【正文】 odels. 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。n, The ideal free distribution as an evolutionarily stable strategy, J. Biol. Dyn. 1 (20xx) 249–271. [11] . Cantrell, C. Cosner, Y. Lou, Movement towards better environments and the evolution of rapid diffusion, Math. Biosci. 204 (20xx) 199–214. [12] . Cantrell, C. Cosner, Y. Lou, Advection mediated coexistence of peting species, Proc. Roy. Soc. Edinburgh Sect. A 137 (20xx) 497–518. [13] . Chen, Y. Lou, Principal eigenvalue and eigenfunction of elliptic operator with large convection and its application to a petition model, Indiana Univ. Math. J. 57 (20xx) 627–658. [14] C. Cosner, A dynamic model for the idealfree distribution as a partial differential equation, Theor. Pop. Biol. 67 (20xx) 101–108. [15] C. Cosner, Y. Lou, Does movement toward better environments always benefit a population?, J. Math. Anal. Appl. 277 (20xx) 489–503. [16] J. Dockery, V. Hutson, K. Mischaikow, M. Pernarowski, The evolution of slow dispersal rates: A reaction–diffusion model, J. Math. Biol. 37 (1998) 61–83. [17] . Fretwell, . Lucas, On territorial behavior and other factors influencing habitat distribution in birds. I. Theoretical development, Acta Biotheor. 19 (1970) 16–36. [18] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equation of Second Order, second ed., SpringerVerlag, Berlin, 1983. [19] P. Grindrod, Models of individual aggregation or clustering in single and multispecies munities, J. Math. Biol. 26 (1988) 651–660. 具適應(yīng)性的人口疏散模型的 整體解 24 [20],On the global txistence of solutions to an aggregation madol,(20xx)379398 [21] and ,Broundness in a quasitinear parabolicparabolic KellerSegel System with subticat sensitivity, Equations ,252 (20xx)692715 [22] . McPeek, . Holt, The evolution of dispersal in spatially and temporally varying environments, Am. Nat. 140 (1992) 1010– 1027. [23] M. Kshatriya, C. Cosner, A continuum formulation of the ideal free distribution and its applications for population dynamics, Theor. Pop. Biol. 61 (20xx) 277– 284. [24] S. Kirkland, . Li, . Schreiber, On the evolution of dispersal in patchy environments, SIAM J. Appl. Math. 66 (20xx) 1366– 1382. [25] V. Hutson, K. Mischaikow, P. Pol225。 u(x,0)、 v(x,0)是連續(xù)的，非負(fù)的 ，不衡等零的。 。要做到這一點，我們會考慮雙物種模型，它們生態(tài)相同但是采用不用的擴散策略。 單一物種模型 在本文中，我們將考慮對上述模型 的變化，包括人口的增長和沿著定向的適應(yīng)性梯度運動的擴散。 研究問題 一個理想自由分布的關(guān)鍵想法是，個體們用這樣一個方式 —為了優(yōu)化他們的適應(yīng)性，將它們定位?！?3， 4】中兩個種群模型被用來代替反應(yīng)移流分布模型。 具適應(yīng)性表示人口向著資源密集的地方移動 (遷移 )。在那種情況下它遵循 McPeek和 Holt【 22】和區(qū)分非條件、有條件擴散之間的區(qū)別是有用的。假設(shè)一個種群有一個固有的人均增長率，m(x),該 m 在空間上不同但是經(jīng)歷增加的死亡率和或減少的繁殖成功率由于擁擠在整個環(huán)境一致得變化。同時，通過將分布和人口動態(tài)包含進(jìn)模型，我們能夠把它放進(jìn)一個框架，允許我們將它與其他已經(jīng)在擴散演化【 1113,16,25】背景下研究過的模型相比較。例如， g=0，對應(yīng)于通過簡單擴散的無條件擴散。s u p ( , )Ttk Q k Ixu u x??????. 在 ,( ) 2 ()kk TCQ???? 中， 我們依次以下方式引入半模 : ? ? 。 引理 假設(shè) (u,v)是 ()的在時間區(qū)間 ? ?T,0 上的一個古典解，則