AbstractStochastic modelling of biological processes seems complex and mathematically intractable at first. However, previous research has shown that uncertainty and randomness often simplify the assumptions for theoretical models, and various fields in mathematics provide useful tools to analyse and extract information that is otherwise hidden in deterministic mean-field approaches. In this thesis we describe population dynamics in ecosystems and evolutionary processes through the lens of stochastic mathematical biology and develop methods to analyse phenomena that emerge from fluctuations and uncertainty.
First, we model the dynamics of large random ecosystems in terms of randomly distributed interaction parameters and stochastic differential equations. We then compute the power spectral density for this process, where we utilise the random nature of the underlying interaction network and derive solutions in dependence of statistical properties of the system, rather than biological presumptions. The scaling laws we obtain for the fluctuation spectra reveal information about the dominant interaction types within the ecological network and open questions with regards to temporal stability of ecosystems as opposed to local stability in deterministic models. Furthermore, we discuss the effects of ecosystem fluctuations on the equilibrium dynamics of an embedded focal species.
In the second part we combine ecological dynamics at the population level with random mutations, and show how macroscopic structure emerges in the resulting evolutionary process. Here we consider an isogamous species, where the number of compatible mating types for sexual reproduction is not necessarily limited to two. Unlike in a model with neutral mutations, we find that fitness differences damp the growth of the average number of mating types and derive predictions independent of the underlying fitness distribution. Finally, we further discuss potential models for the evolution of sexual reproduction in the context of ecosystem dynamics by combining the modelling approaches we present throughout this thesis.
|Date of Award
|16 Nov 2022
|The Royal Society
|Tim Rogers (Supervisor), Ben Ashby (Supervisor) & George Constable (Supervisor)