Aims & Learning Objectives:
Aims: The aim of the course is to introduce students to applications of partial differential equations to model problems arising in biology. The course will complement Mathematical Biology I where the emphasis was on ODEs and Difference Equations.
Students should be able to derive and interpret mathematical models of problems arising in biology using PDEs. They should be able to perform a linearised stability analysis of a reaction-diffusion system and determine criteria for diffusion-driven instability. They should be able to interpret the results in terms of the original biological problem.
Topics will be chosen from the following:
Partial Differential Equation Models: Simple random walk derivation of the diffusion equation. Solutions of the diffusion equation. Density-dependent diffusion. Conservation equation. Reaction-diffusion equations. Chemotaxis. Examples for insect dispersal and cell aggregation.
Spatial Pattern Formation: Turing mechanisms. Linear stability analysis. Conditions for diffusion-driven instability. Dispersion relation and Turing space. Scale and geometry effects. Mode selection and dispersion relation.
Applications: Animal coat markings. "How the leopard got its spots". Butterfly wing patterns.