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September  2019, 18(5): 2511-2528. doi: 10.3934/cpaa.2019114

## Effects of dispersal for a predator-prey model in a heterogeneous environment

 1 School of Science, Xi'an Polytechnic University, Xi'an, Shaanxi, 710048, China 2 School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710071, China 3 College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China

Received  June 2018 Revised  November 2018 Published  April 2019

Fund Project: The work is supported by the Natural Science Foundation of China (11801431, 61672021), the Postdoctoral Science Foundation of China (2018T111014, 2018M631133), the Natural Science Foundation of Shaanxi Province (2018JQ1004, 2018JQ1017), Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 18JK0343).

In this paper, we study the stationary problem of a predator-prey cross-diffusion system with a protection zone for the prey. We first apply the bifurcation theory to establish the existence of positive stationary solutions. Furthermore, as the cross-diffusion coefficient goes to infinity, the limiting behavior of positive stationary solutions is discussed. These results implies that the large cross-diffusion has beneficial effects on the coexistence of two species. Finally, we analyze the limiting behavior of positive stationary solutions as the intrinsic growth rate of the predator species goes to infinity.

Citation: Yaying Dong, Shanbing Li, Yanling Li. Effects of dispersal for a predator-prey model in a heterogeneous environment. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2511-2528. doi: 10.3934/cpaa.2019114
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