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This course consists of twenty 50-minute lectures given over the first five weeks of the MRes year.

The aim is to introduce the student to some of the mathematical techniques most commonly used in the construction of models in the Life Sciences. However, time does not permit a deep and rigorous development of the underlying mathematical theories involved. Rather, the main results in each area are sketched out and illustrated by the discussion of selected examples. The course avoids methods that rely on heavy computational power, however important these are in practice. 

Topics covered include:

• Dynamical systems (discrete and continuous). Bifurcations. Reaction kinetics (if there is time). Systems biology.

• Models using PDEs (spatial effects and reaction-diffusion equations).

• Stochastic modelling. Markov processes.

Dynamical systems

• The logistic map, route to chaos, Lyapunov exponents. 
• Simple models of continuous-time dynamical systems such as the Lotka Volterra equations. 
• Sketch of the basic theory of local stability analysis, and introduction to the standard local bifurcations.

• Examples from enzyme kinetics. 
• The mass-action law. 
• The Michaelis-Menten and Briggs-Haldane approximations for an enzyme-substrate reaction. 
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• Switch systems that use autocatalytic stimulation of production of some product. 
• Activator-inhibitor systems and stable limit cycles. 

• Simple models from systems biology. 
• Gene and cellular signalling networks.

Reaction-diffusion equations

• Fisher's equation for invading organisms in a 1-dimensional spatial domain. Stability of equilibria. 
• Existence of a travelling wave solution and its stability. Approximate solution for the travelling wavefront.
• Switch systems with diffusion.
• Activator-inhibitor systems with spatial diffusion. 
• Turing instabilities and spatial patterning via 'morphogens'.
• Chemotaxis.

Stochastic modelling

• Markov chains. 
• Illustration of possible asymptotic states for a 2-state MC.
• Sketch of theory for general finite MCs. 
• The martingale property. 
• Illustration using the Wright-Fisher model from population genetics.