♠♦♦t ♥ ♦rr t♦ ♦t♥ t r② ♦r♠t♦♥ ♦ t s②st♠ ♦♥sr φ, ψ tst ♥t♦♥s s tt

• ϕ, ψ r s♠♦♦t ϕ, ψ ∈ C2C([0,+∞)× Ω)

❲ r ♦♥sr♥ ♥t♦♥s tt r C2 ♥ [0,+∞) × Ω ♣ t♦ t ♦♥r②♥ t ♦♠♣t s♣♣♦rt ③r♦ ♦r t→ +∞

• ϕ, ψ sts② ♦♠♦♥♦s ♠♥♥ ♦♥r② ♦♥t♦♥s

∇xϕ · n|∂Ω = 0, ∇xψ · n|∂Ω = 0.

❲ ♠t♣② qt♦♥ ♦r tst ♥t♦♥ ♥ ♥trt ♦r t qt♦♥ ♦t♥

∫ ∞

0

∫

Ω

(∂tN − d1∆xN)ϕ =

∫ ∞

0

∫

Ω

f(N,P )ϕ, ∀ϕ.

♥trt♥ ② ♣ts t s ♥ r♠♠r♥ tt ϕ s ♦♠♣t s♣♣♦rt ∫ ∞

0

∫

Ω

(∂tN − d1∆xN)ϕ = −∫ ∞

0

∫

Ω

N∂tϕ−∫

Ω

ϕ(0, ·)Nin

+ d1

∫ ∞

0

∫

Ω

∇xϕ∇xN − d1

∫ ∞

0

∫

∂Ω

ϕ∇xN · n, ∀ϕ.

st tr♠ ♥ss t♦ t ♦♠♦♥♦s ♠♥♥ ♦♥r② ♦♥t♦♥s ♥trt ② ♣rts t tr tr♠

∫ ∞

0

∫

Ω

∇xϕ∇xN = −∫ ∞

0

∫

Ω

N∆xϕ+

∫ ∞

0

∫

∂Ω

N∇xϕ · n,

r t s♦♥ tr♠ ♥ ♥ss t♦ t ♣r♦♣rts ♦ t tst ♥t♦♥s ❲♦t♥ t r② ♦r♠t♦♥ ♦ t rst qt♦♥

A Very-weak formulation 141

−∫ ∞

0

∫

Ω

N∂tϕ−∫

Ω

ϕ(0, ·)Nin − d1

∫ ∞

0

∫

Ω

N∆xϕ =

∫ ∞

0

∫

Ω

f(N,P )ϕ, ∀ϕ.

♥♦♦s② ♦r t s♦♥ qt♦♥

∫ ∞

0

∫

Ω

(∂tP −∆x(µ(N)P ))ψ =

∫ ∞

0

∫

Ω

g(N,P )ψ, ∀ψ.

♥trt♥ ② ♣rts t s ♥ r♠♠r♥ tt ψ s ♦♠♣t s♣♣♦rt ∫ ∞

0

∫

Ω

(∂tP −∆x(µ(N)P ))ψ = −∫ ∞

0

∫

Ω

P∂tψ −∫

Ω

ψ(0, ·)Pin

+

∫ ∞

0

∫

Ω

∇xψ∇x(µ(N)P )−∫ ∞

0

∫

∂Ω

ψ∇x(µ(N)P ) · n, ∀ψ.

st tr♠ ♥ss t♦ t ♦♠♦♥♦s ♠♥♥ ♦♥r② ♦♥t♦♥s ♥trt ② ♣rts t tr tr♠∫ ∞

0

∫

Ω

∇xψ∇x(µ(N)P ) = −∫ ∞

0

∫

Ω

µ(N)P∆xψ +

∫ ∞

0

∫

∂Ω

µ(N)P∇xψ · n, ∀ψ,

r t s♦♥ tr♠ ♥ ♥ss t♦ t ♣r♦♣rts ♦ t tst ♥t♦♥s ❲♦t♥ t r② ♦r♠t♦♥ ♦ t s♦♥ qt♦♥

−∫ ∞

0

∫

Ω

P∂tψ −∫

Ω

ψ(0, ·)Pin −∫ ∞

0

∫

Ω

µ(N)P∆xψ =

∫ ∞

0

∫

Ω

g(N,P )ψ, ∀ψ.

