Spatial Population Dynamics in Changing Ecosystems

Most classical models of interacting populations neglect the spatial structure of ecosystems. You may have seen, for instance, the Lotka-Volterra model of predator-prey dynamics, or the competitive variant modelling two species competing for resources. These ODE models can give rise to rich behaviour, especially when considering more than two species or trophic levels, such as chaos in tritrophic predator-prey dynamics. But in order to model the impact of environmental changes, due to climate or human fragmentation of ecosystems, it is often necessary to consider spatial extensions of these models.

This project will be an introduction to spatial modelling in population biology. You will study either spatially discrete models where the populations move on networks (graphs), or spatially continuous models which lead to partial differential equations (some species, such as annual plants, are better modelled with difference or integrodifference equations). You will be given a few starting points in contemporary literature of simple models that include spatial population dynamics, and then you will consider extensions of these to account for environmental change. You are also welcome to develop your own model of a particular ecosystem. It is expected that you will be able to do simple analytical investigations (such as phase plane analysis, or asymptotic solutions) and will more thoroughly explore these models numerically.

Please do email me at krause@maths.ox.ac.uk if you have any questions, and I'd be happy to chat in more detail about the project.

Prerequisites

None. Some familiarity with basic dynamical systems modelling (phase plane analysis, systems of ODEs) is helpful. Some familiarity with scientific programming (MATLAB, Python, Julia, C++ etc) and numerical solutions of differential equations is also very helpful. The mathematical modelling or numerical analysis courses include some components of Python programming, and this should be more than sufficient.

Resources

Murray's Mathematical Biology - Volume I contains helpful material on non-spatial models, and Vol II some examples of spatial modelling. There are many free online discussions of models as well, such as Hal Smith's analysis of RM, and others which I can suggest. Sections 3 and 4 of this blog post also nicely illustrate spatial and spatially discrete modelling frameworks.

Spatial Ecology is a good collection of frameworks for modelling spatial dispersal.

Many papers have been written on these kinds of models studying many different questions. Examples include: PDE 1, PDE 2, Networks, Hybrid Spatial Models

Integrodifference Equations in Spatial Ecology (mostly applicable for particular kinds of organisms, such as seasonal plants)