Natural selection is central to the understanding of how species evolve and adapt. We consider the accumulation of beneficial and deleterious mutations in populations of moderate size, which depends strongly on the effects of natural selection. The rate of adaptation, i.e. the long-term speed of increase/decrease of the average fitness of the population, is affected by the total mutation rate, proportion of beneficial mutations, and population size. We describe a weak selection model that converges to an infinite system of interacting stochastic differential equations after proper scaling. We use a Girsanov transformation of probability measures to obtain an expansion formula for the rate of adaptation, with each term dependent only on what happens in a corresponding neutral model without the selection mechanism. Furthermore, we propose an extension to our model that takes into account the effects of recombination, and give a heuristic argument for the advantage of sexual reproduction (compared to asexual reproduction) in this setting.

Keywords: Rate of adaptation; Natural selection; Interacting stochastic differential equations; Fisher's fundamental theorm

Biography: Dr Feng Yu received a Ph.D. in mathematics in 2005 from the University of British Columbia, under the supervision of Prof Ed Perkins. He subsequently performed postdoctoral work at the University of Oxford under the supervision of Prof Alison Etheridge, before taking up his current appointment of a lectureship at the University of Bristol in 2007. His main research interest is in stochastic processes arising in population genetics.