♥ t r② ♦r♠t♦♥ ♦ s②st♠ s

−∫ ∞

0

∫

Ω

N∂tϕ−∫

Ω

ϕ(0, ·)N0 − d1

∫ ∞

0

∫

Ω

N∆xϕ =

∫ ∞

0

∫

Ω

f(N,P )ϕ

−∫ ∞

0

∫

Ω

P∂tψ −∫

Ω

ψ(0, ·)P0 −∫ ∞

0

∫

Ω

µ(N)P∆xψ =

∫ ∞

0

∫

Ω

g(N,P )ψ

∀ϕ, ψ ∈ C2C([0,+∞)× Ω),∇xϕ · n|∂Ω = 0, ∇xψ · n|∂Ω = 0.

♦ ♥t t♦ ♣r♦ t ♦r♥ ♦ t r② ♦r♠t♦♥rst ♣ ϕ, ψ ∈ C2

C(]0,+∞) × Ω) ❲ ♦♥sr t s②st♠ ♥ ♥trt r ② ♣rts ❲ t ♦r tst ♥t♦♥s ϕ, ψ

∫ ∞

0

∫

Ω

(∂tN − d1∆xN)ϕ =

∫ ∞

0

∫

Ω

f(N,P )ϕ,

142 8 Concluding remarks

∫ ∞

0

∫

Ω

(∂tP −∆x(µ(N)P ))ψ =

∫ ∞

0

∫

Ω

g(N,P )ψ, ∀ϕ, ψ.

t♥r t♦r♠ ♦ strt♦♥s s t♦ t s②st♠ ♥ t str♦♥ ♦r♠t♦♥ ♦ tr② t♦ t t ♥t t ♦t ♦ t r② ♦r♠t♦♥❲ ♣ ϕ, ψ ∈ C2

C([0,+∞)×Ω)r♦♠ tt s

−∫ ∞

0

∫

Ω

N∂tϕ−∫

Ω

ϕ(0, ·)Nin − d1

∫ ∞

0

∫

Ω

N∆xϕ =

∫ ∞

0

∫

Ω

f(N,P )ϕ,

♥trt♥ r ② ♣rts t rst ♥ t tr tr♠s

−∫ ∞

0

∫

Ω

N∂tϕ =

∫ ∞

0

∫

Ω

ϕ∂tN +

∫

Ω

N(0, ·)ϕ(0, ·),∫ ∞

0

∫

Ω

N∆xϕ =

∫ ∞

0

∫

Ω

ϕ∆xN,

♦t♥

−∫ ∞

0

∫

Ω

N∂tϕ−∫

Ω

(N(0, ·)−Nin)ϕ(0, ·)− d1

∫ ∞

0

∫

Ω

N∆xϕ =

∫ ∞

0

∫

Ω

f(N,P )ϕ.

t N stss t qt♦♥ ♦ t str♦♥ ♦r♠t♦♥ tt s t♦

∫ ∞

0

∫

Ω

(∂tN − d1∆xN − f(N,P ))ϕ = 0,

t♥ ∫

Ω

(N(0, ·)−Nin)ϕ(0, ·) = 0, ∀ϕ.

❲ ♥t t♦ ♣r♦ tt t st qt♦♥ s ts q♥t t♦ N(0, ·) = Nin Φ ∈ C2

c (Ω) ♥ ♥ ♥ ϕ(t, x) = Φ(x)χ(t) r χ ∈ C∞(R) s tt

χ(t) =

1, 0 ≤ t ≤ 1

h(t), 1 < t < 2

0, t ≥ 2,

t h s st s♠♦♦t ♥t♦♥ ♠s χ ∈ C∞(R) h′(1) = h′(2) = 0♥ ϕ ∈ C2

c ([0,+∞)×Ω) ♥ ϕ(0, ·) = Φ s ♠♣s tt

∀Φ ∈ C2c (Ω)

∫

Ω

(N(0, ·)−Nin)Φ = 0,

A Very-weak formulation 143

t♥ N(0, ·) = Nins ♣r♦r ♥ ♣♣ s♦ ♦r t qt♦♥ ♥ ♦rr t♦ r♦r

P (0, ·) = Pin♥ ♥♦ tr② t♦ t t ♠♥♥ ♦♥r② ♦♥t♦♥s ♦t ♦ t r②

♦r♠t♦♥ ❲ ♣ tst ♥t♦♥s

ϕ, ψ ∈ C2C(]0,+∞)× Ω), ∇xϕ · n|∂Ω = 0, ∇xψ · n|∂Ω = 0.

