Dynamics of a generalized Gause-type predator–prey model with a seasonal functional response

Abstract We extend a previous Gause-type predator–prey model to include a general monotonic and bounded seasonally varying functional response. The model exhibits rich dynamical behaviour not encountered when the functional response is not seasonally forced. A theoretical analysis is performed on the model to investigate the global stability of the boundary equilibria and the existence of periodic solutions. It is shown that, under certain well-defined conditions, the Poincare map of the model undergoes a Hopf bifurcation leading to the appearance of a quasi-periodic solution. Numerical results are given for the Poincare sections and bifurcation diagrams for Holling-types II and III functional responses, using the amplitude of seasonal variation as bifurcation parameter. The model shows a rich variety of behaviour, including period doubling, quasi-periodicity, chaos, transient chaos, and windows of periodicity.

[1]  Robert M. May,et al.  Theoretical Ecology: Principles and Applications , 1977 .

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

[3]  L. Perko.  Differential Equations and Dynamical Systems , 1991 .

[4]  Murray E. Alexander,et al.  A non-standard numerical scheme for a generalized Gause-type predator–prey model , 2004 .

[5]  Jitsuro Sugie.  Two-Parameter Bifurcation in a Predator–Prey System of Ivlev Type , 1998 .

[6]  Joseph W.-H. So,et al.  Global stability and persistence of simple food chains , 1985 .

[7]  Maynard Thompson,et al.  Deterministic Mathematical Models in Population Ecology. , 1982 .

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

[9]  Lansun Chen,et al.  Quasiperiodic Solutions and Chaos in a Periodically Forced Predator-prey Model with Age Structure for predator , 2003, Int. J. Bifurc. Chaos.

[10]  R. Seydel.  From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis , 1988 .

[11]  M. Lakshmanan,et al.  Occurrence of multiple period-doubling bifurcation route to chaos in periodically pulsed chaotic dynamical systems , 2003 .

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

[13]  Seyed M. Moghadas,et al.  Existence of limit cycles for predator–prey systems with a class of functional responses , 2001 .

[14]  P. Ricciardi,et al.  Lyapunov functions for a generalized Gause-type model , 1995 .

[15]  Mary R. Myerscough,et al.  Stability, persistence and structural stability in a classical predator-prey model , 1996 .

[16]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[17]  Jitsuro Sugie,et al.  On a predator-prey system of Holling type , 1997 .

[18]  S. M. Moghadas.  Some Conditions for the Nonexistence of Limit Cycles in a Predator-Prey System , 2002 .

[19]  Xianning Liu,et al.  Complex dynamics of Holling type II Lotka–Volterra predator–prey system with impulsive perturbations on the predator ☆ , 2003 .

[20]  Y. Kuznetsov,et al.  BIFURCATIONS AND CHAOS IN A PERIODIC PREDATOR-PREY MODEL , 1992 .

[21]  Aaron M. King,et al.  The rainbow bridge: Hamiltonian limits and resonance in predator-prey dynamics , 1999, Journal of mathematical biology.

[22]  Jim M Cushing,et al.  ESTIMATING CHAOS AND COMPLEX DYNAMICS IN AN INSECT POPULATION , 2001 .

[23]  G. F. Simmons.  Differential Equations With Applications and Historical Notes , 1972 .

[24]  Robert M. May,et al.  Theoretical Ecology: Principles and Applications , 1981 .

[25]  R. Seydel.  Practical bifurcation and stability analysis : from equilibrium to chaos , 1994 .