*(English)*Zbl 1042.35016

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

where $x\in {\Omega}$ and $t\ge 0$, ${\Omega}$ denotes a smooth bounded domain in ${\mathbb{R}}^{N}$ $(N\ge 2)$, ${d}_{1},{d}_{2},{a}_{1},{a}_{2},b,c,d,e$ 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 $b\left(x\right)$ 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.

##### MSC:

35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |

35B40 | Asymptotic behavior of solutions of PDE |

35K57 | Reaction-diffusion equations |

92D25 | Population dynamics (general) |