s ♠♣s tt ϕ(0, ·) = 0, ψ(0, ·) = 0 ❲t ts ♦ t r② ♦r♠t♦♥♦♠s

−∫ ∞

0

∫

Ω

N∂tϕ− d1

∫ ∞

0

∫

Ω

N∆xϕ =

∫ ∞

0

∫

Ω

f(N,P )ϕ

−∫ ∞

0

∫

Ω

P∂tψ −∫ ∞

0

∫

Ω

µ(N)P∆xψ =

∫ ∞

0

∫

Ω

g(N,P )ψ

♥trt♥ ② ♣rt t rst qt♦♥

∫ ∞

0

∫

Ω

∂tNϕ+ d1

∫ ∞

0

∫

Ω

∇xN∇xϕ− d1

∫ ∞

0

∫

∂Ω

N∇xϕ · n =

∫ ∞

0

∫

Ω

f(N,P )ϕ

tr tr♠ ♥ t s ♥ss t♦ t ♣r♦♣rts ♦ t tst ♥t♦♥s♥trt♥ t s♦♥ tr♠ ② ♣rts

∫ ∞

0

∫

Ω

∂tNϕ− d1

∫ ∞

0

∫

Ω

ϕ∆xN + d1

∫ ∞

0

∫

∂Ω

ϕ∇xN · n =

∫ ∞

0

∫

Ω

f(N,P )ϕ

♥ rrtt♥ s∫ ∞

0

∫

Ω

(∂tN − d1∆xN − f(N,P ))ϕ+ d1

∫ ∞

0

∫

∂Ω

ϕ∇xN · n = 0,

♥ s♥ N s str♦♥ s♦t♦♥ ♦t♥

∫ ∞

0

∫

∂Ω

ϕ∇xN · n = 0, ∀ϕ.

❲ ♥t t♦ r♦r tt ∇xN · n = 0❲ ♦♥sr t s♠♣ s Ω = R× R+

∗ ♥ (x1, x2) ∈ Ω ♥ ∂Ω = R× 0♥ n = (0,±1) qt♦♥ ♦♠s

∫ ∞

0

∫

R

ϕ(t, x1, 0)∂N

∂x2(t, x1, 0)dx1dt = 0, ∀ϕ.

144 8 Concluding remarks

❲ ♥t t♦ ♣r♦ tt Φ(t, x1) ♥ ♥ Φ(t, x1) ∈ C2c (]0,+∞) × R) tr

①sts ϕ ∈ C2c (]0,+∞)× R× R+

∗ ) s tt

Φ(t, x1) = ϕ(t, x1, 0),∂ϕ

∂x2(t, x1, 0) = 0.

❲ ♥ ϕ(t, x1, x2) := Φ(t, x1)χ(x2) r χ ∈ C∞(R) s tt

χ(x2) =

1, 0 ≤ x2 ≤ 1

h(x2), 1 < x2 < 2

0, x2 ≥ 2,

t h s st s♠♦♦t ♥t♦♥ s ♥ t ♣r♦s s ♥

∂ϕ

∂x2(t, x1, x2) = Φ(t, x1)χ

′(x2)

s♦ tt∂ϕ

∂x2(t, x1, 0) = Φ(t, x1)χ

′(0) = 0

♥ϕ(t, x1, 0) = Φ(t, x1)χ(0) = Φ(t, x1).

s ♠♣s tt∂N

∂x2= 0.

s r♠♥ts ♥ ♥r③ t♦ ♥r ♦♠♥ Ω

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P r♠s ♥ ♥③r ♥tr ♦ ♣rt♦♥ ♣r② ♣♥♥t rt♦ ♣♥♥t ♦r♥tr r♥s ♥ ♦♦② ♦t♦♥ ♣♣

♠s♦♥ ♥ ❨ ♦r♦③♦❲♥ ♥ trst ♦r ♠♦ ♣rt♦♥s ❯♥rt♥ strtrs♥stt② ♥ ♦♦ s②st♠s Pr♦♥s ♦ t ♦② ♦t② ♦ ♦♥♦♥ t♠t P②s♥ ♥♥r♥ ♥s ♣

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P rr ♥r ss ♦ ♣rt♦♥ ♠♦s t ♠t♣t t ♦♥♥r ②♥♠s ♣♣

P rr ♦rs ♥ ♦♥③á③rs rt♦♥s ♥ ♦ ②♥♠s ♥ ♣rt♦r♣r② ♠♦ t str♦♥ t ♦♥ t ♣r② ♥ rt♦♣♥♥t ♥t♦♥ rs♣♦♥s♦♥♥r ♥②ss ❲♦r ♣♣t♦♥s ♣♣

