C. R. Acad. Sci. Paris, t. 3??, S erie I, p. ??–??, 2002 - PXMA???.TEX -Equations diff erentielles/ Ordinary Differential Equations

Bifurcations of a predator-prey model withnon-monotonic response functionH.W. Broer a, Vincent Naudot a, Robert Roussarieb, Khairul Saleh a

a University of Groningen, Department of Mathematics, P.O. Box 800 9700 AV Groningen, The NetherlandsE-mail: broer@math.rug.nl, v.naudot@math.rug.nl, khairul@math.rug.nl

b Institut Math ematiques de Bourgogne, 9, Av Alain Savary, B.P. 47870, 21078 Dijon Cedex, FranceE-mail: roussari@u-bourgogne.fr

(Recu le jour mois annee, accepte apres revision le jour mois annee)

Abstract. A 2-dimensional predator-prey model with five parameters isinvestigated, adapted from theVolterra-Lotka system by a non-monotonic response function. A description of the variousdomains of structural stability and their bifurcations is given. The bifurcation structure isreduced to four organising centres of codimension 3. Research is initiated on time-periodicperturbations by several examples of strange attractors.

c© 2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS

Bifurcations dans un systeme predateur-proie avec reponse fonctionnelle non-monotone

Resum e. On considere un modele predateur-proie en dimension 2 d´ependant de cinq parametresadapte du systeme Volterra-Lotka par une reponse fonctionnelle non-monotone. Une des-cription des differents domaines de stabilite structurelle est presentee ainsi que leurs bifur-cations. La structure de l’ensemble de bifurcation se reduit a quatre centres organisateursde codimension 3. Nous presentons quelques examples d’attracteurs etranges obtenus parune pertubation periodique non autonome.c© 2002 Academie des sciences/Editions scien-tifiques et medicales Elsevier SAS

1. Introduction

This paper deals with a particular family of planar vector fields which models the dynamics of the popu-lations of predators and their prey in a given ecosystem. Thesystem is a variation of the classical Volterra-Lotka system [7, 12] given by

x = x(a − λx) − yP (x), y = −δy − µy2 + cyP (x), (1)

where the variablesx andy denote the density of the prey and predator populations respectively, whileP (x) is a non-monotonic response function [1] given byP (x) = mx/(αx2 + βx + 1), where0 ≤ α, 0 <

Note presentee par

S0764-4442(00)0???-?/FLAc© 2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS. Tous droits reserves. 1

1

H.W. Broer, V. Naudot, R. Roussarie, K. Saleh

δ, 0 < λ, 0 ≤ µ andβ > −2√

α are parameters. The coefficienta represents the intrinsic growth rate ofthe prey, whileλ > 0 is the rate of competition or resource limitation of prey. The natural death rate of thepredator is given byδ > 0. The functioncP (x) wherec > 0 is the rate of conversion between prey andpredator. The non-negative coefficientµ is the rate of the competition amongst predators [2]. See [3,5] fora more detailed discussion concerning system (1).

Our goal is to understand the structurally stable dynamics of (1) and in particular the attractors with theirbasins where we have a special interest for multi-stability. We also study the bifurcations between the openregions of the parameter space that concern such dynamics thereby giving a better understanding of thefamily.

We briefly address the modification of this system, where a small parametric forcing is applied in theparameterλ, i.e.,λ = λ0(1+ε sin(2πt)), (as suggested by Rinaldi et al. [11]) whereε < 1 is a perturbationparameter. Our main interest is with large scale strange attractors. For several phase portraits of the Poincarereturn map (or stroboscopic map) see Figure 3.

2. Sketch of results

The investigation concerns the dynamics of (1) in the closedfirst quadrantclos(Q) whereQ = {x >0, y > 0} with boundary∂Q = {x = 0, y ≥ 0}∪ {y = 0, x ≥ 0}, which are both invariant under the flowassociated to system (1). Since limit cycles are hard to detect mathematically, our approach is to reduce,by surgery [8, 9], the structurally stable phase portraits to new portraits without limit cycles. In [3, 5]with help of topological means (Poincare-Hopf Index Theorem, Poincare-Bendixson Theorem [8, 10]) acomplete classification of all Reduced Morse-Smale Portraits is found, which is of great help to understandthe original system (1).

