Short Biomaths Modules

An educational module exploring the modelling of a single-species chemostat system using two coupled equations: a bacterial equation describing changes in bacterial density within the chemostat over time, and a nutrient equation describing changes in nutrient concentration.

Analysis of the model reveals two steady states: one in which no bacteria remain (washout) and another in which bacteria persist (coexistence).

The course follows the model through its derivation, nondimensionalisation, and stability analysis, before considering harvest optimisation as a practical application and a principled alternative to trial-and-error approaches. Worked examples throughout the course illustrate and reinforce the key aspects of the model.

An educational module focused exclusively on finding solutions to first-order partial differential equations using the method of characteristics. The course begins by classifying equations as linear, semilinear, quasilinear, or fully nonlinear, and then examines how the method of characteristics is formulated and applied to each class.

Worked examples are worked through to explicit closed-form solutions wherever possible, covering all four classes of equations. The examples include functions of both two and three variables, equations with variable coefficients, and non-trivial hypersurfaces, demonstrating the generality of the method beyond simpler, more specialised cases.

An educational module focused exclusively on finding solutions to linear second-order partial differential equations using the method of characteristics. The course considers the three classes into which linear second-order PDEs are classified—hyperbolic, parabolic, and elliptic—and shows how the method of characteristics can be used to transform each equation into a canonical form, making its underlying solution theory more transparent.

Worked examples include equations with variable coefficients and non-trivial hypersurfaces, leading to either general or particular solutions depending on the prescribed initial data, and applying integration techniques and ordinary differential equation methods where appropriate to solve the resulting canonical forms.

An educational module exploring a particular class of solutions to partial differential equations—travelling wave solutions of reaction-diffusion equations. The course focuses on few basic epidemic models, using the Fisher-KPP equation as a worked example before demonstrating how travelling waves can be applied to understand disease spread and persistence.

Rather than following the traditional approach of seeking explicit closed-form solutions, the course focuses on determining whether travelling wave solutions exist and, where they do, characterising their form. This is achieved through phase-plane analysis, drawing on appropriate analytical techniques.