Mathematical Sciences: Reaction-Diffusion Equations and Evolutionary Ecology

1937年，费雪写了一篇经典的论文，论述了一种有利基因在生活在同一单维栖息地的种群中的传播。费雪的模型是高度理想化的。例如，他假设个体的适应度和种群规模是恒定的。这使得模型难以应用。首席研究员将在没有这些假设的情况下研究这个模型，从而使模型更加现实。这就提出了许多具有挑战性的数学问题，主要研究者打算使用拓扑方法，如度理论、射击论证和康利指数理论来克服这些问题。得到的结果将清楚地说明应用数学对生物科学的影响，特别是自然选择中的非线性效应。

In 1937, Fisher wrote a classic paper on the spread of an advantageous gene in a population living in a homogeneous one- dimensional habitat. Fisher's model is highly idealized. For example, he assumed that the fitnesses of the individuals and the population size are constant. This makes the model difficult to apply. The principal investigator will to study this model without such assumptions, thus making the model more realistic. This poses many challenging mathematical problems which the principal investigator intends to surmount using topological methods such as degree theory, shooting arguments and Conley's index theory. Results obtained will clearly illustrate the impact of applied mathematics on biological sciences, in particular the effects of nonlinearities in natural selection.