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Rich dynamics of Gause-type ratio-dependent predator-prey system. (English) Zbl 0920.92032
Ruan, Shigui (ed.) et al., Differential equations with applications to biology. Proceedings of the international conference, Halifax, Canada, July 25–29, 1997. Providence, RI: American Mathematical Society. Fields Inst. Commun. 21, 325-337 (1999).
Summary: Ratio-dependent predator-prey models are increasingly favored by field ecologists as an alternative or more suitable ones for predator-prey interactions where predation involves the searching process. However, in the past such models were not well studied mathematically. In our recent work [J. Math. Biol. 28, No. 4, 463-474 (1990; Zbl 0742.92022)] we have shown that such models exhibit much richer dynamics than the traditional ones. This is especially true in boundary dynamics. For example, the ratio-dependent models can exhibit dynamics such as for some parameters and initial conditions, both species can become extinct.
In this paper, we consider the global behaviors of solutions of the rather general Gause-type ratio-dependent predator-prey system. In addition, to confirm that Gause-type ratio dependent predator-prey models are rich in boundary dynamics we shall also present very sharp sufficient conditions to assure that if the positive steady state of the ratio-dependent predator-prey system is locally asymptotically stable, then the system has no nontrivial positive periodic solutions. We also give sufficient conditions for each of the possible three steady states to be globally asymptotically stable. We note that for ratio-dependent systems, paradox of enrichment can not occur. In general, local asymptotic stability of the positive steady state does not even guarantee the so-called persistence of the system, and therefore does not imply global asymptotic stability.
For the entire collection see [Zbl 0903.00038].

92D40 Ecology
34D05 Asymptotic properties of solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations