How to Learn Mathematical Biology

Study exponential and logistic growth models, understand carrying capacity, and analyze single-species dynamics. Learn to interpret phase line diagrams and identify stable and unstable equilibria in one-dimensional systems.

Explore multi-species models including Lotka-Volterra predator-prey and competition models. Master phase plane analysis techniques: nullclines, fixed points, linearization, and classification of equilibria using the Jacobian matrix.

Learn the SIR, SIS, and SEIR compartmental models. Compute and interpret the basic reproduction number (R0), analyze disease-free and endemic equilibria, and study the effects of vaccination and intervention strategies.

Study Michaelis-Menten enzyme kinetics, the law of mass action, and gene regulatory network models. Apply quasi-steady-state approximations and explore bistability, oscillations, and feedback loops in intracellular systems.

Move to partial differential equations: diffusion equations, Fisher's equation for traveling waves, and Turing's reaction-diffusion theory of morphogenesis. Analyze conditions for pattern instability and simulate spatial dynamics.

Learn stochastic modeling with birth-death processes, the chemical master equation, and the Gillespie algorithm. Study discrete-time models including Leslie matrices for age-structured populations and Boolean networks for gene regulation.

Explore evolutionary game theory, multi-scale modeling, delay differential equations, agent-based models, and the integration of data-driven methods with mechanistic models. Read current research papers and apply models to real biological data sets.