Coexistence and stability of predator–prey model with Beddington–DeAngelis functional response and stage structure

Abstract We present a predator–prey model of Beddington–DeAngelis type functional response with stage structure on prey. The constant time delay is the time taken from birth to maturity about the prey. By the uniform persistence theories and monotone dynamic theories, sharp threshold conditions which are both necessary and sufficient for the permanence and extinction of the model as well as the sufficient conditions for the global stability of the coexistence equilibria are obtained. Biologically, it is proved that the variation of prey stage structure can affect the permanence of the system and drive the predator into extinction by changing the prey carrying capacity: Our results suggest that the predator coexists with prey permanently if and only if predator's recruitment rate at the peak of prey abundance is larger than its death rate; and that the predator goes extinct if and only if predator's possible highest recruitment rate is less than or equal to its death rate; furthermore, our results also show that a sufficiently large mutual interference by predators can stabilize the system.

[1]  Y. Kuang,et al.  A Stage Structured Predator-Prey model and its Dependence on Through-Stage Delay and Death Rate , 2003 .

[2]  Qui Zhipeng,et al.  The asymptotic behavior of a chemostat model with the Beddington-DeAngelis functional response. , 2004, Mathematical biosciences.

[3]  Shengqiang Liu,et al.  A Stage-structured Predator-prey Model of Beddington-DeAngelis Type , 2006, SIAM J. Appl. Math..

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

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

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

[7]  K. Schmitt,et al.  Permanence and the dynamics of biological systems. , 1992, Mathematical biosciences.

[8]  Paul Waltman,et al.  Competing Predators , 2007 .

[9]  C. S. Holling.  Some Characteristics of Simple Types of Predation and Parasitism , 1959, The Canadian Entomologist.

[10]  William W. Murdoch,et al.  Consumer-resource dynamics , 2003 .

[11]  Shigui Ruan,et al.  Intraspecific interference and consumer-resource dynamics , 2004 .

[12]  C. S. Holling.  The components of prédation as revealed by a study of small-mammal prédation of the European pine sawfly. , 1959 .

[13]  Rong Yuan,et al.  Stability and bifurcation in a delayed predator–prey system with Beddington–DeAngelis functional response , 2004 .

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

[15]  M. Hassell,et al.  New Inductive Population Model for Insect Parasites and its Bearing on Biological Control , 1969, Nature.

[16]  Horst R. Thieme,et al.  Persistence under relaxed point-dissipativity (with application to an endemic model) , 1993 .

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

[18]  Philip H. Crowley,et al.  Functional Responses and Interference within and between Year Classes of a Dragonfly Population , 1989, Journal of the North American Benthological Society.

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

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

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

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

[23]  Jack K. Hale,et al.  Persistence in infinite-dimensional systems , 1989 .

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

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

[26]  Shengqiang Liu,et al.  Extinction and permanence in nonautonomous competitive system with stage structure , 2002 .

[27]  Yang Kuang,et al.  Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response , 2004 .

[28]  Guilie Luo,et al.  Asymptotic behaviors of competitive Lotka–Volterra system with stage structure , 2002 .