Chapter 1
What is a mathematical model?
The modelling cycle; units and dimensions; first ODEs from rate laws; difference equations.
Chapter 2
Dimensional analysis and nondimensionalisation
Characteristic scales; nondimensionalising an ODE; solution collapse.
Chapter 3
The Buckingham Π-theorem and the dimension matrix
Dimension matrices; finding Π-groups systematically; similarity and scaling laws.
Chapter 4
Ordinary differential equations
Separable equations; integrating factor; Bernoulli; second-order constant-coefficient ODEs; simple harmonic motion; damped oscillations.
Chapter 5
Single species models
Exponential and logistic growth; equilibria and stability; harvesting; bifurcation diagrams.
Chapter 6
Numerical solution of ODEs
Euler’s method; Heun’s method; error analysis; systems of ODEs; Lotka–Volterra and Van der Pol worked examples.
Chapter 7
Data-driven modelling with neural networks
When equations aren’t available; function approximation; feedforward networks; training; mechanistic vs data-driven models.