Computing the heteroclinic bifurcation curves in predator-prey systems with ratio-dependent functional response

Shigui Ruan; Yilei Tang; Weinian Zhang

doi:10.1007/s00285-007-0153-z

Computing the heteroclinic bifurcation curves in predator-prey systems with ratio-dependent functional response

Shigui Ruan, Yilei Tang, Weinian Zhang

Mathematics

Research output: Contribution to journal › Article

9 Scopus citations

Abstract

Predator-prey models with Michaelis-Menten-Holling type ratio- dependent functional response exhibit very rich and complex dynamical behavior, such as the existence of degenerate equilibria, appearance of limit cycles and heteroclinic loops, and the coexistence of two attractive equilibria. In this paper, we study heteroclinic bifurcations of such a predator-prey model. We first calculate the higher order Melnikov functions by transforming the model into a Hamiltonian system and then provide an algorithm for computing higher order approximations of the heteroclinic bifurcation curves.

Original language	English (US)
Pages (from-to)	223-241
Number of pages	19
Journal	Journal of Mathematical Biology
Volume	57
Issue number	2
DOIs	https://doi.org/10.1007/s00285-007-0153-z
State	Published - Aug 2008

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Keywords

Approximation
Beta functions
Hamiltonian system
Heteroclinic loop
Melnikov functions
Predator-prey system

ASJC Scopus subject areas

Agricultural and Biological Sciences (miscellaneous)
Mathematics (miscellaneous)

Cite this

Ruan, S., Tang, Y., & Zhang, W. (2008). Computing the heteroclinic bifurcation curves in predator-prey systems with ratio-dependent functional response. Journal of Mathematical Biology, 57(2), 223-241. https://doi.org/10.1007/s00285-007-0153-z

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abstract = "Predator-prey models with Michaelis-Menten-Holling type ratio- dependent functional response exhibit very rich and complex dynamical behavior, such as the existence of degenerate equilibria, appearance of limit cycles and heteroclinic loops, and the coexistence of two attractive equilibria. In this paper, we study heteroclinic bifurcations of such a predator-prey model. We first calculate the higher order Melnikov functions by transforming the model into a Hamiltonian system and then provide an algorithm for computing higher order approximations of the heteroclinic bifurcation curves.",

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AB - Predator-prey models with Michaelis-Menten-Holling type ratio- dependent functional response exhibit very rich and complex dynamical behavior, such as the existence of degenerate equilibria, appearance of limit cycles and heteroclinic loops, and the coexistence of two attractive equilibria. In this paper, we study heteroclinic bifurcations of such a predator-prey model. We first calculate the higher order Melnikov functions by transforming the model into a Hamiltonian system and then provide an algorithm for computing higher order approximations of the heteroclinic bifurcation curves.

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KW - Melnikov functions

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10.1007/s00285-007-0153-z