Generalized flip and strong resonances bifurcations of a predator–prey model

Abstract

In this paper, bifurcation analysis of a predator–prey discrete model equipped with Allee effect has been carried out both analytically and numerically. Stability circumstances of three fixed points of this model is represented briefly. In this study is shown that this model undergoes codimension one (codim-1) bifurcations such as the transcritical, fold, flip and Neimark–Sacker. Besides, codimension two (codim-2) bifurcations including the generalized flip, resonances 1:2, 1:3, 1:4 have been achieved. The non-degeneracy is one of the conditions to check for bifurcation analysis. Therefore the computing the critical normal form coefficients to verify the non-degeneracy of the listed bifurcations are needed. Using the critical normal form coefficients method to examine the bifurcation analysis makes it to avoid calculating the central manifold and converting the linear part of the map into Jordan form. This is one of the most effective methods in the bifurcation analysis that has not received much attention so far. So in this article our attention are turned to this method. For each bifurcation, normal form coefficients along with its scenario are investigated thoroughly. The bifurcation curves of fixed points under variation of one and two parameters and all codim-1,2 bifurcations curves are computed by using numerical methods in the numerical software matcontm. In the following, our represented analysis is proved by numerical simulation and displays more complex behaviours of model.