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المرحلة الجامعية المتقدمةApplied Mathematicsمُخطَّط له

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

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Ordinary differential equationsLinear algebra

عدد المتطلبات السابقة الإلزامية (حتى الأساس): 37

ستتعلّم

  • 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.

تعريف

  • 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.

مبرهنة

  • 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.

مفاهيم خاطئة شائعة

  • Treating model predictions as forecasts independent of their assumptions.
  • Thinking R₀ is a fixed biological constant; it depends on behaviour and environment.

اكتشافات اختيارية

مسارات التعلّم

قراءات إضافية

  • J. D. Murray, Mathematical Biology (1989)
  • Leah Edelstein-Keshet, Mathematical Models in Biology (1988)
  • Steven H. Strogatz, Nonlinear Dynamics and Chaos (1994)