Applications of Moving Mesh Finite Elements to Population Dynamics

The project concerns mathematical modelling of ecological systems on temporal and spatial scales in quantitative population ecology.
The purpose of the project is to exploit certain two-dimensional mathematical models of competition-diffusion systems, with aggregation and clustering effects, in dynamic domains. A version of the partilal differential Lotka-Volterra competition equations will be studied that describes a two-phase reaction-diffusion system. Of special interest is multi-phase systems, particularly those segregated with a dynamic interface between phases, for example between different physical states or between warring species. An effective numerical technique is the moving mesh finite element method (MMFEM) of Baines, Hubbard and Jimack, that uses a geometric conservation approach to mesh adaptation. This technique has recently been exploited by Watkins in her PhD thesis (Reading 2017) which considers the application of the MMFEM to systems of intraspecies and interspecies interactions with aggregating dynamics and clustering. A touchstone is the Stefan problem with dynamical interface behaviour, as in the work of Hilhorst.

Following on from familiarisation with the background and the numerical technique, the student will compare the behaviour of the models against empirical data sets. The models lend themselves to dynamical adaptations in the sizes and shapes of the domains, as well as to alterations to the logistic terms and changes in parameters, without the need for further calculations. This adaptability means there is a wide range of realistic biological and ecological systems to which the models can be applied and validated. Comparison with data sets for species which show competition-diffusion-aggregation behaviour will be a particular objective of the research.
The models are highly suitable for tackling how changes in the resource space might alter behaviour in a dynamical context. An aim will be to understand these requirements from both a mathematical and quantitative perspective. The subsequent development work will be in the direction of the research requirements of those ecological systems which would most benefit from a study which has access to this modelling capability.
Key References:
Baines, Hubbard and Jimack, (i) A Moving Mesh Finite Element Algorithm for the Adaptive Solution of Time-Dependent Partial Differential Equations with Moving Boundaries}, Appl. Numer. Math., 54, pp. 450-469, 2005, (ii) A moving-mesh finite element method and its application to the numerical solution of phase-change problems), Commun. in Comput.Phys., 6. pp. 595-624 (2009). (with R.Mahmoud)
Watkins, A Moving Mesh Finite Element Method and its Application to Population Dynamics, PhD thesis, University of Reading, UK (2017).
Hilhorst, Vanishing latent heat limit in a Stefan-like problem arising in biology, (Nonlinear analysis: real world applications), 4, pp.261-285, 2003.