P rr ♦♥③á③rs ♥ á③ r ♠t ②s ♥ s♦r ♣rt♦r♣r② ♠♦ t t t ♦r♥ ♦♥ ♣♣ t♠ts ♣♣

♦ ♠t ②s ♥ s♦r ♣rt♦r♣r② ♠♦ t t t ♦♥♥r♥②ss ❲♦r ♣♣t♦♥s ♣♣

❲ ♥♠ rt♦♥s t② ♥ ♥r ♦♦♦② ❯♥rst② ♦ ♦ Prss♦

❲ Pr ♠rs♦♥ Pr P ♠t t Pr♥♣s ♦ ♥♠♦♦② ❲ ♥r ♦ t P♣

♦♥s♦ rt♠s ♥ t♥ t ♥trr♥ t♥ ♣rt♦rs ♥ rst♦ r♥ s♣t ♣ttr♥s ♦♦② ♣♣

rt ♥ ♥③r ♦♣♥ ♥ ♣rt♦r♣r② ②♥♠s rt♦♣♥♥ ♦r♥ ♦♦rt ♦♦② ♣♣

❱ r♥♦ tstr♦♣ ♦r② ♣r♥r❱r r♥ ③②♥ ♦♥♥r ②♥♠s ♦ ♥trt♥ P♦♣t♦♥s ♦ ❲♦r ♥t ♥♣♦r

♥t♦♥ t ♥trr♥ t♥ ♣rsts ♦r ♣rt♦rs ♥ ts t ♦♥ sr♥

♥② ♦r♥ ♦ ♥♠ ♦♦② ♣♣ r③♦s② r ♥ rt Pr♠tr ♥②ss ♦ t rt♦♣♥♥t ♣rt♦r

♣r② ♠♦ ♦r♥ ♦ t♠t ♦♦② ♣♣ rr②♠♥ ♦r♥s ♥ ♦t♦♥ ♦ ♣rt♦r♣r② t♦r② ♦♦② ♣♣

♦t ♥ Prr ♥st♥t♥♦s ♠t ♦r rt♦♥s♦♥ s②st♠s t st rr

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♦ ♥ r ♦♥ ♠t♥♥ ts ♥ ♣♦♣t♦♥s ②♥♠s t♣♣t♦♥s ♥ ♣st ♦♥tr♦ P♦♣t♦♥ ♦♦② ♣♣

♦ ❲ s ♥ r ♦ ♣rt♦r ♥t♦♥ rs♣♦♥ss ♥ ts♥ ♣r② t t ♣r♦① ♦ ♥r♠♥t ♥ ♣♦♣t♦♥ ♦♣ss ♦rt P♦♣t♦♥ ♦♦② ♣♣

r♥ stts ♥ ♥r ♠♦♦t♥ss ♦ ♠♦♠♥ts ♦ t s♦t♦♥s ♦ srt♦t♦♥ qt♦♥s t s♦♥ r❳ ♣r♣r♥t r❳

♦♥ P ss♥r ♥ r♦♣♣ ♦♥ ♦ ♣rt♦r♣r② tr♦♣ ♥trt♦♥sPrt tr tr♦♣ s ♦r♥ ♦ t♠t ♦♦② ♣♣

♦♥ P ss♥r r♦♣♣ ♥ r ♦♥ ♦ ♣rt♦r♣r② tr♦♣♥trt♦♥s Prt t♦ tr♦♣ s ♦r♥ ♦ t♠t ♦♦② ♣♣

♦♥ ♥ ♦ ♠♣ ♣r♠tr ♠♦ ♦r r♥ ♣rt♦r♣r② ♣♦♣t♦♥ ♥trt♦♥s ♦♦ ♦♥ ♣♣

♦♥ r♦♣♣ ♥ ♦rs♥ ts ♦ ♣r② ♦r♥rr♦♥ ♥ ♣rt♦r♣r② s②st♠s t ♣r②♣♥♥t tr♦♣ ♥t♦♥s ♦♥♥r ♥②ss ❲♦r ♣♣t♦♥s ♣♣

②♥♠s ♦ ♣rt♦r♣r② ♠♦s t str♦♥ t ♦♥ t ♣r② ♥ ♣rt♦r♣♥♥ttr♦♣ ♥t♦♥s ♦♥♥r ♥②ss ❲♦r ♣♣t♦♥s ♣♣

ñ③♦ stts ♥ ♥r ♠♣r♦ t② st♠ts ♥ ♣♣t♦♥s t♦rt♦♥s♦♥ qt♦♥s ♦♠♠♥t♦♥s ♥ Prt r♥t qt♦♥s ♣♣

♦♥♦rt♦ ♥ stts ♦r♦s ♣ss t♦ t ♠t ♥ s②st♠ ♦ rt♦♥s♦♥qt♦♥s t♦rs s②st♠ ♥♥ r♦ss s♦♥ ♦♠♠♥t♦♥s ♥ t♠t ♥s ♣♣

