On the Stability of Periodic Solutions in the Perturbed ...malisoff/papers/AMS-Slides.pdf · On the Stability of Periodic Solutions in the Perturbed Chemostat MICHAEL MALISOFF Department

On the Stability of Periodic Solutionsin the Perturbed Chemostat

MICHAEL MALISOFFDepartment of MathematicsLouisiana State University

Joint work withFrederic Mazenc(INRIA-INRA)andPatrick De Leenheer(University of FL)

SIAM Minisymposium on Mathematical Modelingof Complex Systems in Biology I

AMS Joint Meetings, New Orleans, LA, January 6, 2007

GENERAL n SPECIES CHEMOSTAT MODEL

S = D(S0 − S)−n∑

i=1

µi(S)xi/γi; xi = xi(µi(S)−D) (Σn)

xi = concentration ofith species,S = concentration of limiting nutrient,

µi = ith per-capita growth rate,γi ∈ (0, 1) = constantith yield factor.

Controls:dilution rateD and input nutrient concentrationS0.


S = D(S0 − S)−n∑

i=1





Derivation:Use mass-balance equations for total amounts of nutrient and

each of the species, assuming the reactor content is well-mixed.


S = D(S0 − S)−n∑

i=1







Importance:Chemostat models provide the foundation for much of

current research inbioengineering, ecology, andpopulation biology.


S = D(S0 − S)−n∑

i=1







Importance:Chemostat models provide the foundation for much of


Competitive Exclusion:WhenS0 andD are constant and theµi’s are

increasing,at most one species survives. (There is a steady state with at

most one nonzero species concentration, which attracts a.a. solutions.)

OVERVIEW of LITERATURE

Coexistence:In real ecological systems,many species can coexist, so

much of the literature aims at choosingS0 and/orD to force coexistence.

“The Paradox of the plankton,” Hutchinson,American Naturalist, 1961.





Time-Varying Controls:Have competitive exclusion ifn = 2 and one of

the controls is fixed and the other is periodic. See Hal Smith (SIAP’81),

Hale-Somolinos (JMB’83), Butler-Hsu-Waltman (SIAP’85).








State-Dependent Controls:A feedback control perspective based on

mathematical control theorywas pursued e.g. in De Leenheer-Smith

(JMB’03) to generate a coexistence equilibrium forn = 2, 3.








State-Dependent Controls:A feedback control perspective based on

mathematical control theorywas pursued e.g. in De Leenheer-Smith

(JMB’03) to generate a coexistence equilibrium forn = 2, 3.

Intra-Specific Competition:This can be modeled with growth rates

µi(S, xi) that decrease inxi. See Mazenc-Lobry-Rapaport (EJDE’07),

Grognard-Mazenc-Rapaport (DCDS’07).

OUR WORK for ONE SPECIES CASE

TakingS0 to be constant and rescaling gives

S = D(1− S)− µ(S)x, x = x(µ(S)−D) (Σ1)

evolving onX = (0,∞)2. We assume a Monod growth rate

µ(S) =mS

a + S, m > 4a + 1 . (G)



S = D(1− S)− µ(S)x, x = x(µ(S)−D) (Σ1)


µ(S) =mS

a + S, m > 4a + 1 . (G)

Summary of Our Work:

• Instead of studying coexistence, we prove thestabilityof aprescribed

periodic solutionusing a Lyapunov-type analysis; i.e.,tracking.



S = D(1− S)− µ(S)x, x = x(µ(S)−D) (Σ1)


µ(S) =mS

a + S, m > 4a + 1 . (G)




• Lyapunov functions are useful for robustness analysis but have

infrequently been used in chemostat research.



S = D(1− S)− µ(S)x, x = x(µ(S)−D) (Σ1)


µ(S) =mS

a + S, m > 4a + 1 . (G)




• Lyapunov functions are useful for robustness analysis but have

infrequently been used in chemostat research.

• Most chemostat/Lyapunov results usenonstrictLyapunov functions

and LaSalle invariance which are not suited to robustness analysis.

MAIN TRACKING RESULT for Σ1

Statement of Main Tracking Result: Given any componentwise positive

trajectory(S, x) : [0,∞) → X for (Σ1) and the dilution rate

D(t) =sin(t)

2 + cos(t)+

m(2− cos(t))4a + 2− cos(t)

(D)

andµ as in (G) withm > 4a + 1, the corresponding deviation

(S(t), x(t)) := (S(t)− Sr(t), x(t)− xr(t)) (E)

of (S, x) from the reference trajectory

(Sr(t), xr(t)) :=(

12− 1

4cos(t),

12

+14

cos(t))

(R)

for (Σ1) asymptotically approaches(0, 0) ast → +∞.

