Modelling persistence in spatially-explicit ecological and epidemiological systems

In this thesis, we consider the problem of long-term persistence in ecological and epidemiological systems. This is important in conservation biology for protecting species at risk of extinction and in epidemiology for reducing disease prevalence and working towards elimination. Understanding how to predict and control persistence is critical for these aims. In Chapter 2, we discuss existing ways of characterising persistence and their relationship with the modelling paradigms employed in ecology and epidemiology. We note that data are often limited to information on the state of particular patches or populations and are modelled using a metapopulation approach. In Chapter 3, we define persistence in relation to a pre-specified time horizon in stochastic single-species and two-species competition models, comparing results between discrete and continuous time simulations. We find that discrete and continuous time simulations can result in different persistence predictions, especially in the case of inter-specific competition. The study also serves to illustrate the shortcomings of defining persistence in relation to a specific time horizon. A more mathematically rigorous interpretation of persistence in stochastic models can be found by considering the quasi-stationary distribution (QSD) and the associated measure of mean time to extinction from quasi-stationarity. In Chapter 4, we investigate the contribution of individual patches to extinction times and metapopulation size, and provide predictors of patch value that can be calculated easily from readily available data. In Chapter 5, we focus directly on the QSD of heterogeneous systems. Through simulation, we investigate possible compressions of the QSD that could be used when standard numerical approaches fail due to high system dimensionality, and provide guidance on appropriate compression choices for different purposes. In Chapter 6, we consider deterministic models and investigate the effect of introducing additional patch states on the persistence threshold. We suggest a possible model that might be appropriate for making predictions that extend to stochastic systems. By considering a family of models as limiting cases of a more general model, we demonstrate a novel approach for deriving quantities of interest for linked models that should help guide modelling decisions. Finally, in Chapter 7, we draw out implications for conservation biology and disease control, as well as for future work on biological persistence.