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Predator-Prey Dynamics in Models of Prey Dispersal in Two-Patch Environments*



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Predator Prey Dynamics in Models of Prey Dispersal in Two Patch Environments YANG KUANG Department of Mathematics Arizona State University Tempe Arizona 85287 1804 AND YASUHIRO Department Hamamatsu Received TAKEUCHI of Applied 432 Japan Mathematics Faculty of Engineering Shizuoka Uniuersity 16 September 1992 revised 9 March 1993 ABSTRACT Models are presented for a single species that disperses between two patches of a heterogeneous environment with barriers between patches and a predator for which the dispersal between patches does not involve a barrier Conditions are established for the existence uniform persistence and local and global stability of positive steady states In particular an example that demonstrates both the stabilizing and destabilizing effects of dispersion is presented This example indicates that a stable migrating predator prey system can be made unstable by changing the amount of migration in both directions 1 INTRODUCTION Interest has been growing in the study of mathematical models of populations dispersing among patches in a heterogeneous environment l 5 8 9 11 13 17 20 24 26 31 and references cited therein Many of the existing models deal with a single population dispersing among patches Some of them deal with competition and predator prey interactions in patchy environments The analysis of these models has been centered around the coexistence of populations and the stability local and global of equilibria There are also a few papers that focus on the existence and location of positive steady states when the dispersal rate among patches is relatively Dedicated to Paul Waltman on the occasion of his 60th birthday MATHEMATICAL BIOSCIENCES 120 77 98 1994 OElsevier Science Inc 1994 655 Avenue of the Americas New York NY 10010 77 0025 5564 94 7 00 78 YANG KUANG AND YASUHIRO TAKEUCHI small ll 131 The single species dynamics in a patchy environment has been well studied Recently by applying cooperative system theory cf 2 5 Takeuchi 28 succeeded in



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