Modelling the Growth and Decline of Populations
This first set of lectures introduces a range of mathematical functions commonly used to summarise the growth and decline of populations of molecules, cells and organisms in a range of biological applications. A second aim is to show how models can be constructed from biological assumptions, and to introduce the mathematical skills necessary to analyse the behaviour of these models. Models considered are in both discrete and continuous time, and key concepts are equilibrium behaviour and stability for these systems, together with analytic solution using calculus and/or recurrence equation techniques.
These lectures introduce fundamental concepts of probability and randomness, and how to build probabilistic models of biological systems. Such models are essential and widely used tools for understanding complex biological phenomena, and are particularly important in genetics and genome sequence analysis. The lectures also introduce basic population genetics and the Hardy-Weinberg equilibrium as an important example of a probabilistic model.
Modelling Interacting Populations
This part of the course extends the mathematical toolkit to allow analysis of more complicated biological models (in particular the dynamics of populations that are coupled together). The main focus is an examination of the dynamics of non-linear coupled differential equations, both analytically and by graphical "phase plane" techniques, taking an extension of differentiation to multiple dimensions in along the way.
Some knowledge of basic probability and statistics is absolutely necessary to interpret experimental results. This block of lectures introduces the language and notation of statistics, and describes key hypothesis tests to determine significance. Techniques covered include t-tests, chi-squared, ANOVA (analysis of variance) and linear regression. Practicals using the popular statistical package R consolidate the theoretical work.
A matrix is a convenient way to represent many numbers in one compact notation, but is pervasive in university level mathematics. Surprisingly such abstract mathematical entities can have real biological application, in particular after using the powerful ideas of Eigenvalue-Eigenvector decomposition to simplify products of matrices. We illustrate this here using a selection of models for populations.
This final lecture series builds on topics covered earlier in the course. The first lecture series explored the dynamics of single populations. Then Physiological Modelling provided further examples of using differential equation models in biology. Finally in the Lent term the mathematical methods needed to approach interacting population problems were covered. In this part of the course, we use these methods to look at various of coupled population systems, including predator-prey systems, competition within and between species, and the mathematical modeling of epidemics in plants, humans and animals.