# American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2021095
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## Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension

 1 School of Science, Chang'an University, Xi'an 710064, China 2 School of Mathematics and Statistics, Huaiyin Normal University, Huaian 223300, China 3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

* Corresponding author: Yongli Cai

Received  May 2020 Revised  December 2020 Early access March 2021

Fund Project: The authors thank all referees for their careful reading and valuable feedback leading to an improvement of this paper. This research of Q. Cao was supported by the Fundamental Research Funds for the Central Universities (300102120103). This research of Y. Cai was supported by the National Science Foundation of China (No. 61672013 and 12071173)

Resorting to M.G. Crandall and P.H. Rabinowitz's well-known bifurcation theory we first obtain the local structure of steady states concerning the ratio–dependent predator–prey system with prey-taxis in spatial one dimension, which bifurcate from the homogeneous coexistence steady states when treating the prey–tactic coefficient as a bifurcation parameter. Based on this, then the global structure of positive solution is established. Moreover, through asymptotic analysis and eigenvalue perturbation we find the stability criterion of such bifurcating steady states. Finally, several numerical simulations are performed to show the pattern formation.

Citation: Qian Cao, Yongli Cai, Yong Luo. Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021095
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Pattern formation of system (6) where $x\in(0,100)$ with grid number 1000 and $t \in(0, 8000)$ with grid number 1000