Mathematical models of host-parasite co-evolution

An important driver of evolution is the selection pressure that results from interactions between species. One such interaction is the antagonistic relationship of hosts and parasites, characterised by a strong disadvantage for the host and an often essential gain for the parasite. This thesis extends various mathematical models on host-parasite co-evolution by detailing verbally formulated theories and testing model robustness. The models illuminate biological mechanisms and provide long term evolutionary dynamics that are hard to capture experimentally. Within the same population hosts can be differentially susceptible and parasites are often specific to certain host types. Therefore, those parasites that can infect the most common host are temporarily the best adapted, but later, when another host is more common, a different parasite is the momentary winner. The resulting oscillations of type abundances within a population are called Red Queen dynamics. In the first part of this work, Red Queen dynamics are examined under various mathematical and biological assumptions. Relative changes within a population are displayed using evolutionary game theory, but including population dynamics the equations become comparable to the Lotka-Volterra system for predator-prey dynamics. In the first project, these oscillations show chaotic properties when the number of types (dimensions) increases and when oscillation amplitudes are large. In the second project, both constant and changing population size models are derived from individual interactions and compared for their biological and mathematical properties. The stochastic dynamics result in diversity loss, which is sped up in the changing population size models, but can be counteracted by re-emerging extinct types. The effect is especially unpredictable under intermediate population sizes, when deterministic attractive dynamics are balanced with stochastic diffusion. In the second part, selection pressure on both species either to evade infection by the parasite or to find new ways to exploit the host is examined separately. The first project is concerned with the evolution of a bacterial host. Experiments have shown that the success of phage (parasite) infecting bacteria (host) depends on the physiological status of the host. Including important dependencies in an established model for phage infection introduces a bi-stability for the presence of phage. A numerical invasion analysis reveals that a slow growing bacterial mutant can invade a resident population of fast growing hosts and in some cases even remove the phage infection completely. In the final project, a model for bacterial persistence, a survival strategy of the parasite, is developed together with empiricists. Bacteria related to Crohn’s disease are compared to harmless bacteria. The strategies of bacteria to form dormant stages and evade stress are formulated using ordinary differential equations. The model is fitted to laboratory data to reveal important growth and switch parameters. The pathogenic bacteria are more stress-resistant and withstand antibiotic treatment by forming more dormant stages within the human immune system than their harmless relatives. Overall, the models presented here showcase the diverse utility of mathematics and theory in evolutionary biology. Red Queen dynamics are tested for robustness and discussed for their applicability and implications. The evolutionary advantage of both host and parasite strategies are mechanistically founded and the final integration of laboratory data and theoretical reasoning presents a successful interdisciplinary collaboration