THEOREM 2.1. – (GENERAL PROPERTIES) System(1) has the following properties :

1. (TRAPPING DOMAIN) The domainBp = {(x, y) | 0 ≤ x, 0 ≤ y, x + y ≤ p}, wherep >1/λ

(

(1 − δ)2/(4δ) + 1)

is a trapping domain, meaning that it is invariant for positive time evolutionand also captures all integral curves starting inclos(Q);

2. (NUMBER OF SINGULARITIES) There are two singularities on the boundary∂Q, namely(0, 0) whichis a hyperbolic saddle-point andC = (1/λ, 0), which is(semi-) hyperbolic with{x > 0, y = 0} ⊂W s(C). In Q there can be no more than three singularities and the cases with zero, one, two andthree singularities all occur ;

3. (CLASSIFICATION OF THE REDUCED MORSE-SMALE CASE) Exactly six topological types of Re-duced Morse-Smale vector fields occur, listed in Figure1.

The following theorem is illustrated by Figure 2.

THEOREM 2.2. – (ORGANISING CENTRES) In the parameter spaceR5 = {α, β, µ, δ, λ} consider theprojectionΠ : ∆ × W → ∆, where∆ = {0 < δ, 0 < λ} andW = {α ≥ 0, β > −2

√α, µ ≥ 0}. There

exists a smooth curveC that separates∆ into two open regions∆1 and∆2.For all (δ, λ) ∈ ∆1 the corresponding3-dimensional bifurcation set inW has four organising centres of

codimension3 :

1. One transcritical point(TC3) ;

2. Two nilpotent-focus type points(NFa3

andNFb3) connected by a smooth Hopf curve(H2) and by a

smooth cusp curve(SN2) containingTC3 ;

3. One Bogdanov-Takens point(BT3) connected toNFb3 by a smooth Bogdanov-Takens curve(BT2).

Furthermore, the pointsNFa3 , NFb

3 collide when(δ, λ) approachC and disappear for(δ, λ) ∈ ∆2. Theorganising centresTC3 andBT3 remain.

2

[ -1][ -2] [ -2][ -1] [ -3][ -3]

(a) The case where C is a sink (b) The case where C is a saddle-point

C CCCCC

AA S

S

S

S A0A0A0

R

R

O OOOOOaa a bbb

x

y

FIG. 1 –Reduced Morse-Smale portraits occurring in system (1) ;A is a sinkS is a saddle-point andR a source.C is either a sinkor a saddle. Portraits de phase reduits realises par le systeme (1) ;A est un puit,S est un point de scelle,R une source.C est soit unpuit soit un point de scelle.

0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

1.75

20.2

0.4

0.6

0.8

1

0

0.01

0.02

0.03

0.3

0.1

0.2

-2

-1

0

1

0.2

0.4

0.6

0.8

0

0.1

0.2

0.3

0.4

x

y

y

x

x

y

x

y

8

604020

4

80

12

16

60 8020 40

2

6

10

20 40 60 80

4

8

12

16

y

x

x

y

y

x

20 40 60 80

4

8

12

16

20 40 60 80

5

10

15

0

20 40 60 800

5

10

15

20 40 60 800

5

15

10

5

40 60 800 20

10

15

20

25

x

y

60 8020 400

5

10

15

20

25

30

y

x

H1

SN1

1

2

4

3

8

TC1

5

SNLC1

aL

2

BT2

7

5

SN2

0.01

0

0.2 0.4 0.6

A0A0

A0

A0

C

C

C

C

C

C

C

C

C

14

15-0.004

0.004

60 8020 400

10

15

20

25

30

y

x

A0

C

60 8020 400

5

10

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25

30

y

xC

60 8020 400

5

10

15

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25

30

y

xC

0

10

9

11

12

13

SN1H

2

L1

a

L2

L1

b

b

-0.1

DL2

b

a

α

β

αα

ββ

µµ

λ

δ

(a) ∆ = {δ > 0, λ > 0} (b) Region∆1 (c) Region∆2

(d) Bifurcation diagram inS1

C

Region12 [a-2]

