Global asymptotic behavior in single-species discrete diffusion systems

We consider a single-species dynamical system which is composed of several patches connected by discrete diffusion. Based on recently developed cooperative system theory and the property of a cooperative matrix, we obtain sufficient and necessary conditions for the system with linear diffusion to be extinct and for one with nonlinear diffusion to be globally stable. We also obtain a critical patch number in the system with linear diffusion for the species to go extinct. These results extend some recent known ones for discrete diffusion systems.

[1]  Thomas G. Hallam,et al.  Persistence in food webs—I Lotka-Volterra food chains , 1979 .

[2]  Josef Hofbauer,et al.  The theory of evolution and dynamical systems , 1988 .

[3]  L. Allen,et al.  Persistence, extinction, and critical patch number for island populations , 1987, Journal of mathematical biology.

[4]  Hans F. Weinberger,et al.  Spatial patterning of the spruce budworm , 1979 .

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

[6]  James F. Selgrade,et al.  Asymptotic behavior of solutions to single loop positive feedback systems , 1980 .

[7]  H. I. Freedman,et al.  Persistence in a model of three competitive populations , 1985 .

[8]  E. Kamke.  Zur Theorie der Systeme gewöhnlicher Differentialgleichungen. II. , 1932 .

[9]  Hal L. Smith.  On the asymptotic behavior of a class of deterministic models of cooperating species , 1986 .

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

[11]  J. F. Selgrade.  On the existence and uniqueness of connecting orbits , 1983 .

[12]  Paul Waltman,et al.  Uniformly persistent systems , 1986 .

[13]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[14]  Hal L. Smith.  Cooperative systems of differential equations with concave nonlinearities , 1986 .

[15]  Cooperative Systems Theory and Global Stability of Diffusion Models , 1989 .