Mathematical Modelling of Evolution in Complex Biological Systems

Modelling biological evolution is a growing area of research with different approaches used with ever-growing applications, proved to be beneficial in various real-world applications such as engineering, economics, biology, machine learning, optimal control. However, many aspects of modelling evolution remain understudied, especially concerning the situation where the life history trait/behaviour is described by a set of functions whose shape is unknown and/or the underlying system is highly complex, for example, infinite-dimensional. This thesis is comprised of two main parts, each of which investigates a different mathematical approach to reveal behaviours and life-history traits that emerged as a result of long-term evolution.

Here, we attempt to formalise Darwin’s fundamental ideas of survival of the fittest to develop a new framework to obtain evolutionarily optimal life-history traits/behavioural patterns based on the reconstruction of evolutionary fitness using underlying equations for population dynamics, applicable to Hilbert spaces with infinitely high dimensional spaces for life-history traits. This inspired the novel method, based on the principles of evolution, of stochastic global optimisation in high or even infinite-dimensional Hilbert spaces, named Survival of the Fittest Algorithm (SoFA). We test the novel algorithm on a phenomenon of particular interest, the mass synchronised diel vertical migration (DVM) of zooplankton, whose fitness is highly dependent on the trajectories of these movements. Using this ecologically relevant case study, we demonstrate that for maximising fitness in high-dimensional spaces, our proposed novel evolutionary algorithm, SoFA, provides better performances compared to other stochastic global optimisation algorithms.

We then apply current game-theoretical approaches to evolutionary optimisation to various complex models. Among many insightful results are the following. Considering an SI model with the infected subpopulation described by a von Förster-type model, in which the infection load plays the role of age. We demonstrate that in this infinite-dimensional system, for simple trade-off functions between virulence, disease transmission and parasite growth rates, multiple evolutionary attractors are possible. A result not observed in the case of unstructured models, indicating the benefits of additional complexity within modelling pathogens. Furthermore, theoretically exploring the co-evolution of life-history traits in a generic host-parasite-hyperparasite system, we find that in the presence of hyperparasites, the evolutionarily optimal pathogen virulence generally shifts towards more virulent strains. However, the use of hyperparasites in biocontrol is still justifiable since overall host mortality decreases. An intriguing possible outcome of the evolution of the hyperparasite can be its evolutionary suicide. The presented results help warrant biocontrol agents and programs in pathogen management and provide a better understanding of pathogenic infections.