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 | |
| 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
Computing the heteroclinic bifurcation curves in predator-prey systems with ratio-dependent functional response. / Ruan, Shigui; Tang, Yilei; Zhang, Weinian.
In: Journal of Mathematical Biology, Vol. 57, No. 2, 08.2008, p. 223-241.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Computing the heteroclinic bifurcation curves in predator-prey systems with ratio-dependent functional response
AU - Ruan, Shigui
AU - Tang, Yilei
AU - Zhang, Weinian
PY - 2008/8
Y1 - 2008/8
N2 - 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.
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.
KW - Approximation
KW - Beta functions
KW - Hamiltonian system
KW - Heteroclinic loop
KW - Melnikov functions
KW - Predator-prey system
UR - http://www.scopus.com/inward/record.url?scp=43949121090&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=43949121090&partnerID=8YFLogxK
U2 - 10.1007/s00285-007-0153-z
DO - 10.1007/s00285-007-0153-z
M3 - Article
C2 - 18193425
AN - SCOPUS:43949121090
VL - 57
SP - 223
EP - 241
JO - Journal of Mathematical Biology
JF - Journal of Mathematical Biology
SN - 0303-6812
IS - 2
ER -