# American Institute of Mathematical Sciences

December  2020, 25(12): 4641-4657. doi: 10.3934/dcdsb.2020117

## Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone

 1 Department of Mathematics and Institute of Natural Sciences, Shanghai Jiaotong University, Shanghai 200240, China 2 Department of Mathematics, Dalian Minzu University, Dalian 116600, China 3 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author: Sining Zheng

Received  October 2019 Published  March 2020

Fund Project: The first author is supported by National Natural Science Foundation of China No.11701067 and Natural Science Foundation of Liaoning 2019-ZD-0180

In this paper we study the protection zone problem to a predator-prey model subject to Beddington-DeAngelis functional responses and small prey growth rate. This is a successive work to a previous paper of the authors [X. He, S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol. 75 (2017) 239-257], where the model with large prey growth rate was considered. At first we establish the existence and multiplicity of positive steady state solutions, and then give the dynamic behavior of the evolution problem. It is proved that there may be no positive steady state, or may have at leat one, two, or even three positive steady states, depending on the parameters involved such as the growth rate, the predation rate, and the food handling time of the predators, the growth rate and the refuge ability of the preys, and the sizes of the habitat with protection zone. In addition, it is shown that the dynamics of the solutions rely on the initial state as well, e.g., though there could be multiple positive steady states, the prey will go to extinction as time tends to infinity if its initial value is small.

Citation: Xiao He, Sining Zheng. Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4641-4657. doi: 10.3934/dcdsb.2020117
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