Multiplicity and stability of a predator–prey model with non-monotonic conversion rate

Abstract

This paper is concerned with a predator–prey model possessing a non-monotonic conversion rate. The main purpose is to determine the multiple existence and stability of positive steady-state solutions to this system. The results show that if the parameter d is suitably large, then the system contains an S -shaped global bifurcation curve with respect to a bifurcation parameter. That is, the system has two or three positive solutions for a suitable range of parameters. Moreover, the stability of positive solutions on this curve is also given. If d is properly small, both uniqueness and non-uniqueness results can occur. The main tools used here include the bifurcation theory, the Lyapunov–Schmidt procedure, and the perturbation technique.