A probabilistic approach to fractional reaction-diffusion equations

这个研究项目将使用概率和分析技术来回答生物学中出现的问题。特别是，种群如何扩大其范围的问题为我们提供了复杂的数学模型来研究。标准模型（所谓的Fisher-KPP方程）中有几个关键的缺陷，这将提供有趣的研究课题。例如，Fisher-KPP方程不能解释在种群中观察到的某些现象。这种现象的一个例子是“Allee效应”，在这种效应下，当人口密度小时，可能不会出现对一个人来说最佳的条件（例如，在人口密度低的地区，对资源的竞争可能会减少，但其他优势，如免受牛群的保护，也会受到损害）。Allee效应可以通过多种方式影响种群的范围扩展。我们希望研究的一个主题是“膨胀负荷”。这是对在不断扩大的人口的前端积累有害突变的名称。在不存在Allee效应的情况下，扩展负荷归因于个体从处于前沿（其中种群密度小且生长速率最大）中获得的优势，而不是抵消有害突变所带来的不利。在存在噪声的情况下（对应于有限种群中由于繁殖而产生的随机性），最适合的类型可能会从扩展的种群前沿完全丢失。然而，在Allee效应下，人口增长有利于更高密度的地区，而人口前沿的有害突变不再享有这种优势。在存在阿利效应的情况下，人口的最大增长率位于扩张前沿的后面。因此，人们可能会期望人口中更健康的个体能够加入前线。我们将研究有害突变在不同的种群扩张模型中的积累（在穆勒棘轮的空间模拟中）。在数学上，我们将占人口中的阿利效应的存在，将一个“合作项”到我们的模型。特别感兴趣的是理解选择，范围扩张和所谓的遗传漂变（有限种群中繁殖的随机性）之间的相互作用。对这种情况的模拟呈现出一幅有趣的画面，将有助于我们更好地理解这些相互作用。

This research project will use techniques from probability and analysis to answer questions arising in biology. In particular, the question of how populations expand their range provides us with sophisticated mathematical models to study. There are several critical deficiencies in the standard model (the so-called Fisher-KPP equation) that would provide interesting research topics. For instance, the Fisher-KPP equation does not account for certain phenomena observed in populations. An example of one such phenomenon is the "Allee effect", under which the optimal conditions for an individual may not occur when the population density is small (for example, competition for resources might decrease in areas of low population density, but other advantages, such as protection from the herd, will also be compromised). The Allee effect can impact the range expansion of a population in a number of ways. One topic that we should like to investigate is that of `expansion load'. This is the name given to the accumulation of deleterious mutations at the front of an expanding population. In the absence of an Allee effect, expansion load is attributed to the advantage that an individual enjoys from being at the front (where population density is small and growth rates are maximal) more than offsetting the disadvantage conferred by a deleterious mutation. In the presence of noise (corresponding to the randomness due to reproduction in a finite population), the fittest types can be completely lost from the expanding population front. However, under an Allee effect, population growth favours areas of higher density, and deleterious mutations at the population front no longer enjoy this advantage. In the presence of an Allee effect, the maximum growth rate of the population sits behind the expanding front. One might therefore expect fitter individuals in the population's bulk to recolonise the front. We shall investigate the accumulation of deleterious mutations in different models of expanding populations (in a spatial analogue of Muller's ratchet). Mathematically, we will account for the presence of the Allee effect in a population by incorporating a `cooperation term' into our models. Of particular interest is to understand the interplay between selection, range expansion and what is known as genetic drift (the randomness due to reproduction in a finite population). Simulations of this scenario present an intriguing picture that will help us better understand these interactions.