Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey System

In this paper, we consider a delayed diffusive Leslie–Gower predator–prey system with homogeneous Neumann boundary conditions. The stability/instability of the coexistence equilibrium and associated Hopf bifurcation are investigated by analyzing the characteristic equations. Furthermore, using the upper and lower solutions method, we give a sufficient condition on parameters so that the coexistence equilibrium is globally asymptotically stable.

[1]  S. Ruan.  DELAY DIFFERENTIAL EQUATIONS IN SINGLE SPECIES DYNAMICS , 2006 .

[2]  Sanling Yuan,et al.  Bifurcation and stability analysis for a delayed Leslie-Gower predator-prey system , 2009 .

[3]  P. H. Leslie.  SOME FURTHER NOTES ON THE USE OF MATRICES IN POPULATION MATHEMATICS , 1948 .

[4]  C. V. Pao,et al.  Systems of Parabolic Equations with Continuous and Discrete Delays , 1997 .

[5]  Xiao-Qiang Zhao,et al.  A non-local delayed and diffusive predator—prey model , 2001 .

[6]  Wan-Tong Li,et al.  Hopf bifurcation analysis for a delayed predator–prey system with diffusion effects , 2010 .

[7]  Teresa Faria,et al.  Stability and Bifurcation for a Delayed Predator–Prey Model and the Effect of Diffusion☆ , 2001 .

[8]  Yinnian He,et al.  Diffusion effect and stability analysis of a predator–prey system described by a delayed reaction–diffusion equations☆ , 2008 .

[9]  S. Ruan,et al.  On the zeros of transcendental functions with applications to stability of delay differential equations with two delays , 2003 .

[10]  Rui Peng,et al.  Positive steady states of the Holling–Tanner prey–predator model with diffusion , 2005, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[11]  Rui Xu,et al.  Global stability of a reaction-diffusion predator-prey model with a nonlocal delay , 2009, Math. Comput. Model..

[12]  Thomas W. Schoener,et al.  STABILITY AND COMPLEXITY IN MODEL ECOSYSTEMS , 1974 .

[13]  Jianhua Wu,et al.  Convergence of solutions for Volterra-Lotka prey-predator systems with time delays , 2009, Appl. Math. Lett..

[14]  Andrei Korobeinikov,et al.  A Lyapunov function for Leslie-Gower predator-prey models , 2001, Appl. Math. Lett..

[15]  J. Hainzl.  Multiparameter bifurcation of a predator-prey system , 1992 .

[16]  Kwang Ik Kim,et al.  Asymptotic behavior of an SEI epidemic model with diffusion , 2008, Math. Comput. Model..

[17]  Xiang-Ping Yan,et al.  Stability and Hopf bifurcation for a delayed prey-predator system with diffusion effects , 2007, Appl. Math. Comput..

[18]  Sze-Bi Hsu,et al.  Global Stability for a Class of Predator-Prey Systems , 1995, SIAM J. Appl. Math..

[19]  Wonlyul Ko,et al.  Non-constant positive steady-states of a diffusive predator–prey system in homogeneous environment , 2007 .

[20]  C. V. Pao,et al.  Convergence of solutions of reaction-diffusion systems with time delays , 2002 .

[21]  J. Sugie.  Uniqueness of limit cycles in a Predator—Prey system with Holling-type functional response , 2000 .

[22]  Meike J. Wittmann,et al.  Mathematical Ecology , 2006 .

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

[24]  Shigui Ruan,et al.  On Nonlinear Dynamics of Predator-Prey Models with Discrete Delay ⁄ , 2009 .

[25]  James T. Tanner,et al.  THE STABILITY AND THE INTRINSIC GROWTH RATES OF PREY AND PREDATOR POPULATIONS , 1975 .

[26]  Sanling Yuan,et al.  Stability and Hopf bifurcations in a delayed Leslie – Gower predator – prey system ✩ , 2009 .

[27]  Sze-Bi Hsu,et al.  A diffusive predator–prey model in heterogeneous environment , 2004 .

[28]  Zhou Yu-yuan.  Global Stability of a Class Predator-Prey System , 2004 .

[29]  Jianhong Wu.  Theory and Applications of Partial Functional Differential Equations , 1996 .

[30]  Sanling Yuan,et al.  RETRACTED: Stability and Hopf bifurcations in a delayed Leslie–Gower predator–prey system , 2009 .

[31]  Yuan-Ming Wang.  Asymptotic behavior of solutions for a class of predator–prey reaction–diffusion systems with time delays , 2007 .

[32]  J. Gower,et al.  The properties of a stochastic model for the predator-prey type of interaction between two species , 1960 .

[33]  C. V. Pao,et al.  Dynamics of Nonlinear Parabolic Systems with Time Delays , 1996 .

[34]  Rui Peng,et al.  Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model , 2007, Appl. Math. Lett..

[35]  Xiang-Ping Yan,et al.  Asymptotic stability of positive equilibrium solution for a delayed prey–predator diffusion system , 2010 .

[36]  Xin Lu,et al.  Harmless Delays for Permanence in a Class of Population Models with Diffusion Effects , 1997 .