# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021136
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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:

show all references