【正文】 a single species to a twospecies petition model. Both species have the same population dynamics, but one species adopts a bination of random and fitnessdependent dispersal and the other adopts random dispersal. Global existence of smooth solutions to the timedependent quasilinear parabolic system is studied. When a single species has a strong tendency to move up its fitness gradient, it results in a stable equilibrium that can approximate the spatial distribution predicted by the ideal free distribution (Cantrell et al., 20xx [9]). For the twospecies petition model, if one species has strong tendency to move up its fitness gradient, such approximately ideal free dispersal is advantageous relative to random dispersal. Bifurcation analysis shows that two peting species can coexist when one species has only an intermediate tendency to move up its fitness gradient and the other species has a smaller random dispersal rate. 1. Introduction This work extends our previous work [9] on fitnessdependent dispersal for a single species to a twospecies petition model, with one species adopting a bination of random and fitnessdependent dispersal and the other adopting random dispersal. 具適應(yīng)性的人口疏散模型的 整體解 27 The model we considered in [9] has the form ? ?? ? ? ?? ?? ???? ???????????? 0, ,nuxfuu uxufuxfuuu t ?? ?? () Where ? ???? ,0R , and ? ? ? ? uxmuxf ??, () The function u(x, t) represents the density of a single species with random diffusion coefficient μ, and α measures the tendency of the species to move upward along the gradient of the fitness of the species, measured by f (x, u). We assume that μ is a positive constant and α is a nonnegative constant. Ω is a bounded region in RN with boundary ?Ω, and n denotes the outward unit normal vector on ?Ω. Throughout this paper we assume that m ∈ C2,γ (Ω) for some γ ∈ (0, 1) and m is positive somewhere in Ω, and u(x, 0) is continuous, nonnegative and not identically zero in Ω. We briefly summarize some of the main results in [9] as follows: ? (Global existence in time) Suppose that μ 0 and α _ 0. Then () has a unique solution u ∈ C2,1(Ω (0,∞)) ∩ C(Ω [0,∞)). ? (Existence of positive steady state) If u = 0 is linearly unstable, then () has at least one positive steady state. Note that if _Ω m 0, u = 0 is linearly unstable for any μ 0 and α _ 0. ? (Global attractor) If m 0 in Ω, then for large α/μ, () has a unique positive steady state which is also globally asymptotically stable. To study the evolution of dispersal, a mon approach, initiated by Hastings [24] for reaction– diffusion models, is to consider models of two populations that are ecologically identical but use different dispersal strategies. In general, using such a modeling approach would lead to a system of the form ? ?? ? ? ?? ?? ?? ? ? ?? ????????????????????????????????0,),(),(,nvuxguunvuxfuuvuxvfvuxgvvvvuxufvuxufuutt???????? () where f is as in (), and g represents part of an alternate dispersal strategy. For example, g = 0 具適應(yīng)性的人口疏散模型的 整體解 28 would correspond to unconditional dispersal of anisms by simple diffusion, g = m would correspond to advection up resource gradient without consideration of crowding, while g = ?(u + v) would correspond to avoidance of crowding without reference to resource distribution. We refer to [3–5,7–11,13,14,17,23,27,30–32,42,38,46] for recent progress in this direction for reaction–diffusion models. In this paper we will focus on system () with g = 0, ., ? ?? ? ? ?? ?? ???????????????????????????0,),(,nvnvuxfuuvuxvfvvvuxufvuxufuutt????? () where the initial conditions u(x, 0) and v(x, 0) are nonnegative and not identically zero in Ω, and μ, ν, α are all positive constants. We consider reaction–diffusion–advection models for spatially distributed populations that have a tendency to disperse up the gradient of fitness, where fitness is defined as a logistic local population growth rate. We show that in temporally constant but spatially varying environments such populations have equilibrium distributions that can approximate those that would be predicted by a version of the ideal free distribution incorporating population dynamics. The modeling approach shows that a dispersal mechanism based on local information about the environment and population density can approximate the ideal free distribution. The analysis suggests that such a dispersal mechanism may sometimes be advantageous because it allows populations to approximately track resource availability. The models are quasilinear parabolic equations with nonlinear boundary conditions. Our analysis is partially motivated by an interest in understanding the evolution of dispersal in spatially varying but temporally constant environments. In that context it is useful to follow McPeek and Holt [28] and distinguish between unconditional and conditional dispersal. Unconditional dispersal refers to dispersal without regard to the environment or the presence of other anisms. Pure diffusion and diffusion with 具適應(yīng)性的人口疏散模型的 整體解 29 physical advection (. due to winds or currents) are examples of unconditional dispersal. Conditional dispersal refers to dispersal that is influenced by the environment or the presence of other anisms. It has been shown that in the framework of spatially explicit population models on spatially varying but temporally constant environments with only unconditional dispersal that evolution favors slow dispersal [16,21,28]. A reason why unconditional dispersal is not favored is that it leads to a mismatch between the distribution of population and the distribution of resources. However, for certain type