Complex dynamics of a discrete predator–prey system with a strong Allee effect on the prey and a ratio-dependent functional response

Abstract

Abstract In this paper, we consider the complex dynamics of a discrete predator–prey system with a strong Allee effect on the prey and a ratio-dependent functional response, which is the discrete version of the continuous system in (Nonlinear Anal., Real World Appl. 16:235–249, 2014). First, by giving several examples to display the limitations and errors of the local stability of the equilibrium point p obtained in (Nonlinear Anal., Real World Appl. 16:235–249, 2014), we provide an easily verified and complete discrimination criterion for the local stability of this equilibrium. Then we study some properties of its discrete version, especially for the stability and bifurcation for the equilibrium point E1 $E_{1}$, which has not been considered in any literature to the best of our knowledge. By using the center manifold theorem and bifurcation theory, we consider the flip bifurcation of this system at E1 $E_{1}$ and obtain the stability of the closed orbits bifurcated from E1 $E_{1}$. The numerical simulations not only show the correctness of our theoretical analysis, but also we find some new and interesting dynamics of this system.

Keywords

• Discrete predator–prey model
• Strong Allee effect
• Flip bifurcation
• Center Manifold Theorem