The course will consider the mathematical modelling and analysis of the behaviour of populations of individual organisms. We will discuss the analysis of long-term dynamics of:
(1) Models with finite type space, which are realised as a system of non-linear ordinary differential equations (ODEs). In particular we shall consider the Lotka-Volterra type of predator-prey models and extensions thereof. They have a particularly nice and rich abstract mathematical structure that allows for a general approach to studying the long term behaviour of their solutions. Another example in this class is furnished by elementary models for the spread of infectious disease, in particular the widely used SIR-model (from
Susceptible’,Infectious’ and `Resistant’ individuals).
(2) Models with infinite (continuum) state space, which consist of systems of partial differential equations (PDEs) and integro-partial differential equations. As a running example we shall discuss the modelling of chemotaxis: a biological process in which individuals move in response to an observed concentration of a chemical compound in their environment and adjust their movement characteristics accordingly, resulting in pattern formation, e.g. aggregation. We introduce the Patlak-Keller-Segel class and the class of kinetic models for chemotaxis. We discuss their relationship, in particular how the former can be derived from the latter through diffusive scaling, and long-term behaviour.
Analysis 1-3, Linear Algebra 1-2; Linear Analysis, an introduction to PDEs and/or Dynamical Systems will be greatly helpful, but is not required.
The course is built on various books and research papers. No particular book needs to be purchased.
Multiple individual assignments
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Lecturer’s home page