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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 TAKEUCHI Department of Applied Mathematics, Faculty of Engineering, Shizuoka Uniuersity Hamamatsu 432, Japan Received 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 inter- actions in patchy environments. The analysis of these models has been centered around the coexis- tence 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 77 655 Avenue of the Americas, New York, NY 10010 0025-5564/94/$7.0078 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 showing that in a system composed of several patches connected by diffusion and occupied by a single species, if the species is able to survive at a globally stable equilibrium point when the patches are isolated, then it continues to do so for any diffusion rate at a different equilibrium (depending on the diffusion rate). In other words, diffusion among patches will not destabilize single-population dynamics. This result greatly generalizes some previously known results [2-4, 13, 141. Existing results on dynamics of two interacting populations in a patchy environment have largely been restricted to persistence and extinction analyses due to the increased complexity of global analysis. The known results on global stability are usually too general to be of any use in real mathematical or biological applications. Stabilizing and/or destabilizing effects of dispersions remain largely unknown due to difficulties involved in local stability analyses. Nevertheless, it is generally accepted among both mathematicians and biologists that discrete diffusion tends to stabilize ecosystems. The main objective of this paper is to prove or disprove this well- adopted point of view for some particular classes of models. We will see in Section 4, through an example, that discrete diffusions are capable of both stabilizing and destabilizing a given ecosystem. Specifically, we are able to show that for that example, the dispersion stabilizes the system when the dispersal rate is small and then destabilizes the system when the dispersal rate is further increased. Thus, the assertion that “migration is a stabilizing influence in population interactions” is not true without qualification. In two recent articles [29, 301, Takeuchi proved that introducing refuge patches in patchy environments tends to stabilize the population interaction, for example, allowing species to coexist. Our study here further confirms such a point of view. The model we chose to study here is a special case of the one introduced and analyzed in Freedman and Takeuchi [9]. The model deals with a single species that disperses between two patches (instead of the n in [91) in a heterogeneous environment with barriers between patches with a predator for which the dispersal between patches does not involve a barrier. Systems where there are barriers for dispersal among prey but not among predators are well known in nature. One example is described in [161, where cyclamen mites disperse among strawberry plants with the space between plants as a barrier to their dispersion. There, predators are mainly several species of small wasps, which do not consider the space between plants as barriers.PREDATOR-PREY DYNAMICS WITH DISPERSAL 79 A second example is given in 1201. The prey are the members of the porcupine caribou herd that dwell in‘the northern Yukon Territory of Canada. Their dispersion carries them across rivers and mountains. These are clearly barriers to their dispersion. One of the predator populationfor this herd is the golden eagle. For these birds, the rivers and mountains do not form barriers. The paper is organized as follows. In the next section we describe our model in detail. Results on boundedness of solutions, existence of boundary and interior equilibria, and persistence and extinction of predator species are presented. In Section 3, we obtain criteria for local and global stability of positive equilibria via Routh-Hurwitz criteria and Lyapunov functions, respectively. We also consider the special case when one patch is free of predators. In Section 4, we present an example with a Michaelis-Menten type of predator functional response, which demonstrates both the


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