An introduction to prey-predator Models

• Lotka-Volterra model• Lotka-Volterra model with prey logistic growth• Holling type II model

• Generic Model

),()(

),()(

yxehygdtdy

yxhxfdtdx

• f(x) prey growth term• g(y) predator mortality term• h(x,y) predation term• e prey into predator biomass conversion coefficient

• Lotka-Volterra Model

bxymydtdy

axyrxdtdx

• r prey growth rate : Malthus law• m predator mortality rate : natural mortality• Mass action law • a and b predation coefficients : b=ea• e prey into predator biomass conversion coefficient

• Lotka-Volterra nullclines

Direction field for Lotka-Volterra model

Local stability analysis

• Jacobian at positive equilibrium

0

0*

*

*

by

axJ

• detJ*>0 and trJ*=0 (center)

• Linear 2D systems (hyperbolic)

Local stability analysis

• Proof of existence of center trajectories (linearization theorem)

• Existence of a first integral H(x,y) :

aybxyrxmyxH )ln()ln(),(

Lotka-Volterra model

Lotka-Volterra model

Hare-Lynx data (Canada)

• Logistic growth (sheep in Australia)

• Lotka-Volterra Model with prey logistic growth

bxymydtdy

axyKx

rxdtdx

1

Nullclines for the Lotka-Volterra model with prey logistic growth

• Lotka-Volterra Model with prey logistic growth

bxymydtdy

axyKx

rxdtdx

1

• Equilibrium points : (0,0) (K,0) (x*,y*)

Local stability analysis

• Jacobian at positive equilibrium

0*

**

*

by

axKrx

J

• detJ*>0 and trJ*<0 (stable)

• Condition for local asymptotic stability

Lotka-Volterra model with prey logistic growth : coexistence

Lotka-Volterra with prey logistic growth : predator extinction

• Transcritical bifurcation

*xK

*xK (K,0) stable and (x*,y*) unstable and negative

(K,0) and (x*,y*) same

*xK (K,0) unstable and (x*,y*) stable and positive

• Loss of periodic solutions

bxymydtdy

axyKx

rxdtdx

1

x-y

0 0,3 0,6 0,9 1,2 1,5

x

0

1,6

3,2

4,8

6,4

8

y

x-y

0 0,3 0,6 0,9 1,2 1,5

x

0

4

8

12

16

20

y

coexistence Predator extinction

Functional response I and II

• Holling Model

xDbxy

mydtdy

xDaxy

Kx

rxdtdx

1

• Existence of limit cycle (Supercritical Hopf bifurcation)

22

22

yxyxdtdy

yxxydtdx

• Polar coordinates

1

2

dtd

rrdtdr

• Stable equilibrium

• At bifurcation

• Existence of a limit cycle

• Supercritical Hopf bifurcation

Poincaré-Bendixson Theorem

A bounded semi-orbit in the plane tends to :• a stable equilibrium• a limit cycle• a cycle graph

Trapping region

Trapping region : Annulus

Example of a trapping region

xdtdy

xxy

dtdx

3

3

• Van der Pol model (>0)

• Holling Model

xDbxy

mydtdy

xDaxy

Kx

rxdtdx

1

Nullclines for Holling model

Poincaré box for Holling model

Holling model with limit cycle

Paradox of enrichment

When K increases :

• Predator extinction• Prey-predator coexistence (TC)• Prey-predator equilibrium becomes unstable (Hopf)• Occurrence of a stable limit cycle (large variations)

Other prey-predator models• Functional responses (Type III, ratio-dependent …)• Prey-predator-super-predator…• Trophic levels

Routh-Hurwitz stability conditions

0... 1

1

2

2

1

1

n

n

n

nnn aaaa

00)(, ik HRk

11 aH

• Characteristic equations

• Stability conditions : M* l.a.s.

2

31

2 1 a

aaH

31

42

531

3

0

1

aa

aa

aaa

H

Routh-Hurwitz stability conditions

032

2

1

3 aaa

011 trAaH

• Dimension 2

• Dimension 3

0det2 AtrA

0det3212 AaaaH

011 aH

03212 aaaH

033 aH

3-trophic example

dyzzzdtdz

cyzbxymydtdy

axyrxdtdx

)1(

• Interspecific competition Model

2

121

2

222

2

1

212

1

111

1

1

1

Kx

aKx

xrdtdx

Kx

aKx

xrdtdx

• Transformed system

buvvddv

avuuddu

1

1

Competition model