Existence and Uniqueness of the Positive Steady State Solution for a Lotka-Volterra Predator-Prey Model with a Crowding Term

Abstract

This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation. We obtain a critical value λ 1 (Ω0), and demonstrate that the existence of the predator in $${\overline \Omega _0}$$ only depends on the relationship of the growth rate μ of the predator and λ 1 (Ω0), not on the prey. Furthermore, when μ < λ 1 (Ω0), we obtain the existence and uniqueness of its positive steady state solution, while when μ ≥ λ 1 (Ω0), the predator and the prey cannot coexist in $${\overline \Omega _0}$$ . Our results show that the coexistence of the prey and the predator is sensitive to the size of the crowding region $${\overline \Omega _0}$$ , which is different from that of the classical Lotka-Volterra predator-prey model.