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In this paper, we demonstrate some special behavior of steady-state solutions to a predator-prey model due to the introduction of spatial heterogeneity. We show that positive steady-state solutions with certain prescribed spatial patterns can be obtained when the spatial environment is designed suitably. Moreover, we observe some essential differences of(More)
We consider bounded solutions of the Cauchy problem { ut −∆u = f(u), x ∈ R , t > 0, u(0, x) = u0(x), x ∈ R , where u0 is a nonnegative function with compact support and f is a C1 function on R with f(0) = 0. Assuming a minor nondegeneracy condition on f , we prove that, as t→∞, the solution u(·, t) converges to an equilibrium φ locally uniformly in RN .(More)
We study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. For simplicity, we assume that the environment and the solution are radially symmetric. In the(More)
We present several recent results obtained in our attempts to understand the influence of spatial heterogeneity in the predator-prey models. Two different approaches are taken. The first approach is based on the observation that the behavior of many diffusive population models is very sensitive to certain coefficient functions becoming small in part of the(More)
This is Part II of our study on the positive steady state of a quasi-linear reactiondiffusion system in one space dimension introduced by Klausmeier and Litchman for the modelling of the distributions of phytoplankton biomass and its nutrient. In Part I, we proved nearly optimal existence and nonexistence results. In Part II, we obtain complete descriptions(More)