Bifurcations of a predator-prey model with non-monotonic response function

H.W. Broer; Vincent Naudot; Robert Roussarie; Khairul Saleh

doi:10.1016/j.crma.2005.09.033

Ordinary Differential Equations

Bifurcations of a predator-prey model with non-monotonic response function
[Bifurcations dans un système prédateur-proie avec réponse fonctionnelle non-monotone]

Présenté par : Étienne Ghys

H.W. Broer ¹ ; Vincent Naudot ¹ ; Robert Roussarie ² ; Khairul Saleh ¹

¹ University of Groningen, Department of Mathematics, P.O. Box 800, NL-9700 AV Groningen, The Netherlands
² Institut mathématiques de Bourgogne, 9, avenue Alain-Savary, B.P. 47870, 21078 Dijon cedex, France

Comptes Rendus. Mathématique, Volume 341 (2005) no. 10, pp. 601-604.

Résumé
Abstract

On considère un modèle prédateur-proie en dimension 2 dépendant de cinq paramètres adapté du système Volterra–Lotka par une réponse fonctionnelle non-monotone. Une description des différents domaines de stabilité structurelle est présentée ainsi que leurs bifurcations. La structure de l'ensemble de bifurcation se réduit à quatre centres organisateurs de codimension 3. Nous présentons quelques examples d'attracteurs étranges obtenus par une pertubation périodique non autonome.

A 2-dimensional predator-prey model with five parameters is investigated, adapted from the Volterra–Lotka system by a non-monotonic response function. A description of the various domains of structural stability and their bifurcations is given. The bifurcation structure is reduced to four organising centres of codimension 3. Research is initiated on time-periodic perturbations by several examples of strange attractors.

Reçu le : 2005-06-30
Accepté le : 2005-09-11
Publié le : 2005-10-26

DOI : 10.1016/j.crma.2005.09.033

Affiliations des auteurs :

H.W. Broer ¹ ; Vincent Naudot ¹ ; Robert Roussarie ² ; Khairul Saleh ¹

¹ University of Groningen, Department of Mathematics, P.O. Box 800, NL-9700 AV Groningen, The Netherlands
² Institut mathématiques de Bourgogne, 9, avenue Alain-Savary, B.P. 47870, 21078 Dijon cedex, France

@article{CRMATH_2005__341_10_601_0,
     author = {H.W. Broer and Vincent Naudot and Robert Roussarie and Khairul Saleh},
     title = {Bifurcations of a predator-prey model with non-monotonic response function},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {601--604},
     publisher = {Elsevier},
     volume = {341},
     number = {10},
     year = {2005},
     doi = {10.1016/j.crma.2005.09.033},
     language = {en},
}

TY  - JOUR
AU  - H.W. Broer
AU  - Vincent Naudot
AU  - Robert Roussarie
AU  - Khairul Saleh
TI  - Bifurcations of a predator-prey model with non-monotonic response function
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 601
EP  - 604
VL  - 341
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2005.09.033
LA  - en
ID  - CRMATH_2005__341_10_601_0
ER  -

%0 Journal Article
%A H.W. Broer
%A Vincent Naudot
%A Robert Roussarie
%A Khairul Saleh
%T Bifurcations of a predator-prey model with non-monotonic response function
%J Comptes Rendus. Mathématique
%D 2005
%P 601-604
%V 341
%N 10
%I Elsevier
%R 10.1016/j.crma.2005.09.033
%G en
%F CRMATH_2005__341_10_601_0

H.W. Broer; Vincent Naudot; Robert Roussarie; Khairul Saleh. Bifurcations of a predator-prey model with non-monotonic response function. Comptes Rendus. Mathématique, Volume 341 (2005) no. 10, pp. 601-604. doi : 10.1016/j.crma.2005.09.033. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.09.033/

Bibliographie
Cité par

[1] J.F. Andrews A mathematical model for the continuous of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng., Volume 10 (1968), pp. 707-723

[2] A.D. Bazykin; F.S. Berezovskaya; G. Denisov; Y.A. Kuznetsov The influence of predator saturation effect and competition among predators on predator-prey system dynamics, Ecol. Modelling, Volume 14 (1993), pp. 39-57

[3] H.W. Broer, V. Naudot, R. Roussarie, K. Saleh, Dynamics of a predator-prey model with non-monotonic response function, Preprint, 2005

[4] F. Dumortier; R. Roussarie; J. Sotomayor; H. Zoladek Bifurcations of Planar Vector Fields, Lecture Notes in Math., vol. 1480, Springer-Verlag, 1991

[5] K. Saleh, Organising centres in semi-global analysis of dynamical systems, PhD Thesis, University of Groningen, 2005

[6] Y.A. Kuznetsov Elements of Applied Bifurcations Theory, Springer-Verlag, 1995

[7] A.J. Lotka Elements of Physical Biology, Williams and Wilkins, Baltimore, MD, 1925

[8] J.W. Milnor Topology from Differential Viewpoint, The University Press of Virginia, 1990

[9] J.R. Munkres Elementary Differential Topology, Princeton University Press, 1963

[10] J. Palis; W. de Melo Geometric Theory of Dynamical System, Springer-Verlag, 1982

[11] S. Rinaldi; S. Muratori; Y.A. Kuznetsov Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities, Bull. Math. Biol., Volume 55 (1993), pp. 15-35

[12] V. Volterra Variazioni e fluttuaziono del numero di individui in specie animali conviventi, Mem. Accad. Lincei, Volume 2 (1926), pp. 31-113

Cité par Sources :

Commentaires - Politique