Bifurcation Difference Induced by Different Discrete Methods in a Discrete Predator-Prey Model (Journal of Nonlinear Modeling and Analysis, 2024)
In this paper, we revisit a discrete predator-prey model with Alleeeffect and Holling type-I functional response. The most important is for us tofind the bifurcation difference: a flip bifurcation occurring at the fixed point $E_3$ in the known results cannot happen in our results. The reason leading tothis kind of difference is the different discrete method. In order to demonstrate this, we first simplify corresponding continuous predator-prey model.Then, we apply a different discretization method to this new continuous model to derive a new discrete model. Next, we consider the dynamics of thisnew discrete model in details. By using a key lemma, the existence and localstability of nonnegative fixed points $E_0,$ $E_1,$ $E_2$ and $E_3$ are completely studied. By employing the Center Manifold Theorem and bifurcation theory, theconditions for the occurrences of Neimark-Sacker bifurcation and transcriticalbifurcation are obtained. Our results complete the corresponding ones in aknown literature. Numerical simulations are also given to verify the existenceof Neimark-Sacker bifurcation.
Keywords: Discrete predator-prey model with Holling type-I funcational response; Flip bifurcation; Neimark-Sacker bifurcation; Transcritical bifurcation; Allee effect.