♥s ♦st♥ ♥ ♠♦ ♦r tr♦♣ ♥trt♦♥ ♦♦② ♣♣

♥♥s ts ♣♦♣t♦♥ r♦t rt ♥st② ♥ t ♥ ♦ ①t♥t♦♥ trs♦r ♦♥ ♣♣

stts ♦t ♥tr♦♣② ♠t♦s ♦r rt♦♥s♦♥ qt♦♥s st t♠t❯♥rstà Pr♠ ♣♣

stts ♥r Prr ♥ ❱♦ ♦ ①st♥ ♦r qrt s②st♠s♦ rt♦♥s♦♥ ♥ ♦♥♥r ts ♣♣

stts ♥ rsss rsts ♦r tr♥r rt♦♥ r♦ss s♦♥ s②st♠♦r♥ ♦ t♠t ♥②ss ♥ ♣♣t♦♥s ♣♣

♦♦ ❲ ♦rts ♥ ❨ ③♥ts♦ t♦♥t ♣ ♦r ♥♠rrt♦♥ ♥②ss ♦ s r♥st♦♥s ♦♥ t♠t ♦tr ♣♣ ♦tr r♦♠ tt♣♠t♦♥t♥t

rrtt ♥ ♥ ♠♣♦rt♥ ♦ ♥ srt ♥ s♣t ♦rt P♦♣t♦♥♦♦② ♣♣

♥ ♥ ♥ Ptt r♥ ♥stts ♥ rt♦♥s♦♥ s②st♠s tr♦ss s♦♥ r♦♣♥ P②s ♦r♥ ♣♣

♦rs ♥ ♦♥③á③rs ②♥♠s ♦ ♣rt♦r♣r② ♠♦ t t ♦♥♣r② ♥ rt♦♣♥♥t ♥t♦♥ rs♣♦♥s ♦♦ ♦♠♣①t② ♣♣

r♠♥ ♥ ♦ ♦ stt② ♥ ♣rsst♥ ♦ s♠♣ ♦♦ ♥s t♠t♦s♥s ♣♣

♠♥♦ ♦♠r♦ ♥ ♠♠rt♥♦ r♥ ♥stt② ♥ tr♥ r♦♥ts ♦r ♥♦♥♥r rt♦♥s♦♥ s②st♠ t r♦sss♦♥ t♠ts ♥ ♦♠♣trs ♥ ♠t♦♥ ♣♣

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❨ ♦ ♥ ②♥♠s ♦ rt♦♣♥♥t ♣rt♦r♣r② s②st♠ t str♦♥ tsrt ♦♥t♥♦s ②♥♠ ②st♠s rs ♣♣

rt③ ♥ ②♥r ♠♥st rt♦♥ ♦ t ♥s♥t♦♥ ♥t♦♥rs♣♦♥s ♦r♥ ♦ t♦rt ♦♦② ♣♣

♣♥ ♠♦ ♦ t ♣rt♦r♣r② rt♦♥s♣ ♦rt P♦♣t♦♥ ♦♦② ♣♣

♣♥ ♥ ♦é ♥♠♠ ♣♦♣t♦♥s ♣r♦sss ♦ s♣s ①t♥t♦♥ ♥ ♦é ♦♥srt♦♥ ♦♦② t ♥ ♦ rt② ♥ rst② ♥r ss♦ts ♥r♥ ssstts ♣♣

♥③r ♥ ç② ♦♥sq♥s ♦ rt♦♣♥♥t ♣rt♦♥ ♦r st②stt♣r♦♣rts ♦ ♦s②st♠s ♦♦② ♣♣

♦♠♣rt③ ♥ t ♥tr ♦ t ♥t♦♥ ①♣rss ♦ t ♦ ♠♥ ♠♦rtt② ♥ ♦♥ ♥♠♦ ♦ tr♠♥♥ t ♦ ♦♥t♥♥s P♦s♦♣ r♥st♦♥s ♦ t ♦② ♦t②♦ ♦♥♦♥ ♣♣

♦♥③á③rs ♥♦r ♦sP♠ ♥ ♦rs ②♥♠ ♦♠♣①ts ♥ t s♦r ♣rt♦r♣r② ♠♦ s ♦♥sq♥s ♦ t t ♦♥ ♣r② ♣♣t♠t ♦♥ ♣♣

r♦ss ♥ ❯ ♥r③ ♠♦s s ♥rs ♣♣r♦ t♦ t ♥②ss ♦ ♥♦♥♥r②♥♠ s②st♠s P②s ♣

♥♠r ♥ P ♦♠s ♦♥♥r st♦♥s ②♥♠ ②st♠s ♥ rt♦♥s ♦❱t♦r s ♦ ♣r♥r ❱r ❨♦r

trr③ P②s♦♦ ss ♦ rt♦♣♥♥t ♣rt♦r♣r② t♦r② t ♠t♦ ♣♦♦ ♠♦s ♣r♠ ♦♦② ♣♣

trr③ ♥ ♠ärt♥r ttr♦♣ ♠♦s ♦ ♣rt♦r♣r② ♥rts rst ♠♦ ♦ ♣♥tr♦r♣rst♦♣rt♦r ♥trt♦♥s ♥♥ ♥t♦♠♦♦st ♣♣

