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doi: 10.3934/dcdsb.2021218
Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model
Yu Ma 1,, , Chunlai Mu 1, and Shuyan Qiu 2,
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
School of Sciences, Southwest Petroleum University, Chengdu 610500, China
* Corresponding author: Yu Ma
Received May 2021 Revised July 2021 Early access September 2021
This work deals with a Neumann initial-boundary value problem for a two-species predator-prey chemotaxis system
$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = d_1\Delta u-\chi\nabla\cdot(u\nabla w)+u(\lambda-u+av),\quad &x\in \Omega,\quad t>0,\\ v_t = d_2\Delta v+\xi\nabla\cdot(v\nabla w)+v(\mu-v-bu),\quad &x\in \Omega,\quad t>0,\\ 0 = d_3\Delta w-\alpha w+\beta_1 u+ \beta_2 v,\quad &x\in\Omega,\quad t>0,\\ \end{array} \right. \end{eqnarray*} $
in a bounded domain
$ \Omega\subset \mathbb{R}^n \,\,(n = 2,3) $
with smooth boundary
$ \partial\Omega $
, where the parameters
$ d_1, d_2, d_3,\chi, \xi,\lambda,\mu,\alpha,\beta_1,\beta_2, a, b $
are positive. It is shown that for any appropriate regular initial date
$ u_0 $
$ v_0 $
, the corresponding system possesses a global bounded classical solution in
$ n = 2 $
, and also in
$ \chi $
being sufficiently small. Moreover, by constructing some suitable functionals, it is proved that if
$ b\lambda<\mu $
and the parameters
$ \xi $
are sufficiently small, then the solution
$ (u,v,w) $
of this system converges to
$ (\frac{\lambda+a\mu}{1+ab}, \frac{\mu-b\lambda}{1+ab}, \frac{\beta_1(\lambda+a\mu)+\beta_2(\mu-b\lambda)}{\alpha(1+ab)}) $
exponentially as
$ t\rightarrow \infty $
; if
$ b\lambda\geq \mu $
is sufficiently small and
is arbitrary, then the solution
converges to
$ (\lambda,0,\frac{\beta_1\lambda}{\alpha}) $
with exponential decay when
$ b\lambda> \mu $
, and with algebraic decay when
$ b\lambda = \mu $
Keywords: chemotaxis, predator-prey, boundedness, logistic source, asymptotic stability.
Mathematics Subject Classification: Primary: 35A01, 35B40; Secondary: 35Q92, 92C17.
Citation: Yu Ma, Chunlai Mu, Shuyan Qiu. Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021218
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Yu Ma Chunlai Mu Shuyan Qiu