Advanced undergraduateApplied MathematicsPlanned
Mathematical biology
Mathematical models in biology: population growth and the logistic equation, predator–prey and competition models, epidemic models such as SIR, stability analysis by linearisation, and the limits of models.
math.applied.mathematical-biology
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 37
You will learn to
- Analyse the logistic equation and discrete-time population models, including the onset of chaos in the logistic map.
- Find the equilibria of predator–prey and SIR models and classify their stability using the Jacobian.
- Interpret the basic reproduction number R₀ and the assumptions behind it.
Definition
- The logistic equation is dN/dt = rN(1 − N/K).
- The SIR model is S′ = −βSI/N, I′ = βSI/N − γI, R′ = γI.
- In the simple SIR model, R₀ = β/γ is the expected number of new infections caused by one infected individual in a fully susceptible population.
Theorem
- In the SIR model the number of infected individuals initially grows if and only if R₀ S(0)/N > 1.
- An equilibrium at which every eigenvalue of the Jacobian has negative real part is locally asymptotically stable.
- The Lotka–Volterra predator–prey model has a conserved quantity, so its orbits are closed curves.
Common misconceptions
- Treating model predictions as forecasts independent of their assumptions.
- Thinking R₀ is a fixed biological constant; it depends on behaviour and environment.
Optional discoveries
Learning pathways
Further reading
- J. D. Murray, Mathematical Biology (1989)
- Leah Edelstein-Keshet, Mathematical Models in Biology (1988)
- Steven H. Strogatz, Nonlinear Dynamics and Chaos (1994)