MAIN TRACKING RESULT for Σ1

Statement of Main Tracking Result: Given any componentwise positivetrajectory(S, x) : [0,∞) → X for (Σ1) and the dilution rate

D(t) =sin(t)

2 + cos(t)+

m(2− cos(t))4a + 2− cos(t)

(D)

andµ as in (G) withm > 4a + 1, the corresponding deviation

(S(t), x(t)) := (S(t)− Sr(t), x(t)− xr(t)) (E)

of (S, x) from the reference trajectory

(Sr(t), xr(t)) :=(

12− 1

4cos(t),

12

+14

cos(t))

(R)

for (Σ1) asymptotically approaches(0, 0) ast → +∞.

(Similar results hold if we instead pick anyxr(t) s.t.∃` > 0 s.t.∀t ≥ 0,max{`, |xr(t)|} ≤ xr(t) ≤ 3

4 andSr = 1− xr, for suitableD.)

OUTLINE of PROOF of MAIN TRACKING RESULT for Σ1

First transform the error dynamics for (E) into

˙z = −D(t)z,

˙ξ = µ(z − eξ)− µ(1− eξr(t)),

(TE)

wherez := z − 1, z = S + x, ξ := ξ − ξr, ξ := ln(x), andξr := ln(xr).

OUTLINE of PROOF of MAIN TRACKING RESULT for Σ1

First transform the error dynamics for (E) into

˙z = −D(t)z,

˙ξ = µ(z − eξ)− µ(1− eξr(t)),

(TE)

wherez := z − 1, z = S + x, ξ := ξ − ξr, ξ := ln(x), andξr := ln(xr).Next show that (TE) admits the Lyapunov-like function

L3(z, ξ) := eξ − 1− ξ +4m

aDz2 (L)

whereD(t) ≥ D > 0 ∀t. Along the trajectories of (TE), we get

L3 ≤ − ma(eξ − 1)2

16(a + 2 + z2)(a + 1)− 4m

az2 . (DK)

Using a Barbalat’s Lemma argument,(z, ξ) → 0 exponentially.

ROBUSTNESS with respect ton ADDITIONAL SPECIES

S = D(t)(1− S)− µ(S)x−n∑

i=1

νi(S)yi ,

x = x(µ(S)−D(t)), yi = yi(νi(S)−D(t))

(AS)

D is from (D),yi = concentration of theith additional species, eachνi is

continuous and increasing and satisfiesνi(0) = 0 andνi(1) < D.


S = D(t)(1− S)− µ(S)x−n∑

i=1

νi(S)yi ,


(AS)

D is from (D),yi = concentration of theith additional species, eachνi is

continuous and increasing and satisfiesνi(0) = 0 andνi(1) < D.

Multi-Species Result: The error between any componentwise

positive solution(S, x, y1, y2, . . . , yn) of (AS) and

(Sr, xr, 0, . . . , 0) =(

12− 1

4cos(t),

12

+14

cos(t), 0, . . . , 0)

converges exponentially to the zero vector ast → +∞.


S = D(t)(1− S)− µ(S)x−n∑

i=1

νi(S)yi ,


(AS)

D is from (D),yi = concentration of theith additional species, eachνi iscontinuous and increasing and satisfiesνi(0) = 0 andνi(1) < D.

Multi-Species Result: The error between any componentwisepositive solution(S, x, y1, y2, . . . , yn) of (AS) and

(Sr, xr, 0, . . . , 0) =(

12− 1

4cos(t),

12

+14

cos(t), 0, . . . , 0)

converges exponentially to the zero vector ast → +∞.

Significance:The stability of the reference trajectory (R) is robust withrespect to additional species that are exponentially decaying to extinction.

OUTLINE of PROOF of MULTI-SPECIES RESULT

Sinceνi(1) < D for eachi, the form of the dynamics forS along our

componentwise positive trajectories implies that there existε > 0 and

T ≥ 0 such that(i) S(t) ≤ 1 + ε for all t ≥ T and(ii) νi(1 + ε) < D for

all i = 1, 2, . . . , n. We next choose

δ := D − maxi=1,...,n

νi(1 + ε) > 0.

OUTLINE of PROOF of MULTI-SPECIES RESULT

Sinceνi(1) < D for eachi, the form of the dynamics forS along ourcomponentwise positive trajectories implies that there existε > 0 andT ≥ 0 such that(i) S(t) ≤ 1 + ε for all t ≥ T and(ii) νi(1 + ε) < D forall i = 1, 2, . . . , n. We next choose

δ := D − maxi=1,...,n

νi(1 + ε) > 0.

The result now follows using the Lyapunov-like function

L4(z, ξ, y1, ..., yn) = L3(z, ξ) + An∑

i=1

y2i , where A :=

16mn2

aδ.

in conjunction with Barbalat’s Lemma. Along the relevant trajectories,

L4 ≤ − ma(eξ − 1)2

16(a + 1)(a + 2 + z2)− 3m

az2 − 16mn2

a

n∑

i=1

y2i .

