We provide an asymptotic analysis of a nonlinear integro-differential equation which describes the evolutionary dynamics of a population which reproduces sexually and which is subject to selection and competition. The sexual reproduction is modeled via a nonlinear integral term, known as the 'infinitesimal model'. We consider a regime of small segregational variance, where a parameter in the infinitesimal operator, which measures the deviation between the trait of the offspring and the mean parental trait, is small. We prove that in this regime the phenotypic distribution remains close to a Gaussian profile with a fixed small variance and we characterize the dynamics of the mean phenotypic trait via an ordinary differential equation. We also briefly discuss the extension of the method to the study of steady solutions and their stability.[-]

Consider a diploid population living in one spatial dimension. Suppose a particular gene appears in two forms (alleles) A and a, and that individuals carrying AA have a higher fitness than aa individuals, while Aa individuals have a lower fitness than both AA and aa individuals. We can prove that on a suitable timescale, the genealogy of a sample of A alleles taken from near the wavefront converges to a Kingman coalescent as the population density goes to infinity. Joint work with Alison Etheridge.[-]

Start with a rooted tree, and to each vertex attach a label consisting of an i.i.d. uniform(0,1) random variable plus a parameter theta times its distance from the root. Are there paths from the root of the tree to infinity along which the labels are increasing? Clearly this depends on the tree and the value of theta. The aim is to specify a critical theta, depending on the tree, above which the answer is yes with positive probability, and below which the answer is no. We also consider the same question on the integer lattice, where the bounds are worse but the pictures are better. This is a work in progress with Diana De Armas Bellon.[-]