# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021136
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## Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis

 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China 2 Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

*Corresponding author: Bin Liu

Received  March 2021 Revised  July 2021 Early access September 2021

Fund Project: The first author is supported by NSF grant No. 12001214 and the second author is supported by NSF grant No. 11971185

In this work we consider a two-species predator-prey chemotaxis model
 $\left\{ \begin{array}{lll} u_t = d_1\Delta u+\chi_1\nabla\cdot(u\nabla v)+u(a_1-b_{11}u-b_{12}v), &x\in \Omega, t>0, \\[0.2cm] v_t = d_2\Delta v-\chi_2\nabla\cdot(v\nabla u)+v(-a_2+b_{21}u-b_{22}v-b_{23}w), & x\in \Omega, t>0, \\[0.2cm] w_t = d_3\Delta w-\chi_3\nabla\cdot(w\nabla v)+w(-a_3+b_{32}v-b_{33}w), & x\in \Omega, t>0 \\ \end{array}\right.(\ast)$
in a bounded domain with smooth boundary. We prove that if (1.7)-(1.13) hold, the model (
 $\ast$
) admits a global boundedness of classical solutions in any physically meaningful dimension. Moreover, we show that the global classical solutions
 $(u,v,w)$
exponentially converges to constant stable steady state
 $(u_\ast,v_\ast,w_\ast)$
. Inspired by [5], we employ the special structure of (
 $\ast$
) and carefully balance the triple cross diffusion. Indeed, we introduced some functions and combined them in a way that allowed us to cancel the bad items.
Citation: Guoqiang Ren, Bin Liu. Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021136
##### References:

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