Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response

Abstract In this paper, we systematically study the dynamics of a nonautonomous predator–prey system with the Beddington–DeAngelis functional response. The explorations involve the permanence, extinction, global asymptotic stability (general nonautonomous case); the existence, uniqueness and stability of a positive (almost) periodic solution and a boundary (almost) periodic solution for the periodic (almost periodic) case. The paper ends with some interesting numerical simulations that complement our analytical findings.

[1]  Qian Wang,et al.  Dynamics of a non-autonomous ratio-dependent predator—prey system , 2003, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[2]  M. Fan,et al.  Optimal harvesting policy for single population with periodic coefficients. , 1998, Mathematical biosciences.

[3]  H. I. Freedman,et al.  Persistence in predator-prey systems with ratio-dependent predator influence , 1993 .

[4]  A. Gutierrez.  Physiological Basis of Ratio-Dependent Predator-Prey Theory: The Metabolic Pool Model as a Paradigm , 1992 .

[5]  Chris Cosner,et al.  On the Dynamics of Predator–Prey Models with the Beddington–DeAngelis Functional Response☆ , 2001 .

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

[7]  Tzy-Wei Hwang,et al.  Uniqueness of limit cycles of the predator–prey system with Beddington–DeAngelis functional response , 2004 .

[8]  L. Ginzburg,et al.  The nature of predation: prey dependent, ratio dependent or neither? , 2000, Trends in ecology & evolution.

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

[10]  P. Lundberg,et al.  Expected Population Density Versus Productivity in Ratio-Dependent and Prey-Dependent Models , 1995, The American Naturalist.

[11]  Paul M. Dolman,et al.  The intensity of interference varies with resource density: evidence from a field study with snow buntings, Plectrophenax nivalis , 1995, Oecologia.

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

[13]  R. Arditi,et al.  Empirical Evidence of the Role of Heterogeneity in Ratio‐Dependent Consumption , 1993 .

[14]  G. Ruxton,et al.  Interference and Generation Cycles , 1992 .

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

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

[17]  H. Thieme,et al.  On the complex formation approach in modeling predator prey relations, mating, and sexual disease transmission. , 2000 .

[18]  R. Arditi,et al.  Functional responses and heterogeneities: an experimental test with cladocerans , 1991 .

[19]  Ke Wang,et al.  Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system , 2002 .

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

[21]  Chris Cosner,et al.  Practical persistence in ecological models via comparison methods , 1996, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[22]  Donald L. DeAngelis,et al.  A Model for Tropic Interaction , 1975 .

[23]  J. F. Gilliam,et al.  FUNCTIONAL RESPONSES WITH PREDATOR INTERFERENCE: VIABLE ALTERNATIVES TO THE HOLLING TYPE II MODEL , 2001 .

[24]  Tzy-Wei Hwang,et al.  Global analysis of the predator–prey system with Beddington–DeAngelis functional response , 2003 .

[25]  S. Ellner,et al.  Testing for predator dependence in predator-prey dynamics: a non-parametric approach , 2000, Proceedings of the Royal Society of London. Series B: Biological Sciences.

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

[27]  Ke Wang,et al.  Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems with functional responses ✩ , 2003 .

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

[29]  D. DeAngelis,et al.  Effects of spatial grouping on the functional response of predators. , 1999, Theoretical population biology.

[30]  R. Arditi,et al.  From pattern to process: identifying predator–prey models from time-series data , 2001, Population Ecology.

[31]  R. Gaines,et al.  Coincidence Degree and Nonlinear Differential Equations , 1977 .

[32]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[33]  P. Chesson.  Understanding the role of environmental variation in population and community dynamics , 2003 .

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

[35]  Y. Kuang,et al.  RICH DYNAMICS OF GAUSE-TYPE RATIO-DEPENDENT PREDATOR-PREY SYSTEM , 1999 .

[36]  Chris Cosner,et al.  Variability, vagueness and comparison methods for ecological models , 1996 .