How many preys could coexist with a shared predator in the Lotka–Volterra system?: State transition by species deletion/introduction

We analyze the Lotka–Volterra n prey-1 predator system without any interspecific interaction between preys, in which each prey species has the relationship of apparent competition with any other prey species. The system we considered in this paper necessarily has a globally asymptotically stable unique equilibrium state. We find the necessary and sufficient condition to determine which equilibrium states becomes asymptotically stable. Then we consider the effect of the deletion of a native prey species and that of the invasion of an alien prey species on the stability of the system. We prove that the deletion of a prey species could not cause the secondary extinction of any other prey species but could make the predator go extinct. In contrast, if an alien prey species is successfully introduced into the system at the coexistent equilibrium state, the extinction of some native prey species could occur by the apparent competition effect. Moreover, such a successful introduction of an alien prey species causes the reduction of the population size of every surviving native prey species while it always increases the predator's equilibrium population size even when some native prey species go extinct. We show that the strength of the apparent competition effect would be significantly affected by the number and the composition of coexisting prey species. (paper)