ROBUSTNESS with respect to ACTUATOR ERRORS

{S(t) = [D(t) + u1(t)](1 + u2(t)− S(t))− µ(S(t))x(t) ,

x(t) = x(t)[µ(S(t))−D(t)− u1(t)] .(Σp)



x(t) = x(t)[µ(S(t))−D(t)− u1(t)] .(Σp)

If |u| stays below a computable prescribed bound, then there are functions

β ∈ KL andγ ∈ K∞ such that the transformed error vector

y(t; to, yo, α) :=(S(t; t0, (S, x)(0), α)− Sr(t), ln(x(t; t0, (S, x)(0), α))− ln(xr(t)))

for all disturbancesu = (u1, u2) = α and initial conditions satisfies

|y(t; to, yo, α)| ≤ β(|yo|, t− to) + γ(|α|∞) . (ISS)



x(t) = x(t)[µ(S(t))−D(t)− u1(t)] .(Σp)

If |u| stays below a computable prescribed bound, then there are functionsβ ∈ KL andγ ∈ K∞ such that the transformed error vector

y(t; to, yo, α) :=(S(t; t0, (S, x)(0), α)− Sr(t), ln(x(t; t0, (S, x)(0), α))− ln(xr(t)))

for all disturbancesu = (u1, u2) = α and initial conditions satisfies

|y(t; to, yo, α)| ≤ β(|yo|, t− to) + γ(|α|∞) . (ISS)

Under the less stringent condition|u| < 12 min{1, D}, there are functions

δi ∈ K∞ andβ ∈ KL so that the trajectories everywhere satisfy

δ1(|y(t; to, yo, α)|) ≤ β(|yo|, t− to) +∫ t+to

to

δ2(|α(r)|)dr . (iISS)

SIMULATIONS

We simulated (Σp) with m = 10, a = 12 , u1(t) = 0.5e−t, u2(t) ≡ 0,

to = 0, x(0) = 2, andS(0) = 1. Our theory implies that the convergence

of (S(t), x(t)) to (Sr(t), xr(t)) satisfies iISS for disturbancesu that are

valued in[−u, u]2 for any positive constantu < min{1, D} = 1.

SIMULATIONS

We simulated (Σp) with m = 10, a = 12 , u1(t) = 0.5e−t, u2(t) ≡ 0,

to = 0, x(0) = 2, andS(0) = 1. Our theory implies that the convergenceof (S(t), x(t)) to (Sr(t), xr(t)) satisfies iISS for disturbancesu that arevalued in[−u, u]2 for any positive constantu < min{1, D} = 1.

0 2 4 6 8 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Tra

ject

orie

s

(a) S(t) TrackingSr(t)

0 2 4 6 8 100

0.5

1

1.5

2

2.5

Time

Tra

ject

orie

s

(b) x(t) Trackingxr(t)

CONCLUSIONS

• Chemostats provide an important framework for modelingspecies

competingfor nutrients. They provide the foundation for much


CONCLUSIONS




• For the case of one species competing for one nutrient and a suitable

time-varying dilution rate, we proved the stability of an appropriate

reference trajectoryusingLyapunov function methods.

CONCLUSIONS







• The stability is maintained when there are additional species that are

being driven to extinction, ordisturbances of small magnitudeon the

dilution rate and input nutrient concentration.

CONCLUSIONS







• The stability is maintained when there are additional species that are

being driven to extinction, ordisturbances of small magnitudeon the

dilution rate and input nutrient concentration.

• Extensions to chemostats withmultiple competing species, time

delays, limited informationabout the current state, andmeasurement

uncertaintywould be desirable and are being studied.

ACKNOWLEDGEMENTS

• Malisoff was supported byNSF/DMS Grant 0424011. De Leenheer

was supported byNSF/DMS Grant 0500861. The authors thankJeff

SheldonandHairui Tu for assisting with the graphics.

ACKNOWLEDGEMENTS




• A journal version will appear inMathematical Biosciences and

Engineering. The authors thankYang Kuangfor the opportunity to

publish their work in his esteemed journal.

ACKNOWLEDGEMENTS







• Part of this work was done while Mazenc and De Leenheer visited

theLouisiana State University (LSU) Department of Mathematics.

They thank LSU for the kind hospitality they enjoyed.

ACKNOWLEDGEMENTS







• Part of this work was done while Mazenc and De Leenheer visited

theLouisiana State University (LSU) Department of Mathematics.

They thank LSU for the kind hospitality they enjoyed.

• The authors thank the referees,Madalena Chaves, andEduardo

Sontagfor illuminating comments and discussions at the 45th IEEE

Conference on Decision and Control in San Diego, CA.

Documents

On the Stability of Periodic Solutions in the Perturbed ...malisoff/papers/AMS-Slides.pdf · On the Stability of Periodic Solutions in the Perturbed Chemostat MICHAEL MALISOFF Department