Complex dynamics in a food chain with slow and fast processes

Abstract This paper is devoted to the analysis of the dynamic behavior of a three-species food chain model, in which two predators compete for the same prey while one of the predators feeds on the other. Under the assumption that the time responses of the three trophic levels are extremely diversified, the model is proved to have homoclinic orbit. We firstly use geometric singular perturbation method to detect singular homoclinic orbits as well as parameter combinations for which these orbits exist. Then, we show, numerically, that there exist also nonsingular homoclinic orbits that tend toward the singular ones for slightly different parameter values. This analysis is particularly helpful to understanding the chaotic behavior of the food chains.

[1]  Sergio Rinaldi,et al.  Low- and high-frequency oscillations in three-dimensional food chain systems , 1992 .

[2]  A. Hastings,et al.  Chaos in a Three-Species Food Chain , 1991 .

[3]  Stephen Schecter,et al.  Persistent unstable equilibria and closed orbits of a singularly perturbed equation , 1985 .

[4]  Peter Szmolyan,et al.  Extending slow manifolds near transcritical and pitchfork singularities , 2001 .

[5]  M. Stiefenhofer.  Singular perturbation with Limit points in the fast dynamics , 1998 .

[6]  Chaos and Hopf bifurcation of a hybrid ratio-dependent three species food chain , 2008 .

[7]  Peter Szmolyan,et al.  Extending Geometric Singular Perturbation Theory to Nonhyperbolic Points - Fold and Canard Points in Two Dimensions , 2001, SIAM J. Math. Anal..

[8]  C. S. Holling,et al.  Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest , 1978 .

[9]  Vikas Rai,et al.  Chaotic population dynamics and biology of the top-predator , 2004 .

[10]  Bo Deng,et al.  Food chain chaos due to Shilnikov's orbit. , 2002, Chaos.

[11]  Zhujun Jing,et al.  Bifurcation and chaos in discrete-time predator–prey system , 2006 .

[12]  Y. Lenbury,et al.  Low- and high-frequency oscillations in a food chain where one of the competing species feeds on the other , 1994 .

[13]  Yingfei Yi,et al.  Relaxation Oscillations in a Class of Predator-Prey Systems , 2003 .

[14]  S Rinaldi,et al.  Singular homoclinic bifurcations in tritrophic food chains. , 1998, Mathematical biosciences.

[15]  Yuri A. Kuznetsov,et al.  Homoclinic bifurcations in slow-fast second order systems , 1995 .

[16]  Y. Lenbury,et al.  Coexistence of competing microbial species in a chemostat where one population feeds on another , 1993 .

[17]  Bo Deng,et al.  Food chain chaos due to junction-fold point. , 2001, Chaos.

[18]  S. Muratori.  An application of the separation principle for detecting slow-fast limit cycles in a three-dimensional system , 1991 .

[19]  Sergio Rinaldi,et al.  Slow-fast limit cycles in predator-prey models , 1992 .

[20]  Sergio Rinaldi,et al.  Limit cycles in slow-fast forest-pest models☆ , 1992 .

[21]  A. Zvonkin,et al.  Non-standard analysis and singular perturbations of ordinary differential equations , 1984 .

[22]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[23]  S Rinaldi,et al.  Remarks on food chain dynamics. , 1996, Mathematical biosciences.

[24]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .