Schedule for: 16w5113 - Stochastic and Deterministic Models for Evolutionary Biology

A welcome drink will be served at the hotel.

A popular line of research in evolutionary biology is to use time-calibrated phylogenies in order to infer the underlying process of species diversification. Most models of diversification assume that species are exchangeable and lead to phylogenetic trees whose shape is the same in distribution as that of a Yule pure-birth tree. Here, we propose a non-exchangeable, individual-based, point mutation model of diversification where interspecific pairwise competition (rate d) is always weaker than intraspecific pairwise competition (rate c), and is only felt from the part of individuals belonging to younger species. The only important parameter in this model is d/c, which can also be seen as a selection coefficient. In the large population limit, the rescaled abundances of species are deterministic but the phylogeny itself is random, interpolating between the completely asymmetric tree (d/c<<1) and the Kingman coalescent (d/c close to 1).

We consider some mathematical models in advective environments, where individuals are exposed to unidirectional flow, with the possibility of being lost through the boundary. We study the persistence and range for a single species. We also consider the evolution of dispersal in such advective environments.

In this talk, I will present some joint work with Olivier Bonnefon, Jimmy Garnier and Lionel Roques concerning the inside dynamics of positive solutions of some non-local equations. The notion of inside dynamics has been recently used to characterise the neutral genetic structure of a colonizing population modeled by a travelling wave solution of a homogeneous reaction diffusion equation. The notions of pushed and pulled fronts were at this purpose introduced. I will present some extensions of the notions of pushed/pulled fronts to the case of positive solutions of some homogeneous non-local reaction diffusion equations and the classification that we were able to achieve in a monostable situation.

We consider an integro-PDE model for a population structured by the spatial variables and a trait variable affecting the dispersal coefficients. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We focus on the asymptotic profile of positive steady state solutions. Our result shows that in the limit of small mutation rate, the solution remains regular in the spatial variables and yet concentrates in the trait variable and forms Dirac concetrations (i) at one boundary point; (ii) the interior; or (iii) at both boundary points. In particular, evolutionary branching is found in spatially heteogeneous but temporally constant environment. Other connections to notions and concepts in evolutionary game theory will also be discussed. This is joint work with Wenrui Hao (MBI) and Yuan Lou (Ohio State).

Life history theory provides a powerful framework to understand the evolution of pathogens in both epidemic and endemic situations. This framework, however, relies on the assumption that pathogen populations are very large and that one can neglect the effects of demographic stochasticity. Here we explore the effects of finite population size on the evolution of pathogen virulence and transmission. We show that demographic stochasticity introduces additional evolutionary forces that can affect qualitatively the dynamics and the evolutionary outcome. We discuss the importance of the shape of pathogen fitness landscape and host heterogeneity on the balance between mutation, selection and genetic drift. In particular, we discuss scenarios where finite population size can either select for lower or higher virulence. This analysis reconciles adaptive dynamics with population genetics in finite populations and provides a new theoretical framework to study life-history evolution. (Todd L. Parsons, Amaury Lambert, Troy Day and Sylvain Gandon)

I will introduce an ODE model and then a SDE model that describes dynamics of trait distribution including evolutionary branching.

Many different organisms are capable of regulating their gene expression to adjust their physiology and possibly change their phenotype in response to cues from the environment. There is a growing body of evidence that some differentiation processes exhibit excitability and change their behaviour influenced by random processes. I will show examples and discuss different problems of possible interest involving random extensions of deterministic, nonlinear dynamical systems.

We will describe an equation that models the invasion of cane toads in Australia. We will present results about long-time and long-range behavior of the population and connections to Hamilton-Jacobi equations.

We study the balance between local selection, mutation and migration in a two habitat model with asexual reproduction. We introduce a robust method to fully characterize the phenotypic distribution at equilibrium. We show that the distributions are not necessarily superpositions of Gaussian functions and in this way, we improve the predictions of Adaptive Dynamics and Quantitative Genetics. The method can be applied to general habitats with migration in both directions (not necessarily symmetric habitats) and to source-sink models. We discuss the implications of these analytic results for the emergence of local adaptation. This is a joint work with Sylvain Gandon.

In this theoretical study, we aim at explaining the role of growth and dispersal in spatial evolution. It shows that an invasion can succeed even if the 'invader' has a lower growth rate, thanks to greater dispersal abilities. This pattern occurs in many situations, such as tumor expansion, invading species, biofilm growth. The theoretical model is experimentally illustrated in the lab with a bacterial system.

Structured models have raised a lot of interest in multiscale modelling of collective motion and dispersal evolution. A minimal reaction-diffusion model has been proposed by Benichou et al. to model the propagation of some invasive species with a high heterogeneity in dispersal capability among individuals. I will present some recent progresses about the qualitative and quantitative study of spreading for this model and related ones. In particular, we will highlight an acceleration phenomena that has intrigued field biologists in the last decade.

Structured models have raised a lot of interest in multiscale modelling of collective motion and dispersal evolution. A minimal reaction-diffusion model has been proposed by Benichou et al. to model the propagation of some invasive species with a high heterogeneity in dispersal capability among individuals. I will present some recent progresses about the qualitative and quantitative study of spreading for this model and related ones. In particular, we will highlight an acceleration phenomena that has intrigued field biologists in the last decade.

Based on a Cannings-type model that is inspired by the design of the Lenki experiment, we show that very simple assumptions on the mutation-selection mechanism in the adaptive evolution lead to a parabolic form of the long-term relative fitness curve, for which more complicated explanations have previoulsly been given in the literature. This is joint work with Adrian Gonzales Casanova, Noemi Kurt and Linglong Yuan (http://arxiv.org/abs/1505.01751, to appear in SPA).

The aim is to advocate the use of renewal equations by giving examples and highlighting some of the available theory.

In large adapting clonal species, several beneficial mutations can co-occur, affecting the process of adaptation, and especially its rate. Several experimental and theoretical works showed that clonal interference can be an important factor limiting the rate of adaptation. However, models done so far do not embrace the diversity of observed dynamics in experiments, especially non-linear dynamics. We develop here a three-type stochastic birth and death model with explicit competitive interactions between clones and describe the complexity of the emerging dynamics of the population, supposing that two mutants enter a resident population in a single copy at different times. These clones can either get fixed, be lost or be maintained in polymorphism, depending on their competitive abilities. We show that frequency-dependent selection can give rise to unexpected dynamics: competitive interactions between clones can foster adaptation by increasing or decreasing both the fixation probability and time of beneficial mutations.