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Bifurcation Difference Induced by Different Discrete Methods in a Discrete Predator-prey Model
  
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KeyWord:Discrete predator-prey model with Holling type-I funcational response; Flip bifurcation; Neimark-Sacker bifurcation; Transcritical bifurcation; Allee effect
Author NameAffiliation
Wenbo Yao Department of Big Data Science, School of Science, Zhejiang University of Science and Technology, Hangzhou, Zhejiang 310023, China 
Xianyi Li Department of Big Data Science, School of Science, Zhejiang University of Science and Technology, Hangzhou, Zhejiang 310023, China 
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Abstract:
      In this paper, we revisit a discrete predator-prey model with Allee effect and Holling type-I functional response. The most important is for us to find 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 to this 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 this new discrete model in details. By using a key lemma, the existence and local stability 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, the conditions for the occurrences of Neimark-Sacker bifurcation and transcritical bifurcation are obtained. Our results complete the corresponding ones in a known literature. Numerical simulations are also given to verify the existence of Neimark-Sacker bifurcation.