F10AM - Mathematical Biology A

Course leader(s):

Andy White (Edinburgh, September)

Aims

The is a mathematical modelling course that aims to teach the application of ordinary differential equations and difference equations to problems in ecology. It will provide an understanding of the mathematical modelling methods that describe the interactions between populations, infectious disease processes and evolutionary processes in ecological systems. It will provide training in a wide variety of mathematical techniques which are used to describe ecological systems and provide instruction in the biological interpretation of mathematical results.

Syllabus

1. Single species population models (1.1 Continuous and discrete time model formulations and analysis; population growth, period-doubling bifurcations, chaos, graphical stability analysis, cobweb diagrams, harvesting problems, insect population models.)

2. Multi species population models (2.1 Continuous and discrete time model formulations and analysis; nondimensionalisation, linear stability analysis, phase plane methods, symbiotic, competitive, predator-prey and host-parasite ecological interactions; Age-structured models.)

3. Mathematical models of ecological systems (3.1 Develop mathematical models from descriptive information of ecological systems; model analysis and biological interpretation of results.)

4. Epidemiological models (4.1 Models of infectious disease; infection transmission and recovery processes, threshold conditions for epidemic outbreaks, the basic reproductive rate of a disease; vaccination strategies to control infection.)

5. Evolution and evolutionary game theory (5.1 Modelling the evolution of life history parameters; the evolution of reproduction and carrying capacity, the evolution of infection, trade-offs between parameters; Game theoretical approaches to evolution; 2-strategy games Hawk-Dove.)

Learning outcomes

By the end of the course, students should be able to do the following:

determine steady states and their stability for single species models in continuous and discrete time.
determine steady states and their stability for ecological models consisting of two coupled ODEs or two coupled discrete time equations.
demonstrate the ability to interpret infectious disease models and analyse the models using linear stability analysis and phase plane methods.
determine the evolutionary outcome in ODE systems where parameter values can evolve and in 2-strategy evolutionary games.
demonstrate expertise in developing and interpreting the findings of mathematical models that represent a range of ecological situations.
Produce phase plane portraits.
Develop and interpret mathematical models for a range of biological situations.
Understand infectious disease models and the concepts of epidemic, endemic and disease-free states.
Derive R0, the basic reproduction rate for a disease.
Understand vaccination strategies to control infectious disease.
Understand how mutation and invasion can lead to the evolution of life history parameters.
Understand the concept of a trade-off between parameters and the effect of trade-off shape on evolution
Understand the concept of evolutionary game theory and an evolutionarily stable strategy (ESS).
Determine Evolutionarily Stable Strategies in 2-strategy games.
Give biological interpretations for the terms in ordinary differential equation and discrete time models for ecological systems.
Provide a biological interpretation of mathematical results.

Further details

Curriculum explorer: Click here

SCQF Level: 10

Credits: 15