Diffusive logistic equations with indefinite weights: population models in disrupted environments II

The dynamics of a population inhabiting a strongly heterogeneous environment are modeled by diffusive logistic equations of the form $u_1 = \nabla \cdot (d(x,u) + \nabla u) - {\bf b}(x) \cdot \nabla u + m(x)u - cu^2 $ in $\Omega \times (0,\infty )$, where u represents the population density, $d(x,u)$ the (possibly) density dependent diffusion rate, ${\bf b(x)}$ drift, c the limiting effects of crowding, and $m(x)$ the local growth rate of the population. The growth rate $m(x)$ is positive on favorable habitats and negative on unfavorable ones. The environment $\Omega $ is bounded and surrounded by uninhabitable regions, so that $u = 0$ on $\partial \Omega \times (0,\infty )$. In a previous paper, the authors considered the special case $d(x,u) = d$, a constant, and ${\bf b} = 0$, and were able to make an analysis based on variational methods. The inclusion of density dependent diffusion and/or drift makes for more flexible and realistic models. However, variational methods are mathematically insufficient ...