Dispersion in Spacio-Temporally Heterogeneous Environments

这个项目的目的是为了更好地理解扩散速率和环境的时空异质性之间的关系。特别是，研究者想要了解是否有任何基本的关系，在此基础上，这个理论的框架可以发展。因此，为了最大限度地减少外来影响，研究者继续研究可能是最简单的连续模型，它明确地包含了一个空间变量：一个反应扩散方程系统，其中每个物种的反应项是相同的，其中出生率在空间和时间上是异质的。这个模型考虑了各种各样的扰动，通过这些扰动，人们可以研究各种因素对扩散速率演化的影响。当然，这种模式也有一些局限性。第一个假设是将扩散表示为简单扩散。研究人员希望了解出租车对分散率和空间异质性之间关系的影响。在更一般的层面上，研究者打算用一个积分核来代替扩散。人们普遍认为，生态和进化从根本上受到环境空间特征的影响。作为一个例子，我们可以考虑多样性的悖论。不包括任何空间成分的简单数学模型产生了竞争排斥原理：当两个物种为同一有限资源而竞争时，其中一个物种通常会灭绝。另一方面，人们通常观察到，在各种各样的栖息地中，许多物种共存。这可以通过空间效应来解释，至少在一定程度上是这样。当然，一旦引入了空间因素，扩散率就成为一个中心特征。不幸的是，在这种更普遍的情况下，我们对因果关系的理解很差。造成这种情况的原因似乎有四方面。首先，现实的生态和环境模型中变量的数量是巨大的。其次，空间异质性发生在环境的所有尺度上。第三，从实地研究中获得这些变量的精确数据是极其困难的。最后，目前处理包含空间和动态特性的模型的数学技术似乎是不够的。在这种情况下，对简单模型进行了深入的研究，以期阐明基本的生物学原理和确定基本的数学问题。

Mischaikow0107396 The goal of this project is to better understand therelationship between dispersal rates and the spatio-temporalheterogeneity of environments. In particular, the investigatorwould like to understand if there are any fundamentalrelationships upon which a framework for this theory can bedeveloped. Therefore, to minimize extraneous effects theinvestigator continues to study what is perhaps the simplestcontinuous model that explicitly incorporates a spatial variable:a system of reaction diffusion equations with Lotka-Volterrareaction terms, where the reaction term of each species isidentical, and where the birth rate is spatially and temporallyheterogeneous. This model allows for a variety of perturbationsthrough which one can study the impact of various factors in theevolution of dispersal rates. Of course, this model has severallimitations. The first is the assumption that leads to thedispersal being represented as simple diffusion. The investigatorwishes to understand the effects of taxis on the relationshipbetween dispersal rates and spatial heterogeneity. On a moregeneral level the investigator intends to replace diffusion by anintegral kernel. That ecology and evolution are fundamentally influenced bythe spatial characteristics of the environment is well accepted.As an example of this one may consider the paradox of diversity.Simple mathematical models that do not include any spatialcomponent give rise to the principle of competitive exclusion:when two species compete for the same limited resource one of thespecies usually becomes extinct. On the other hand it is commonlyobserved that in a wide variety of habitats a multitude ofspecies coexist. This can be explained, at least in part, byincluding spatial effects. Of course, once spatial components areintroduced, dispersal rates become a central feature.Unfortunately, our understanding of cause and effect in this moregeneral situation is poor. The reasons for this appears to befourfold. First, the number of variables in realistic ecologicaland environmental models is enormous. Second, spatialheterogeneities occur at all scales of the environment. Third,obtaining precise data for these variables from field studies isextremely difficult. Finally, current mathematical techniques forhandling models that incorporate both spatial and dynamicalproperties seem to be inadequate. Given this state of affairs, asimple model is thoroughly investigated in the hopes ofelucidating the basic biological principles and identifying thefundamental mathematical issues.