Coexistence of any number of species in the Lotka-Volterra competitive system over two-patches

Abstract

The competitive exclusion principle, one of the fundamental ideas in population ecology, asserts the impossibility of the stable coexistence of two or more species competing for the same resources in the same restricted area (patch). The population dynamics of several competing species in such an area is conveniently described by a system of ordinary differential equations in a single patch. Butler and Waltman (1986) said that a solution q(t) (i= 1, 2, . . . . J) to such a system, representing the numbers of the individuals of the different spacies Si at time t, is persistent if it satisfies lim inf, _ oD ui( t) > 0 for all i. Here, J denotes the number of species in the system. Based on (the negation of) this definition, we say that a J ( >2) species system in one patch is (weakly) exclusively competitive if any solution beginning with two or more initially existent species is never persistent no matter which combination of initial species is chosen. Such J species are also called exclusively competitive. Not all the systems of competitive species are exclusively competitive. Indeed, the classic Lotka-Volterra two species competitive system has a stable coexistent equilibrium if intraspecific competition is stronger than interspecific competition. Armstrong and McGehee (1980) summed up well the conditions under which J competing species can persist upon a smaller number of resources. Furthermore, Smale (1976) has shown that any dynamics is realizable within the framework of competitive systems. In the present paper, however, we only deal with exclusively competitive systems, concentrating our attention upon the problem of how the competitive exclusion principle may be relaxed over a habitat consisting of two patches. It is evident that, over two isolated patches, any two competing species can coexist permanently if each species remains within a single patch. It is 149 0040-5809/90 $3.00