Dynamical analysis of a delayed predator-prey model with impulsive diffusion between two patches

In this work, we propose a delayed predator-prey model with impulsively diffusive prey between two patches. Using the stroboscopic map of the discrete dynamical system, we obtain the globally attractive condition of predator-extinction periodic solution of the system. We also obtain the permanent condition of the system by the theory of impulsive delay differential equation. Our results indicate that the discrete time delay has influence to the dynamical behaviors of the system. Finally, the numerical analysis is inserted to illustrate the results.

[1]  P K Maini,et al.  Pattern formation in a generalized chemotactic model , 1998, Bulletin of mathematical biology.

[2]  Jianjun Jiao,et al.  GLOBAL ATTRACTIVITY OF A STAGE-STRUCTURE VARIABLE COEFFICIENTS PREDATOR-PREY SYSTEM WITH TIME DELAY AND IMPULSIVE PERTURBATIONS ON PREDATORS , 2008 .

[3]  Y. Takeuchi,et al.  Global stability of single-species diffusion volterra models with continuous time delays , 1987 .

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

[5]  S. Levin.  Dispersion and Population Interactions , 1974, The American Naturalist.

[6]  L. Allen.  Persistence and extinction in single-species reaction-diffusion models , 1983 .

[7]  Eric T. Funasaki,et al.  Invasion and Chaos in a Periodically Pulsed Mass-Action Chemostat , 1993 .

[8]  D. Bainov,et al.  Impulsive Differential Equations: Periodic Solutions and Applications , 1993 .

[9]  Jinghui,et al.  A Single Species Model with Impulsive Diffusion , 2005 .

[10]  H. I. Freedman,et al.  Predator survival versus extinction as a function of dispersal in a predator–prey model with patchy environment , 1989 .

[11]  Lansun Chen,et al.  Permanence and extinction in logistic and Lotka-Volterra systems with diffusion , 2001 .

[12]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[13]  H. I. Freedman,et al.  A time-delay model of single-species growth with stage structure. , 1990, Mathematical biosciences.

[14]  H. I. Freedman,et al.  Mathematical models of population interactions with dispersal II: Differential survival in a change of habitat , 1986 .

[15]  B. Shulgin,et al.  Pulse vaccination strategy in the SIR epidemic model , 1998, Bulletin of mathematical biology.

[16]  Xinzhi Liu,et al.  Impulsive Stabilization and Applications to Population Growth Models , 1995 .

[17]  Bernd Blasius,et al.  Categories of chaos and fractal basin boundaries in forced predator–prey models , 2001 .

[18]  H. I. Freedman,et al.  Global stability and predator dynamics in a model of prey dispersal in a patchy environment , 1989 .

[19]  H. I. Freedman.  Single species migration in two habitats: Persistence and extinction , 1987 .

[20]  Katrin Rohlf,et al.  Impulsive control of a Lotka-Volterra system , 1998 .

[21]  Xinzhi Liu,et al.  A cell population model described by impulsive PDEs—Existence and numerical approximation , 1998 .

[22]  Jianjun Jiao,et al.  A delayed stage-structured predator-prey model with impulsive stocking on prey and continuous harvesting on predator , 2008, Appl. Math. Comput..

[23]  Jinghui,et al.  Impulsive diffusion in single species model , 2007 .

[24]  Y. Takeuchi,et al.  Global Asymptotic Stability of Lotka–Volterra Diffusion Models with Continuous Time Delay , 1988 .

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

[26]  Yasuhiro Takeuchi,et al.  Global stability and periodic orbits for two-patch predator-prey diffusion-delay models , 1987 .

[27]  Wendi Wang,et al.  A predator-prey system with stage-structure for predator , 1997 .

[28]  P. S. Simeonov,et al.  Impulsive Differential Equations , 1989 .