Codimension-one and codimension-two bifurcations of a discrete predator–prey system with strong Allee effect

Abstract

In this paper, the dynamic behaviors of a discrete predator–prey system with strong Allee effect for the prey are investigated. Firstly, we clarify topological types for the fixed points. Then we explore all cases of codimension-one bifurcations associated with transcritical bifurcation, subcritical or supercritical flip bifurcation at the boundary fixed points. Meanwhile, the stabilities of these non-hyperbolic fixed points are explored. At the interior fixed point, using the theory of approximation by a flow, we investigate codimension-two bifurcation associated with 1:2 strong resonance, in which the expressions of nondegenerate conditions are very complicated. By a skillful variable substitution, we convert the nondegenerate conditions into parametric polynomials and determine the signs of these conditions. In order to obtain the bifurcation curves around 1:2 strong resonance, we use several variable substitutions and introduction of new parameters. Meanwhile, these bifurcation curves are returned to the original variables and parameters to express for easy verification. Numerical simulations are made to demonstrate the consistence with our theoretical analyses. Furthermore, our theoretical analyses and numerical simulations are explained from the biological point of view.