# American Institute of Mathematical Sciences

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March  2017, 37(3): 1539-1558. doi: 10.3934/dcds.2017063

## Effect of cross-diffusion in the diffusion prey-predator model with a protection zone

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

Received  April 2016 Revised  October 2016 Published  December 2016

Fund Project: The work is supported by the Natural Science Foundation of China (11271236,11401356,11671243,61672021), the Natural Science Basic Research Plan in Shaanxi Province of China (No.2015JQ1023), the Shaanxi New-star Plan of Science and Technology (No.2015KJXX-21)

In this work, we continue the mathematical study started in [K. Oeda, J. Differential Equations 250 (2011) 3988-4009] on the analytic aspects of the diffusion prey-predator system with a protection zone and cross-diffusion. For small birth rates of two species and large cross-diffusion for the prey, the detailed structure of positive solutions is established by the bifurcation theory and the Lyapunov-Schmidt reduction, which is determined by a finite dimensional limiting system. Moreover, we prove that the stability of positive solutions changes only at every turning point by a spectral analysis for the linearized eigenvalue problem of the limiting system and its perturbation.

Citation: Shanbing Li, Jianhua Wu. Effect of cross-diffusion in the diffusion prey-predator model with a protection zone. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1539-1558. doi: 10.3934/dcds.2017063
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