Author's Personal Copy Journal of Differential Equations Versal Unfoldings of Predator–prey Systems with Ratio-dependent Functional Response

Abstract In this paper we study the versal unfolding of a predator–prey system with ratio-dependent functional response near a degenerate equilibrium in order to obtain all possible phase portraits for its perturbations. We first construct the unfolding and prove its versality and degeneracy of codimension 2. Then we discuss all its possible bifurcations, including transcritical bifurcation, Hopf bifurcation, and heteroclinic bifurcation, give conditions of parameters for the appearance of closed orbits and heteroclinic loops, and describe the bifurcation curves. Phase portraits for all possible cases are presented.

[1]  D. Jordan,et al.  Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers , 1979 .

[2]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

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

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

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

[6]  H. I. Freedman Deterministic mathematical models in population ecology , 1982 .

[7]  Roberto Conti,et al.  Non-linear differential equations , 1966 .

[8]  Yang Kuang,et al.  Heteroclinic Bifurcation in the Michaelis-Menten-Type Ratio-Dependent Predator-Prey System , 2007, SIAM J. Appl. Math..

[9]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

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

[11]  W. A. Coppel,et al.  A new class of quadratic systems , 1991 .

[12]  R Fluck,et al.  Evaluation of natural enemies for biological control: A behavioral approach. , 1990, Trends in ecology & evolution.

[13]  Chih-fen Chang,et al.  Qualitative Theory of Differential Equations , 1992 .

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

[15]  Zhang Zhifen,et al.  Proof of the uniqueness theorem of limit cycles of generalized liénard equations , 1986 .

[16]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[17]  Yilei Tang,et al.  Generalized normal sectors and orbits in exceptional directions , 2004 .

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

[19]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[20]  Robert E. Kooij,et al.  A predator-prey model with Ivlev's functional response , 1996 .

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

[22]  Christian Jost,et al.  About deterministic extinction in ratio-dependent predator-prey models , 1999 .

[23]  W. A. Coppel Quadratic systems with a degenerate critical point , 1988, Bulletin of the Australian Mathematical Society.

[24]  Philip Hartman,et al.  Ordinary differential equations, Second Edition , 2002, Classics in applied mathematics.

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

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

[27]  Yilei Tang,et al.  Computing the heteroclinic bifurcation curves in predator–prey systems with ratio-dependent functional response , 2008, Journal of mathematical biology.

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

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

[30]  Robert M. May,et al.  Stability and Complexity in Model Ecosystems , 2019, IEEE Transactions on Systems, Man, and Cybernetics.

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