∆1

Region13 [a-2]

∆2

Region 1 [a-3]

SN2

SN2

Region 2 [a-3]

BT2 BT2

Region 9 [b-2]

BT3

BT3

Region 11 [b-2]

NFa3

Region 8 [b-2]

NFb3

Region 10 [b-1]

TC2

TC2

Region 4 [a-1]

TC3

TC3

Region 3 [b-3]

H2

Region 5 [b-1]

Ha2

Region 7 [b-1]

Hb2

S1

S1

S2

S3

S4

S5

S6

FIG. 2 – (a) : Region∆ = {δ > 0, λ > 0}. (b) : Bifurcation set inW = {α ≥ 0, β > −2√

α, µ ≥ 0} when(δ, λ) ∈ ∆1. (c) :Similar to (b) for the case(δ, λ) ∈ ∆2. (d) : Bifurcation diagram in 2-dimensional sectionS1 ⊂ {µ = 0.1} of Figure (b), (δ, λ) =(1.01, 0.01) ∈ ∆1. For terminology see Table 1. See [3, 5] for description of the other sections. (a) : Region∆ = {δ > 0, λ > 0}.(b) : L’ensemble de bifurcation dansW = {α ≥ 0, β > −2

√α, µ ≥ 0} lorsque(δ, λ) ∈ ∆1. (c) : Meme figure qu’en (c) lorsque

(δ, λ) ∈ ∆2. (d) : Diagramme de bifurcation pour la sectionS1 de la figure (b), (δ, λ) = (1.01, 0.01) ∈ ∆1. Voir terminologie enTable 1. Voir [3, 5] pour une description dans les autres sections.

3

H.W. Broer, V. Naudot, R. Roussarie, K. Saleh

Notation Name Notation Name

TC1 Transcritical TC2 Degenerate transcriticalTC3 Doubly degenerate transcritical SN1 Saddle-nodeSN2 Cusp BT2 Bogdanov-TakensBT3 Degenerate Bogdanov-Takens NF3 Singularity of nilpotent-focus typeH1 Hopf H2 Degenerate HopfL1 Homoclinic (or Blue Sky) L2 Homoclinic at saddle-nodeDL2 Degenerate homoclinic SNLC1 Saddle-node of limit cycles

TAB . 1 – List of bifurcations occurring in system (1). In all cases the subscript indicates the codimension of the bifurcation. See[4, 6] for details concerning the terminology. Liste des bifurcations qui concernent le systeme (1). Pour chaqu’une d’elles, l’indicecorrespondant indique la codimension de la bifurcation. Voir [4, 6] pour plus de details concernant la terminologie.

50.5 51 51.5 52 52.5 53 53.5 54 54.5 55

12.2

12.4

12.6

12.8

13

13.2

M

55 55.5 56 56.5 57 57.5

9.4

9.6

9.8

10

10.2

10.4

10.6

10.8

N

31 32 33 34 35 36 37

17

17.5

18

18.5

19

L

55 56 57 58 59 60 61

7

8

9

10

11

12

K

FIG. 3 – Phase portraits of the Poincare return map : On the left handside(α, β, µ, δ, λ) = (0.007, 0.036, 0.1, 1.01, 0.01) andε = 0.6. On the right hand side(α, β, µ, δ, λ) = (0.007, 0.036, 0.1, 1.01, 0.01) andε = 0.99. Portrait de Phase de l’applica-tion de retour de Poincare : A gauche(α, β, µ, δ, λ) = (0.007, 0.036, 0.1, 1.01, 0.01) et ε = 0.6. A droite (α, β, µ, δ, λ) =(0.007, 0.036, 0.1, 1.01, 0.01) etε = 0.99.

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