Modelling populations in heterogeneous environments

现在有大量的数学文献致力于模拟生物种群的动态。所采用的模型通常分为两类：生态模型，旨在阐明种群内和种群之间以及种群与环境之间的相互作用;种群遗传学模型，旨在解释在种群样本中观察到的遗传变异模式。生态模型通常考虑到（一些）空间结构，资源竞争，捕食者-猎物的相互作用和不断变化的环境条件。它们通常假设人口数量无限，从而使人们可以专注于增长率的波动，而忽略人口统计的随机性。相比之下，群体遗传学的模型往往集中于人口统计学的随机性（导致所谓的随机遗传漂变），这种随机性来自于有限群体中繁殖的随机性，并试图用一个单一的参数，即“有效群体规模”来捕捉潜在的生态学。当然，这是一个过于简单化，有生态模型，其中包括人口统计随机性，正如有遗传模型，其中包括波动选择（导致波动的增长率）。然而，在这两种情况下，虽然有广泛的模拟研究，相对较少的注意力已经支付给分析模型，占的空间和时间heterogeneity. We的目标在这个项目中的影响是写下来，并分析模型，结合联合收割机环境和人口的随机性，特别是，允许调查的影响异质条件下的人口动态。首先要解决的问题是种群的生存能力和可入侵性。我们将开始与人口细分成有限数量的离散补丁，但最终我们希望解决的空间连续体的（困难）问题，并开发方法，使我们能够保留有关人口中个体祖先的信息，环境条件的波动对种群动态和生物多样性的影响是研究的核心问题，虽然我们的工作本质上是理论性的，但它有可能为更多的应用工作提供信息。面对气候变化，了解栖息地质量变化与物种迁移之间的相关性将在保护中发挥重要作用。同样，预测（和减轻）入侵物种影响的能力需要更好地理解环境变化对种群增长率和入侵速度的影响。

There is now a vast mathematical literature devoted to modelling the dynamics of biological populations. The models employed generally fall into one of two classes: ecological models, that aim to elucidate the interactions within and between populations, and between those populations and the environment; and models of population genetics, that aim to explain the patterns of genetic variation observed in samples from a population. Ecological models typically take into account (some of) spatial structure, competition for resources, predator-prey interactions and changing environmental conditions. Often they assume infinite populations, allowing one to concentrate on fluctuations in growth rates and ignore demographic stochasticity. Models from population genetics, by contrast, often concentrate on the demographic stochasticity (leading to what is known as random genetic drift) which arises from the randomness due to reproduction in a finite population and try to capture the underlying ecology in a single parameter, the `effective population size'. Of course this is an over-simplification and there are ecological models which incorporate demographic stochasticity, just as there are genetic models that incorporate fluctuating selection (leading to fluctuating growth rates). However, in both settings, although there are extensive simulation studies, comparatively little attention has been paid to analytic models which account for the effects of both spatial and temporal heterogeneity.Our goal in this project is to write down and analyse models that combine environmental and demographic stochasticity and, in particular, allow an investigation of the effects of heterogeneous conditions on the dynamics of a population. The first questions to be addressed concern viability and invasibility of the populations. We shall begin with populations that are subdivided into a finite number of discrete patches, but ultimately we wish to tackle the (difficult) problem of spatial continua and to develop approaches that allow us to retain information about the ancestry of individuals in the population, with a view to using genetic data to unravel population history.The effect of fluctuating environmental conditions on population dynamics and biodiversity is a central question in ecology and although our work is theoretical in nature, it has the potential to inform much more applied work. In the face of climate change, understanding the correlations between changing habitat quality and speices movement will play an important role in conservation. Equally, the ability to forecast (and mitigate) the impact of an invading species requires a better undertanding of the effects of environmental variability on population growth rates and invasion speeds.