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 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 E_{1} and obtain the stability of the closed orbits bifurcated from 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.
Highlights
It is well known that one of the most challenging investigation areas for the biology and/or ecology population is the predator–prey interaction among the population
2 Equilibria and their dynamics we first determine the existence of the equilibria of system (1.5), we investigate their dynamics
5 Result and discussion In this paper, we have considered a discrete predator–prey system with strong Allee effect on the prey and ratio-dependent functional response
Summary
It is well known that one of the most challenging investigation areas for the biology and/or ecology population is the predator–prey interaction among the population. The numerical simulation results are generally obtained by discretizing the corresponding continuous-time model. Jacobian matrix Jp undergoes a Hopf bifurcation at the equilibrium point p. The equilibrium point p in system (1.1) cannot undergo a Hopf bifurcation.
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