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| Modelling Biodiversity and Pattern Formation with Evolutionary Games (AND) | |||||||||||||
Understanding the maintenance of biodiversity and the emergence of cooperation is an important topic in 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 uniform) models, it has been shown that the joint effect of noise and spatial degrees of freedom (dof) are important and realistic ingredients to be considered. In this project, we will use nonlinear dynamics, stochastic processes, the theory of differential equations and front propagation, as well as methods of statistical mechanics, to address problems of co-evolutionary dynamics in the presence of intrinsic noise and spatial dof. 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). The goal is to generalise the methods developed in recent works for the analysis of the nonlinear and stochastic evolutionary dynamics and to understand under which circumstances biodiversity is maintained and what sorts of out-of-equilibrium patterns emerge. (ii) Population models 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, as well as the emergence of sustained erratic oscillations in population's abundance, are striking examples of demographic noise's influence. To analyse these phenomena we will notably use methods of diffusion theory and mechanistic approaches that respectively allow to account for 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) Randomness in the environment and heterogeneity in the populations structure are ubiquitous in real ecological and biological systems (e.g., phase variation) and affect the evolutionary dynamics. As there is a lack of models accounting for randomness and heterogeneity in the presence of spatial dof and intrinsic noise, we plan to investigate evolutionary models in fluctuating environments and with spatially varying reaction rates. The goal is to quantitatively assess the influence of randomness and heterogeneity on the populations co-evolution and on the spatial arrangement of individuals. Mathematical Biology and Medicine Modelling biological systems is one of the most challenging and fastest growing research areas in Applied Mathematics. Mathematics and physics are used to describe biology at different levels: genes, proteins, cells and populations. The description can be simple, such as the time evolution of the number of cells, or more complex, such as the description, both in space and time, of the molecules inside a cell. In this group, the five permanent members of staff work with four postdocs and postgraduate students. In the Leeds Mathematical Biology and Medicine group, research is being carried out in theoretical immunology, gene regulatory networks, and synchronisation in neuronal networks. The immune system is one of the most complicated multiscale systems imaginable. The adaptive immune system of a vertebrate is a vast army of cells and molecules that cooperate to seek out, mark, bind to and destroy pathogens. Stochastic modelling is ideally suited to immunology at many scales. For example, cells live in a Brownian world, where motion is partly directed and partly random, so the battle between invading pathogens and the immune system is best described statistically. Similarly, gene expression is inlfuenced by noise and fluctuations: small numbers of molecules as well as the intrinsically stochastic nature of biochemical reactions mean that fluctuations must be taken into account in order to understand cellular function. |
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