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Positive steady states of a ratio-dependent predator-prey system with cross-diffusion

1 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2 Department of Mathematics, University of Texas Rio Grande Valley, Edinburg, Texas 78539, USA

Special Issues: Recent Advances in Mathematical Population Dynamics

In this paper, we study a ratio-dependent predator-prey system with diffusion and cross-diffusion under the homogeneous Neumann boundary condition. By applying the maximum principle and Harnack’s inequality, we present a priori estimates of the positive steady state of the system. The existence and non-existence of non-constant positive steady states are established. Our findings show that under certain hypotheses, non-constant positive steady states can exist due to the emergence of cross-diffusion, which reveals that cross-diffusion can induce stationary patterns but the random diffusion fails.
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Keywords cross-diffusion; priori estimates; non-constant positive steady state; maximum principle; Leray-Schauder degree theory

Citation: Xiaoling Li, Guangping Hu, Xianpei Li, Zhaosheng Feng. Positive steady states of a ratio-dependent predator-prey system with cross-diffusion. Mathematical Biosciences and Engineering, 2019, 16(6): 6753-6768. doi: 10.3934/mbe.2019337

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