Sharp spatial patterns of the diffusive Holling-Tanner prey-predator model in heterogeneous environment. (English) Zbl 1180.35290

Wang, Mingxin; Pang, Peter Y. H.; Chen, Wenyan

IMA J. Appl. Math. 73, No. 5, 815-835 (2008).

The authors deal with the following elliptic system: \[ \begin{cases} -\Delta u=\lambda u-a(x)u^2-\frac{buv}{1+\gamma(x)u}, & x\in\Omega\\ -\Delta v=\mu v\left(1-\frac vu\right), & x\in\Omega\\ \frac{\partial u} {\partial\nu}=\frac{\partial v}{\partial\nu}=0, & x\in\partial\Omega \end{cases}\tag{1} \] where \(\Omega\subset\mathbb{R}^n\) is a bounded domain with smooth boundary \(\partial\Omega\), the parameters \(\lambda,\mu\), and \(b\) are positive constants and \(a(x)\) and \(\gamma(x)\) are non-negative continuous functions on \(\overline\Omega\). The main purpose of this paper is to study the effect of the degeneracies of \(a(x)\) and \((\gamma(x)\) on the steady-state behaviour in heterogeneous environment, that is on positive solutions of (1).

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

35K57	Reaction-diffusion equations
92D25	Population dynamics (general)

Keywords:

Holling-Tanner prey-predator model; heterogeneous environments; diffusion; positive steady states; spatial patterns