【導(dǎo)讀】具適應(yīng)性表示人口向著資源密集的地方移動(遷移)。這里的疏散不僅僅指。人口疏散，還可以表示生物種群的擴散演化。對這些問題的研究在生物學(xué)、社會。學(xué)上有著廣泛的應(yīng)用。證明其在整個時間區(qū)間上的古典解的存在唯一性。所以很自然的要問，種群的擴散過程是如何導(dǎo)致空間分布的模式的，什么樣的模式產(chǎn)生怎樣的過程，在這方面已經(jīng)作出了相當(dāng)大的努。一個版本的理想自由分布連續(xù)空間可以源自于一種平流分布方程，該方程。我們考慮一個在這個模型中同樣包括隨機擴散的分布部分的變。我們將表明隨著比率向上層適應(yīng)性梯度移動變得更大和或擴散比率變小，這。個生物的擴散被我們的模型預(yù)測接近于期望的理想自由棲息地的選擇。中兩個種群模型被用來代替反應(yīng)移流分布模型。那些文本中可以看出。在那種情況下它遵循McPeek和Holt和區(qū)分非條件、有條件擴。散之間的區(qū)別是有用的。純擴散和與物質(zhì)的移流相關(guān)的擴散都是無條件擴散。為什么無條件擴散室不利的是因為它導(dǎo)

　　

【正文】 he 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 types of conditional dispersal, evolution can sometimes favor faster dispersal if that allows the population to track resources more efficiently [11,12,28]. These conclusions were obtained by considering models for two petitors that use different dispersal strategies but otherwise are ecologically identical, and examining the evolutionary stability of the strategies in terms of invasibility. (A strategy is considered evolutionarily stable if a population using that strategy cannot be invaded by a small population using a different strategy.) We plan to consider ideal free dispersal from that viewpoint in future work. To do that, we need to understand well the behavior of a single species using ideal free dispersal。 developing that understanding is the goal of this paper. Further it is worth noting that dispersal processes that result in patterns embodying certain features of the ideal free distribution have been shown to be evolutionarily stable in discrete diffusion models。 see [10,26]. However, it should also be noted that in models with temporal variation in the coefficients or plex dynamics, faster unconditional dispersal may sometimes be favored。 see [23,24,28]. Some of these phenomena and other aspects of the ecological effects of directed versus random movement and the evolution of dispersal are studied in the context of twopatch models in [3,4]. A key idea underlying the ideal free distribution is that individuals will locate themselves in such a way as to optimize their fitness. Thus, at equilibrium, all anisms in the occupied part of the habitat will have equal fitness and there will be no movement of individuals if the population is constant. A continuum model that captures those features was introduced in [25]. Suppose that a population has an 具適應(yīng)性的人口疏散模型的 整體解 30 intrinsic per capita growth rate m(x) that varies in space but experiences increased mortality and/or decreased reproductive success due to crowding uniformly throughout its environment. If the population density is scaled appropriately the local reproductive fitness of an individual at location x in the presence of conspecifics at density u(x) is given by f(x,u)=m(x)u(x) Let F denote the fitness of anisms in the occupied part of the habitat Ω. For a fixed total population U the distribution of the population will be given by u = ? _m(x)? F if m(x) F 。 0 otherwise, where F is made as large as possible subject to the conditions 。)(184。 Udxxu ??? ? ? ? ?? ??? ?????? 0)(: )(0)(: xux UdxxmFxux The first of these conditions simply requires the total population to be conserved. The second condition is obtained by integrating the previous formula for the density u(x). It can be used to determine F and the region where u(x) 0 by viewing it as a constraint a