Some coexistence and extinction results for a $3$-species ecological system

Abstract

This paper considers a reaction-diffusion system that models the situation in which two predators feed on one prey species. A sufficient condition for the existence of a positive steady-state solution and a positive global attractor is given in terms of the natural growth rates and the relative reaction rates of the three species. We prove that when the natural growth rates are in a certain unbounded parameter set $\Lambda$, the corresponding elliptic system has a pair of positive upper-lower solutions which ensures the existence of a positive solution. Two converging monotone sequences are generated from the upper and lower solutions by an iteration scheme and their limits enclose a positive attractor for the reaction-diffusion system. The present paper also gives some explicit information on the limiting behavior of the time-dependent solution in relation to the semitrivial and trivial steady states. Finally, numerical evidence of the long-term coexistence is demonstrated by solving the corresponding finite-difference system.