Understanding the maintenance of biodiversity and the emergence of cooperation are important topics in the Life and Behavioural Sciences. Evolutionary game theory, where the success of one species depends on what the others are doing, provides a promising mathematical framework to study the coexistence dynamics of interacting populations. As paradigmatic examples, the prisoners dilemma and the rock-paper-scissors games have emerged as a fruitful metaphor for cooperative and co-evolutionary dynamics (with applications in microbiology and ecology). While mathematical biology classically deals with deterministic (and often spatially homogeneous) models, it has been shown that the joint effect of noise and spatial degrees of freedom are important and realistic ingredients to be considered. In our research, we use tools of nonlinear dynamics, stochastic processes, differential equations and the theory of front propagation, as well as methods of statistical mechanics, to study the co-evolutionary dynamics of structured and unstructured populations in the presence of intrinsic noise. More specifically, possible lines of investigation are the following:
(i) It has recently been demonstrated that populations movement can have important evolutionary implications. Here, we shall consider evolutionary models with realistic forms of mobility (e.g,. inspired by chemotaxis) and different types of interactions between the species (e.g., to account for long-range interactions between colicinogenic and sensitive bacteria, or mutations) and study the joint influence of movement and stochastic noise on the population's self-organisation and co-evolution.
(ii) Mathematical models of population dynamics are classically formulated in terms of rate equations whose predictions are now recognised to be altered by stochastic effects. The extinction of sub-populations and the fixation of mutants are striking examples of the influence of stochastic noise. To analyse these phenomena we will notably use suitable size expansion methods (diffusion approximation and WBK theory) that respectively allow to account for random (weak) and large fluctuations. It is planned to carry out this line of research notably on complete and complex graphs for ecologically and biologically motivated models.
(iii) In nature, organisms often interact with a finite number of individuals in their neighbourhood. The population is heterogeneously structured and cannot be described by well-mixed models. This often results in patterns observed in ecosystems and whose origin is an intense subject of research. According to Turing's theory, diffusion can yield pattern-forming instabilities in systems with species of different diffusivities. However, pattern formation has also been observed in ecosystems not displaying large separation of diffusivities, and it has recently been proposed that intrinsic noise together with movement can be the generic mechanism responsible for the emergence of patterns. Here, we would like to test this scenario by investigating the origin of pattern formation in paradigmatic examples like the "rock-paper-scissor" model and its variants. An approach will be to adopt an "individual-based" approach for metapopulation models where interacting sub-populations are subdivided in connected islands and can migrate from one patch to another.