q ①st♥ ♦ ♦♠♣① ♣ttr♥s ♥ t ♥t♦♥♥s ♣rt♦r♣r② ♠♦ t♠t ♦s♥s ♣♣

♦rst ♠r ♥ ♥♦♠② st rt♦♥ ♠t ♦ ♦♠♣tt♦♥s♦♥ s②st♠s♥♦♦ ♦ r♥t qt♦♥s ♦t♦♥r② qt♦♥s ♣♣

♦rst ❱♥ r ♦t ♥ Ptr st rt♦♥ ♠t ♦r rt♦♥s♦♥s②st♠ ♦r♥ ♦ t♠t ♥②ss ♥ ♣♣t♦♥s ♣♣

♦rst ❱♥ r ♦t ♥ Ptr ♦♥♥r s♦♥ ♥ t ♣rs♥ ♦ strt♦♥ ♦♥♥r ♥②ss ♦r② t♦s ♣♣t♦♥s ♣♣

♦♥ ♥t♦♥ rs♣♦♥s ♦ ♥rtrt ♣rt♦rs t♦ ♣r② ♥st② ♠♦rs ♦ t♥t♦♠♦♦ ♦t② ♦ ♥ ♣♣

s♠♥ ♥ ♦r ♦r♠ rt♦♥ ♦ t ♥t♦♥ ♥t♦♥ rs♣♦♥s ♦r♥♦ ♦rt ♦♦② ♣♣

♠r ♥ ♥♦♠② s♦♥ r♦sss♦♥ ♥ ♦♠♣tt ♥trt♦♥♦r♥ ♦ t♠t ♦♦② ♣♣

❱ ①♣r♠♥t ♦♦② ♦ t ♥ ♦ ss ♦ ❨ ❯♥rst② Prss ♥

③r ♠r t t♦♥s♦♥ s②st♠ ♣♣r♦①♠t♦♥ t♦ t r♦sss♦♥♦♠♣tt♦♥ s②st♠ r♦s♠ t♠t ♦r♥ ♣♣

♦s ♦♦♣rt♦♥ ♦♣t♠ ♥st② ♥ ♦ ♥st② trs♦s ②t ♥♦tr ♠♦t♦♥ ♦ t♦st ♠♦ ♦♦ ♣♣

♥t P ♦♥str ♥ ♥ ♦♥sq♥s ♦r ♣rt♦rs ♦ rs ♥ ts ♦♥ ♣r② ♦♦ ♦♥ ♣♣

♦♠♦♦r♦ t♦r ❱♦trr ♦tt ♣r sst♥③ ♦r♥ sttt♦ t♥♦ ttr ♣♣

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♥ ♠♥ ♥ r♦ss ②♥♠ ♥②ss ♦ ♦t♦♥ qt♦♥s ♥ ♥r③♠♦s ♦r♥ ♦ ♣♣ t♠ts ♣♣

♠r ♥ ❲ ♦rst♠ ts ♦ r♦ss s♦♥ ♦♥ r♥ rt♦♥s ♥ t♦s♣srt♦♥tr♥s♣♦rt s②st♠s P②s ♣

♥♦ Pr♥♣s ♦ ♣rt♦r♣r② ♥trt♦♥ ♥ t♦rt ①♣r♠♥t ♥ ♥tr ♣♦♣t♦♥s②st♠ ♥s ♥ ♦♦ sr ♣♣

❨ ③♥ts♦ ♠♥ts ♦ ♣♣ rt♦♥ ♦r② ♦ ♣r♥r ❱r ❨♦r

❨ ③♥ts♦ ♥ ♥ ♠rs ♦♥ ♦♦ ♥ ②♥♠s t♠t ♦s♥s ♣♣

♦♥ rt♦♥ ♥ ♥②ss ♦ ♦♠♣♦st ♠♦s ♦r ♥st ♣♦♣t♦♥s ♥ st♠t♦♥ ♥♥②ss ♦ ♥st P♦♣t♦♥s ♦ ♦ tr ♦ts ♥ ttst ♣r♥r ♣♣

