Effects of a degeneracy in the competition model. I: Classical and generalized steady-state solutions (Q1604635)

This paper is devoted to how heterogeneous environmental effects can greatly change the dynamical behaviour of a population model. The author considers the competition model given by \[ \begin{cases} \frac{\partial u}{\partial t}-d_1(x)\Delta u=\lambda a_1(x)u-b(x)u^2-c(x)uv\\ \frac{\partial v}{\partial t}-d_2(x)\Delta v=\mu a_2(x)v-e(x)v^2-d(x)uv\end{cases}\tag{1} \] where \(x\in\Omega\) and \(t\geq 0\), \(\Omega\) denotes a smooth bounded domain in \(\mathbb{R}^N\) \((N\geq 2)\), \(d_1,d_2,a_1,a_2,b,c,d,e\) are nonnegative functions over \(\Omega\), and \(\lambda,\mu\) are positive constants. Throughout this paper the author supposes that \(u\) and \(v\) satisfy homogeneous Dirichlet boundary conditions. The author shows that there exists a critical number \(\lambda_*\) such that (1) behaves as if \(b(x)\) is a positive constant if \(\lambda<\lambda_*\), but if \(\lambda>\lambda_*\), then interesting new phenomena occur. This shows that heterogeneous environmental effects on population models are not only quantitative, but can be qualitative as well.