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

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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.