♦ ❱ Ptr♦s ♥ ❱♥tr♥♦ ♣t♦t♠♣♦r ♣ttr♥s ♥ ♦♦② ♥ ♣♠♦♦② t♦r② ♠♦s ♥ s♠t♦♥ ♣♠♥ ♦♥♦♥

♥♥ ♥ P ❨♦③s ♦♦ ♦♥t♦♥s ♦r ♦s ♥ trs♣s ♦♦ ♥ ♦♦② ♣♣

♥ P♦ó♣③ r♥ ♣ttr♥s ♥ ♠♦ ♦t♦trr ♠♦P②ss ttrs ♣♣

♥s② ❱ Ptr♦s ♦♥♦ ♦ ♥ ♣t♦t♠♣♦r ♦♠♣①t② ♦ ♣♥t♦♥ ♥ s ②♥♠s r ♣♣

❩ ♠r rt♦♥ ♥②ss ♦r t♦♥s♦♥ qt♦♥s ♦ ♣r♥r ♥ s♥ss

t③ ♥ ♠♥♥ ②♥♠s ♦ P②s♦♦② trtr P♦♣t♦♥s ♦ ♣r♥r

r ♦♠ ♦♠♠♥ts ♦♥ ♠t♦♠♣♦♥♥t s♦♥ ♥t ♠♥ tr♠ s♦♥ ♦♥ts s♦♥ ♦♥str♥ts s♦♥t ♦s ♥ rr♥ r♠ tr♥s♦r♠t♦♥s ♦r♥ ♦P②s ♠str② ♣♣

♦ss ♦♠ r♥ts ♦ t ss ♥♦♥s ♠♠ ♦r♥ ♦ ♦t♦♥ qt♦♥s ♣♣

r t♦♥ t♥ r♦sss♦♥ ♥ t♦♥s♦♥ tsr rr② t♠t ♦♦② ♣t ♦s ♥ ♦♠ ♣♣t♦♥s

♥trs♣♥r② ♣♣ t♠ts ❱ ♣r♥r❱r ❨♦r ♥♦r♣♦rt ♥♦ rt ♥ r♦♥r ♥♥ ♦ ♥tr♣rt♦r② ♥trr♥s ♦♥ ♠

♣s ♦♦ ♦♥tr♦ ♥② t♥ ♦ t♠t ♦♦② ♣♣ Pr♦ r♥t qt♦♥s ♥ ②♥♠ ②st♠s ♣r♥r❱r ❨♦r ❱ Ptr♦s ♥ ♦ ♠♥♠ ♠♦ ♦ ♣ttr♥ ♦r♠t♦♥ ♥ ♣r②♣rt♦r

s②st♠ t♠t ♥ ♦♠♣tr ♦♥ ♣♣ ❲ ♦ ♦s ♥ ♠♥s♠ ♦ ♣ttr♥ ♦r♠t♦♥ ♥ s♣t♦t♠♣♦r ♣♦♣t♦♥ ②♥♠s

♦rt ♣♦♣t♦♥ ♦♦② ♣♣ Prr ♥ ♠tt ♦♣ ♥ rt♦♥s♦♥ s②st♠s t ss♣t♦♥ ♦ ♠ss

♣♣ ♦♣ ♥ rt♦♥s♦♥ s②st♠s t ss♣t♦♥ ♦ ♠ss ♣♣

♦♥ ♦ ①st♥ ♥ strt♦♥ ♠t ♥ rt♦♥s♦♥ s②st♠s t r♦ss ts

P tss ♦ ♥♦r♠ s♣érr ♥ ♥ ♥s ❯♥rstät r♠stt♠♥

♦②♠ ♦♠♣rt st② ♦ ♠♦s ♦r ♣rt♦♥ ♥ ♣rsts♠ srs ♦♥ P♦♣t♦♥♦♦② ♣♣

♥ ♥r ♥ ♦r♦③♦ rt♦♥ ♥②ss ♦ rt♦♣♥♥t ♣r②♣rt♦r♠♦ t t t ♦♦ ♦♠♣①t② ♣♣

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♦ ♥ ♥s ♣rt♦r♣r② ♠♦ t ♦♥ t②♣ ♥t♦♥ rs♣♦♥s♥♥ ♣rt♦r ♠t ♥trr♥ ♦r♥ ♦ ♦♥♥r ♥ ♣♣

