Mathematical modelling of predator-prey dynamics in complex environments

The main aim of mathematical ecology is to explore interactions among organisms and the environment where they live, and predator-prey interaction is one of the major type of interactions observed in nature. Models of predator-prey systems - mathematically described by ODEs, PDEs or integro-differential equations - have a long and illustrious history starting from the seminal works by Lotka and Volterra. However, despite a large number of existing publications in the literature, some fundamental questions related to this type of systems still remain open. For example, the spatial heterogeneity of the environment and its role in stabilisation of predator-prey dynamics and persistence of species is still not well understood. Another major challenge is the effect of external forcing (e.g. daily, seasonal, or other variation of model parameters) on long-term dynamics of the predator-prey or host parasite models. Finally, the parameterisation of model functions describing species interactions, for instance, formulation of the functional response of predator, can play a crucial role in the model outcomes. In the present dissertation, we explore the three above challenging issues (i.e. space heterogeneity, external forcing and model parametrisation) on the patterns of spatio-temporal dynamics of predator-prey or/and host-parasite systems and their stability. In particular, we revisit the famous paradox of enrichment which is classical in mathematical biology and explain how the spatial heterogeneity and animal movement on various time scales can stabilise the system characterised by an infinitely large carrying capacity (Chapter 2). Mathematically, we use a system of integro-differential equations and consider a tri-trophic planktonic system as a case study. In the two next chapters, we consider the role of daily and seasonal variation of temperature on the control of pathogenic bacteria by their predators: bacteriophages (i.e. bacterial viruses). As an important ecological case study, we explore seasonable dynamics of the infectious bacteria causing the lethal disease Melioidosis in Thailand. In the beginning we model interaction in the top water of a rice field (Chapter 3). Here we build two different models of host-parasite interactions based on ODEs and DDEs (delay differential equations). In Chapter 4, by using reaction-diffusion framework, we extend the previous model of bacteria-phage interactions to consider bacteria-phage dynamics in soil. Using our modelling approach we can make predictions about disease management of Melioidosis in tropic environments.