A nullcline approach to global stability in discrete-time predator–prey models

Abstract

In this paper, we consider a two-dimensional discrete-time predator–prey model that was recently developed in Ackleh et al. [Persistence and stability analysis of discrete-time predator-prey models: A study of population and evolutionary dynamics, J. Differ. Equ. Appl. 25 (2019), pp. 1568–1603]. Utilizing a novel approach that is based on nullcline analysis, we derive conditions for the global stability of the interior equilibrium. This result significantly expands the parameter ranges under which global stability was shown to hold in Ackleh et al. [Long-term dynamics of discrete-time predator–prey models: Stability of equilibria, cycles and chaos, J. Differ. Equ. Appl. 26 (2020), pp. 693–726] using Lyapunov functions. We then extend these global stability results to a predator–prey model with evolution in the prey to obtain sharper conditions on the persistence of the system and to establish global-stability results for the interior equilibrium of this three-dimensional model. Numerical results corroborating these theoretical findings and demonstrating a relationship between the conditions for local asymptotic stability and global asymptotic stability are also presented.