Mathematical Biology(C)
MATHEMATICAL BIOLOGY (C)
Part II Dynamical Systems is useful.
Introduction to the role of mathematics in biology [1]
Systems without spatial structure: deterministic systems
- Examples: population dynamics, epidemiology, chemical reactions, physiological systems. Continuous and discrete population dynamics governed by deterministic ordinary differential equations or difference equations. Single population models: the logistic model and bifurcation to chaos; systems with time delay; age-structured populations. Two-species models: predator-prey interactions, competition, enzyme kinetics, infectious diseases. Phase-plane analysis, null-clines and stability of equilibrium. Systems exhibiting nonlinear oscillations: limit cycles; excitable systems. [9]
Stochastic systems
- Discrete stochastic models of birth and death processes. Master equations and Fokker-Planck equations. The continuum limit and the importance of fluctuations. Comparison of deterministic and stochastic models, including implications for extinction/invasion. Simple random walk and derivation of the diffusion equation. [6]
Systems with spatial structure: diffusion and reaction-diffusion systems
- The general transport equation. Fundamental solutions for steady and unsteady diffusion. Models with density-dependent diffusion. Fischer-Kolmogorov equation: propagation of reaction-diffusion waves. Chemotaxis and the growth of chemotactic instability. General conditions for diffusion-driven (Turing) instability: linear stability analysis and evolution of spatial pattern. [8]
Appropriate books
- L. Edelstein-Keshet Mathematical Models in Biology. SIAM classics in applied mathematics reprint, 2005
- J.D. Murray Mathematical Biology (3rd edition), especially volume 1. Springer, 2002
- S.P. Ellner and J. Guckenheimer Dynamic Models in Biology. Princeton University Press, 2006
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