Global asymptotic stability of 3-species Lotka-Volterra models with diffusion and time delays

Abstract This paper is concerned with three 3-species time-delayed Lotka–Volterra reaction–diffusion models with homogeneous Neumann boundary condition. Some simple conditions are obtained for the global asymptotic stability of the nonnegative semitrivial constant steady-state solutions. These conditions are explicit and easily verifiable, and they involve only the reaction rate constants and are independent of the diffusion and time delays. The result of global asymptotic stability not only implies the nonexistence of positive steady-state solution but also gives some extinction results of the models in the ecological sense. The instability of some nonnegative semitrivial constant steady-state solutions is also shown. The conclusions for the reaction–diffusion systems are directly applicable to the corresponding ordinary differential systems.

[1]  A. Leung.  A study of three species prey-predator reaction-diffusions by monotone schemes☆ , 1984 .

[2]  Josep Blat,et al.  Bifurcation of steady-state solutions in predator-prey and competition systems , 1984, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[3]  Chia-Ven Pao,et al.  Nonlinear parabolic and elliptic equations , 1993 .

[4]  Xin Lu,et al.  Some coexistence and extinction results for a $3$-species ecological system , 1995 .

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

[6]  Leping Zhou,et al.  Asymptotic behavior of a competition—diffusion system in population dynamics , 1982 .

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

[8]  Sze-Bi Hsu,et al.  Some results on global stability of a predator-prey system , 1982 .

[9]  Anthony W. Leung,et al.  Systems of Nonlinear Partial Differential Equations , 1989 .

[10]  R. Vance,et al.  Predation and Resource Partitioning in One Predator -- Two Prey Model Communities , 1978, The American Naturalist.

[11]  William W. Murdoch,et al.  Switching, Functional Response, and Stability in Predator-Prey Systems , 1975, The American Naturalist.

[12]  Robert Stephen Cantrell,et al.  Permanence in ecological systems with spatial heterogeneity , 1993, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[13]  C. V. Pao,et al.  Global asymptotic stability of Lotka–Volterra 3-species reaction–diffusion systems with time delays , 2003 .

[14]  A Hastings,et al.  Spatial heterogeneity and the stability of predator-prey systems: predator-mediated coexistence. , 1978, Theoretical population biology.

[15]  Philip Korman,et al.  On the existence and uniqueness of positive steady states in the volterra-lotka ecological models with diffusion , 1987 .

[16]  Y. Takeuchi.  Global Dynamical Properties of Lotka-Volterra Systems , 1996 .

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

[18]  R. Gilbert,et al.  Permanence effect in a three—species food chain model , 1994 .

[19]  H. I. Freedman,et al.  Persistence in models of three interacting predator-prey populations , 1984 .

[20]  C. Cosner,et al.  Stable Coexistence States in the Volterra–Lotka Competition Model with Diffusion , 1984 .

[21]  Yoshio Yamada,et al.  Stability of steady states for prey-predator diffusion equations with homogeneous dirichlet conditions , 1990 .

[22]  Xin Lu,et al.  Dynamics and numerical simulations of food-chain populations , 1994 .

[23]  Wei Feng,et al.  Coexistence, Stability, and Limiting Behavior in a One-Predator-Two-Prey Model , 1993 .

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