Fitness Valleys, Metastability and Changing Environments

Abstract

Mathematical modelling of biological processes has become an area of high scientific interest. This field has expanded substantially over the past century and developed a wide variety of mathematical approaches and biological applications. The complexity of biological systems requires models taking random effects into account. <br/> This thesis investigates stochastic individual-based models of adaptive dynamics for asexually reproducing populations with mutation, focussing on the interplay between population dynamics, mutation rates, and environmental changes. Evolution is driven by linear birth rates, density-dependent logistic death rates, and mutations occurring along a finite trait graph. The model partially incorporates time-varying parameters, such as periodic changes in the environment or drug concentrations, which impact the evolutionary process. We investigate the behaviour of mutants and their invasion dynamics under small mutation rates and a simultaneously diverging population size, where environmental changes occur on a moderately diverging time scale. <br/> The results of the first part (Appendix A) provide a detailed analysis of transitions between evolutionary stable conditions (ESC) in a constant environment. Here multiple mutations need to be accumulated to cross fitness valleys. The system exhibits metastable behaviour across multiple time scales which are linked to the widths of these fitness valleys. Introducing a meta-graph framework of ESCs, we describe the multi-scale jump chain resulting from concatenated jumps and prove the convergence of the population process to a Markov jump process that visits only ESCs of sufficiently high stability. <br/> We then turn to the study of periodically changing environments. In the second part (Appendix B), we examine the growth of emergent mutants and their invasion of the resident population with a focus on mesoscopic scaling limits and the effective growth rates of mutants. The dynamics are influenced by an averaging effect of invasion fitness across different phases of the environment. <br/> Additionally, we explore the crossing of fitness valleys in a changing environment in the third part (Appendix B), distinguishing two cases: Under the assumption of a strict fitness valley, we can show that the crossing rates are computed as an average taking into account the ability to survive. A particularly interesting scenario is the pit stop phenomenon, where intermediate mutants within a fitness valley experience phases of positive fitness, allowing them to grow to large sizes before going extinct. This accelerates the traversal of the valley and introduces a novel time scale in the evolutionary process.