The effects of stochasticity and population structure on mathematical descriptions of evolutionary dynamics

研究领域：数学生物学达尔文进化论基于三个基本原则：繁殖，突变和选择。这些决定了特征在种群中如何随时间变化。对于由此产生的进化动力学，有许多数学描述。在以前的工作中，我们展示了这些框架中的几个可以通过两个等价的动态描述来统一：Price方程和复制-增变方程，Page和Nowak，2002。然而，这些描述是确定性的。在小群体中和/或当选择系数的大小很小时，随机效应很重要。有数学描述的随机动力学发生在中性进化，例如木村，1968年。我们的目标是开发一个统一的框架，它可以描述两个随机效应，如遗传漂变，也包括价格/复制-突变框架。我们打算开始与离散的个人的随机描述。从这些，我们打算得到一个宏观的描述，例如使用不同表型的概率/概率密度的方程。这将建立在Champagnat等人的工作基础上，2006年，旨在解释他们的模型如何与以前的数学描述。我们将开始与随机过程模型，并从他们导出福克-普朗克方程/其他宏观描述。我们的目标是描述框架中缺少的内容，例如由空间位置引起的种群结构的影响。将种群结构的影响纳入进化动力学模型是当前非常感兴趣的领域，最近在特殊情况下取得了进展，例如Bolker和Pacala，1996，艾伦和Tarnita，2014，艾伦，Nowak和Dieckmann，2013，巴顿，Depaulis和Etheridge，2002，Chalub和Souza，2009。参考文献：Page & Nowak（2002）Unifying evolutionary dynamics. Journal of Theoretical Biology 219，93- 98. Kimura（1968）Evolutionary rate at the molecular level. Champagnat，Ferriere，Meleard（2006）Unifying evolutionary dynamics：from individual stochastic processes to macroscopic models.理论种群生物学69，297- 321. Bolker和Pacala（1996）使用矩方程来理解生态系统中随机驱动的空间格局形成。理论种群生物学52，179 -197.Allen和Tarnita（2014）具有固定种群大小和结构的一类进化模型的成功措施，数学生物学杂志68，109 -143.Allen，Nowak和Dieckmann（2013）具有相互作用结构的自适应动力学。美国博物学家181，E139- 163.巴顿，Depaulis和Etheridge（2002）空间连续种群中的中性进化。Chalub和Souza（2009）从离散到连续进化模型：漂移扩散和复制动力学的统一方法，理论人口生物学76，268-277。

Research Area: Mathematical BiologyDarwinian evolution is based on three fundamental principles: reproduction, mutation and selection. These determine how traits change in time within a population. There are numerous mathematical descriptions of the resulting evolutionary dynamics. In previous work, we showed how several of these frameworks could be unified by two equivalent descriptions of the dynamics: the Price equation and the replicator-mutator equation, Page and Nowak, 2002. These descriptions are, however, deterministic. In small populations and/or when selective coefficients have small magnitude, random effects are important. There are mathematical descriptions of the stochastic dynamics that occur in neutral evolution, e.g. Kimura, 1968. We aim to develop a unifying framework, which can describe both stochastic effects, like genetic drift, and also incorporate the Price/ replicator-mutator framework.We intend to begin with a stochastic description of discrete individuals. From these we intend to derive a macroscopic description, using e.g. equations for the probabilities/ probability densities of different phenotypes. This will build on work by Champagnat et al., 2006, and aim to explain how their models relate to previous mathematical descriptions.We will begin with stochastic process models and derive from them Fokker-Planck equations/ other macroscopic descriptions. We aim to describe what is missing in the framework, such as effects of population structure, resulting e.g. from spatial location.Incorporating the effects of population structure into models of evolutionary dynamics is an area of great current interest, where recent advances have been made in special cases, see for example Bolker and Pacala, 1996, Allen and Tarnita, 2014, Allen, Nowak and Dieckmann, 2013, Barton, Depaulis and Etheridge, 2002, Chalub and Souza, 2009. We intend to build on these works.References:Page & Nowak (2002) Unifying evolutionary dynamics. Journal of Theoretical Biology 219, 93-98.Kimura (1968) Evolutionary rate at the molecular level. Nature 217, 624-626.Champagnat, Ferriere, Meleard (2006) Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theoretical Population Biology 69, 297-321.Bolker and Pacala (1996) Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theoretical Population Biology 52,179-197.Allen and Tarnita (2014) Measures of success in a class of evolutionary models with fixed population size and structure, Journal of Mathematical Biology 68,109-143.Allen, Nowak and Dieckmann (2013) Adaptive dynamics with interaction structure. American Naturalist 181, E139-163.Barton, Depaulis and Etheridge (2002) Neutral Evolution in Spatially Continuous Populations. Theoretical Population Biology 61, 31-48.Chalub and Souza (2009) From discrete to continuous evolution models: a unifying approach to drift-diffusion and replicator dynamics, Theoretical Population Biology 76, 268-277.