s s ♥ r♠♦t♦ ♣t srt♦♥ ♦ ♥trt♥ s♣s ♦r♥♦ ♦rt ♦♦② ♣♣

P t♣♥s ♥ ❲ tr♥ ♦♥sq♥s ♦ t t ♦r ♦r ♦♦②♥ ♦♥srt♦♥ r♥s ♥ ♦♦② ♦t♦♥ ♣♣

P t♣♥s ❲ tr♥ ♥ P rt♦♥ ❲t s t t ♦s ♣♣

❨ r③ ♥ ♦♦t tt② ♦ ♦♦ ♦♠♠♥ts ♦ Psrs♦s♦

♥♥r ts ♦ ♣♦♣t♦♥ ♥st② ♦♥ r♦t rts ♦ ♥♠ ♣♦♣t♦♥s ♦♦② ♣♣

②♦r ♥ st♥s ts ♥ ♦♦ ♥s♦♥s ♦♦② ttrs ♣♣

P ♦♥ ❲t♠r ♦♥s♦♥ ør♥st ♥ ♦♥s♦♥ s♣ s t ② ♦r♣ rt♦♥ ♥ t str♥t ♦ ts ♦♦② ttrs ♣♣

♠♦ ♦♠r♦ ♥ ♠♠rt♥♦ r♦sss♦♥ r♥ ♥stt② ♥ ♣rt♦r♣r② s②st♠ t r♦sss♦♥ t ♣♣♥ t♠t ♣♣

❱♥ ♦r t♦♠t t♥qs ♦r t qtt ♥②ss ♦ ♦♦♠♦s ♦♥t♥♦s ♠♦s ♦♥srt♦♥ ♦♦② ♣ r♦♠tt♣♦♦②♥s♦t②♦r♦ssrt

❱♥ ❱♦♦r♥ ♠r P ♦r ♥ ❲ ♦♦ tr♦♥ ♦rts ♥t♦r①♣♦tt♦♥ ♥ ♣rt♦r♣r② s②st♠s t str♦♥ t t♠t ♦s♥s ♣♣

❱ ❱♥ ♥ ♣st♥ r♦sss♦♥ ♥ ♣ttr♥ ♦r♠t♦♥ ♥ rt♦♥s♦♥ s②st♠sP②s ♠str② ♠ P②ss ♣♣

P ❱rst ♦t sr ♦ q ♣♦♣t♦♥ st ♥s s♦♥ r♦ss♠♥t ♦rrs♣♦♥♥♠té♠tq t ♣②sq ♣é ♣r tt ♣♣

❱ ❱♦trr ❱r③♦♥ tt③♦♥ ♠r♦ ♥ ♥ ♣ ♥♠ ♦♥♥t ♦ ♠ ♥

P ❲♥s♦r ♦♠♣rt③ r s r♦t r Pr♦♥s ♦ t t♦♥ ♠② ♦♥s ♣♣

❲♦♦ Prt② s♣ ♦♦ ♠♦s ♦♦ ♦♥♦r♣s ♣♣ ❨♦♦ P♦♣t♦♥ ♦♦② Pr♦rss ♥ Pr♦♠s ♦ ts ♦♥ tr P♦♣t♦♥s

Psrs ♦s♦ ❳ ❩♥ ♥ ♥ ❩ ♥ ♣t ②♥♠s ♥ ♣rt♦r♣r② ♠♦ t ♥t♦♥

♥s ♥t♦♥ rs♣♦♥s P②s ♣ ❩♦ ❨ ♥ ❲♥ stt② ♦ ♣rt♦r♣r② s②st♠s st t♦ t

ts ♦rt P♦♣t♦♥ ♦♦② ♣♣ ❩ ♦ qtt ♥②ss ♦ ♣rt♦r♣r② s②st♠ t t ♦♥ t ♣r② s♣s

t♠ts ♥ ♦♠♣trs ♥ ♠t♦♥ ♣♣ ❩ ♥ ♠r ♠♣t ♦ t ♦♥ ♣rt♦r♣r② s②st♠ t ♦♥ t②♣

♥t♦♥ rs♣♦♥s ♣♣ t♠ts ♥ ♦♠♣tt♦♥ ♣♣

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• ♦ s♥ ♥♦ ②♥♠s ♥♥ s②♥r♦♥③t♦♥ ♥ ♥r ♥t♦rs t tr ♦♣♥ r♥ t ♣♦str sss♦♥ ♦ t ♦♦ ♦ r♥ s rts♠♦ ♦ ♠r r r♣♥ t②

• ♠♦♣t♠ ♦♥tr♦ strts ♥ ♣♠s ♠♦s r♥ t ♣♦str sss♦♥ ♦ t ♦rs♦♣ t♠t ♥ ♦♠♣tt♦♥ ♣♠♦♦② st r r♣♥ t②

• ♦ s♥ ♥♦ ②♥♠s ♥♥ s②♥r♦♥③t♦♥ ♥ ♥r ♥t♦rs t tr ♦♣♥ r♥ t ♣♦str sss♦♥ ♦ t ♦♦ t♦s ♥ ♦s ♦♥t ♦r② ♥ P♦rt♦ r♦ r♦sst♦ t②

♥♦ r