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 so-called age-structured population models, where birth and death rates of individuals are allowed to depend on the age of the individual. If time permits, more advanced structured population models are considered, incorporating e.g. size of indivuals, or their spatial location.
In the course we introduce and discuss Laplace transform techniques for the analysis of theses systems and the concept of monotone dynamical systems in relation to population dynamics.
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.
Students hand in solutions to 3-4 assignments during the course. The final grade is determined by the average over the results for the assignments.
For all material and up to date information about the course see the lecturer’s home page
Lecturer’s home page