Modelling trade-offs in mutation-selection dynamics of trait-structured populations

Understanding the factors shaping the trait distribution of a biological population is essential for a predictive theory of evolution. Mathematical modelling provides a parsimonious and rigorous means to interpret empirical data, and to investigate plausible scenarios. In this context, the Replicator-Mutator Equation is a general tool to model mutation-selection dynamics of trait-structured populations. In such populations, individuals differ in the expression of one (or more) phenotypic traits. When trade-offs occur between different traits, then the corresponding trait distributions may reveal interesting, non-trivial behaviours. In this thesis, I will focus on the modelling of two such trade-offs. First, I will introduce the degeneracy-selection trade-off, according to which fitter phenotypes are expected to be less degenerate (thus more prone to mutation disruption). In the context of trait-structured populations, this trade-off is generally treated by means of effective formulations, rather than explicitly considering degenerate fitness landscapes. Here, I compare the two approaches and discuss the limits of effective formulations in the (inevitable) presence of asymmetries between traits. In the second part, I will show an application of the Replicator-Mutator Equation in the context of evolutionary epidemiology, where pathogen display heterogeneous levels of virulence and transmission. I will frame the problem in the context of agricultural practice, where the aim is to understand how to calibrate control strategies, in order to optimise pesticide use, and show how a proper trait-dependent control strategy can harness heterogeneity in pathogen populations to our advantage. Finally, I will discuss future lines of research based on the merging of the two parts, and on the potential for a general mathematical theory of trait-dependent control strategies of heterogeneous pathogens