Persistent prey species in the Lotka-Volterra apparent competition system with a single shared predator

We analyze the Lotka-Volterra n prey-1 predator system with no direct interspecific interaction between prey species, in which every prey species undergoes the effect of apparent competition via a single shared predator with all other prey species. We prove that the considered system necessarily has a globally asymptotically stable equilibrium, and we find the necessary and sufficient condition to determine which of feasible equilibria becomes asymptotically stable. Such an asymptotically stable equilibrium shows which prey species goes extinct or persists, and we investigate the composition of persistent prey species at the equilibrium apparent competition system. Making use of the results, we discuss the transition of apparent competition system with a persistent single shared predator through the extermination and invasion of prey species. Our results imply that the long-lasting apparent competition system with a persistent single shared predator would tend toward an implicit functional homogenization in coexisting prey species, or would transfer to a 1 prey-1 predator system in which the predator must be observed as a specialist (monophagy).