Mathematical Treatment of Evolutionary Ecology

本项目研究自然选择和扩散对由单一进化物种组成的种群的影响，该种群具有一个位点和两个等位基因的遗传系统。该模型是进化生态学的一个例子，它是种群生态学与种群遗传学的融合。该模型比经典种群遗传学理论中遇到的模型更现实，因为它允许密度依赖选择。假设种群规模的增加会降低种群中个体的适合度。未来的目标将是研究几个相互作用的物种的情况，或共同进化理论与扩散。从数学上讲，这是一个反应扩散方程组。有必要证明在不同不变流形上连接两个静止点的行波的存在性。每一个流形对应一个等位基因的缺失。这些波的稳定性将被检验。这些结果将更全面地说明主要研究者关于反应-扩散系统解的渐近行为的结论。它将使我们深入了解如何处理难以应用最大原则的系统。

This project is concerned with the study of the effects of natural selection and diffusion on a population consisting of a single evolving species with a genetic system of one locus with two alleles. The model is an example from evolutionary ecology, which is a fusion of population ecology with population genetics. The model is more realistic than those encountered in the classical theory of population genetics because it allows density dependent selection. It is assumed that increase in population size decreases the fitness of the individuals in the population. A future goal will be to study the case of several interacting species, or coevolutionary theory with diffusion. Mathematically, this is a system of reaction-diffusion equations. It is necessary to prove the existence of travelling waves connecting two rest points lying on different invariant manifolds. Each of these manifolds corresponds to the absence of one allele. The stability of these waves will be examined. The results will give a more complete picture of what has been shown by the principal investigator on the asymptotic behavior of solutions to the reaction-diffusion system. It will give insight into the ways to handle systems where the maximum principle is hard to apply.