Biological systems are seldom well-mixed, but rather have spatial variations. In such cases, it is important to consider transport processes within the system, for instance in the spread of an invasive species, the swimming of bacteria towards nutrients, or the morphogenesis of a tiger's stripes. This module will cover the main mathematical techniques for modelling biological systems with transport, and will be focused around systems of coupled advection-diffusion-reaction partial differential equations, as well as agent-based equations.
The aims of this module are:
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To develop and understand a range of models for transport processes in biology.
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To articulate commonality in these models across systems, and elucidate their differences.
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Develop the partial differential equations relating to agent-based transport models, understanding when these are valid.
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Quantify a range of wave-like and self-similar transport behaviours displayed in various biological systems.
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Understand spatial pattern formation and diffusion-driven instability.
- Measures, Carathéodory's construction, integration and convergence theorems.
- Riesz representation theorem, weak* convergence and Prokhorov's theorem.
- Hardy-Littlewood maximal inequality and Rademacher’s theorem.
The second part provides an introduction to geometric measure theory. Time permitting, we will cover some of the following topics:
- Hausdorff distance.
- Hausdorff measure, rectifiable and purely unrectifiable sets.
- Sard's theorem.
- The Besicovitch projection theorem.
Material to be covered:
Reminder of measure theory
modes of convergence
law of large numbers
central limit theorem (via characteristic functions, Lindeberg principle, Stein's method)
stable laws
large deviations
martingales
References:
S.R.S. Varadhan, Probability Theory (Courant lecture notes), online notes
L. Breiman, Probability theory
F. den Hollander, Large Deviations
N. Zygouras, Discrete stochastic analysis
Notes on Large Deviations