Heteroclinic Bifurcation in the Michaelis-Menten-Type Ratio-Dependent Predator-Prey System

The existence of a heteroclinic bifurcation for the Michaelis–Menten-type ratio-dependent predator-prey system is rigorously established. Limit cycles related to the heteroclinic bifurcation are also discussed. It is shown that the heteroclinic bifurcation is characterized by the collision of a stable limit cycle with the origin, and the bifurcation triggers a catastrophic shift from the state of large oscillations of predator and prey populations to the state of extinction of both populations. It is also shown that the limit cycles related to the heteroclinic bifurcation originally bifurcate from the Hopf bifurcation.

[1]  R. Arditi,et al.  Underestimation of mutual interference of predators , 1990, Oecologia.

[2]  Yilei Tang,et al.  Heteroclinic bifurcation in a ratio-dependent predator-prey system , 2005, Journal of mathematical biology.

[3]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[4]  R Arditi,et al.  The biological control paradox. , 1991, Trends in ecology & evolution.

[5]  Dongmei Xiao,et al.  Global dynamics of a ratio-dependent predator-prey system , 2001, Journal of mathematical biology.

[6]  Donald L. DeAngelis,et al.  A MODEL FOR TROPHIC INTERACTION , 1975 .

[7]  C. Huffaker.  Experimental studies on predation : dispersion factors and predator-prey oscillations , 1958 .

[8]  R Arditi,et al.  Parametric analysis of the ratio-dependent predator–prey model , 2001, Journal of mathematical biology.

[9]  R. Arditi,et al.  Variation in Plankton Densities Among Lakes: A Case for Ratio-Dependent Predation Models , 1991, The American Naturalist.

[10]  S. Hsu,et al.  Global analysis of the Michaelis–Menten-type ratio-dependent predator-prey system , 2001, Journal of mathematical biology.

[11]  Yang Kuang,et al.  Global qualitative analysis of a ratio-dependent predator–prey system , 1998 .

[12]  S. Chow,et al.  Normal Forms and Bifurcation of Planar Vector Fields , 1994 .

[13]  Alan A. Berryman,et al.  The Orgins and Evolution of Predator‐Prey Theory , 1992 .

[14]  H. Resit Akçakaya,et al.  Population Cycles of Mammals: Evidence for a Ratio‐Dependent Predation Hypothesis , 1992 .

[15]  L. Slobodkin,et al.  Community Structure, Population Control, and Competition , 1960, The American Naturalist.

[16]  L. Luckinbill,et al.  Coexistence in Laboratory Populations of Paramecium Aurelia and Its Predator Didinium Nasutum , 1973 .

[17]  J. Beddington,et al.  Mutual Interference Between Parasites or Predators and its Effect on Searching Efficiency , 1975 .

[18]  M. Scheffer,et al.  Geometric Analysis of Ecological Models with Slow and Fast Processes , 2000, Ecosystems.

[19]  M. Rosenzweig.  Paradox of Enrichment: Destabilization of Exploitation Ecosystems in Ecological Time , 1971, Science.

[20]  R. Arditi,et al.  Coupling in predator-prey dynamics: Ratio-Dependence , 1989 .

[21]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .