Simulating a partial differential equation — reaction-diffusion systems and Turing patterns

Partial Differential Equations (PDEs) describe the evolution of dynamical systems involving both time and space. Examples in physics include sound, heat, electromagnetism, fluid flow, and elasticity, among others. Examples in biology include tumor growth, population dynamics, and epidemic propagations.

PDEs are hard to solve analytically. Therefore, PDEs are often studied via numerical simulations.

In this recipe, we will illustrate how to simulate a reaction-diffusion system described by a PDE called the FitzHugh–Nagumo equation. A reaction-diffusion system models the evolution of one or several variables subject to two processes: reaction (transformation of the variables into each other) and diffusion (spreading across a spatial region). Some chemical reactions can be described by this type of model, but there are other applications in physics, biology, ecology, and other disciplines...