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Global asymptotic behavior in single-species discrete diffusion systems. (English) Zbl 0799.92014
A single-species dynamical system is considered which is composed of several patches connected by discrete diffusion. Based on cooperative system theory and the property of a cooperative matrix, sufficient and necessary conditions are obtained for the system with linear diffusion to become extinct and for one with nonlinear diffusion to be globally stable. The authors obtain a critical patch number in the system with linear diffusion for the species to become extinct.
MSC:
92D25Population dynamics (general)
92D40Ecology
References:
[1]Allen, L. J. S.: Persistence and extinction in single-species reaction-diffusion models. Bull. Math. Biol. 45, 209-227 (1983)
[2]Allen, L. J. S.: Persistence, extinction, and critical patch number for island populations. J. Math. Biol. 24, 617-625 (1987)
[3]Berman, A., Plemmons, R. J.: Nonnegative matrices in the mathematical sciences. New York London: Academic Press 1979
[4]Butler, G., Freedman, H. I., Waltman, P.: Uniformly persistent system. Proc. Am. Math. Soc. 96, 425-430 (1986) · doi:10.1090/S0002-9939-1986-0822433-4
[5]Freedman, H. I., Takeuchi, Y.: Global stability and predator dynamics in a model of prey dispersal in a patchy environment. Nonlinear Anal., Theory Methods Appl 13, 993-1002 (1989) · Zbl 0685.92018 · doi:10.1016/0362-546X(89)90026-6
[6]Freedman, H. I., Waltman, P.: Persistence in models of three competitive populations. Math. Biosci. 73, 89-101 (1985) · Zbl 0584.92018 · doi:10.1016/0025-5564(85)90078-1
[7]Gard, T. C., Hallam, T. G.: Persistence in food webs. I. Lotka-Volterra food chains. Bull. Math. Biol. 41, 877-891 (1979)
[8]Hofbauer, J., Sigmund, K.: The theory of evolution and dynamical systems. Cambridge: Cambridge University Press 1988
[9]Kamke, E.: Zur Theorie der Systeme gewöhnlicher Differentialgleichungen. II. Acta Math. 58, 57-85 (1932) · Zbl 02550335 · doi:10.1007/BF02547774
[10]Ludwig, D., Aronson, D. G., Weinberger, H. F.: Spatial patterning of the spruce budworm. J. Math. Biol. 8, 217-258 (1979)
[11]Selgrade, J. F.: Asymptotic behavior of solutions to single loop positive feedback systems. J. Differ. Equations 38, 80-103 (1980) · Zbl 0438.34052 · doi:10.1016/0022-0396(80)90026-1
[12]Selgrade, J. F.: On the existence and uniqueness of connecting orbits. Nonlinear Anal., Theory Methods Appl. 7, 1123-1125 (1983) · Zbl 0524.58039 · doi:10.1016/0362-546X(83)90021-4
[13]Smith, H. L.: On the asymptotic behavior of a class of deterministic models of cooperating species. SIAM J. Appl. Math. 46, 368-375 (1986) · Zbl 0607.92023 · doi:10.1137/0146025
[14]Smith, H. L.: Cooperative systems of differential equations with concave nonlinearities. Nonlinear Anal., Theory Methods Appl. 10, 1037-1052 (1986) · Zbl 0612.34035 · doi:10.1016/0362-546X(86)90087-8
[15]Takeuchi, Y.: Cooperative system theory and global stability of diffusion models. Acta Appl. Math. 14, 49-57 (1989) · Zbl 0665.92017 · doi:10.1007/BF00046673