Method of lines for reaction-diffusion systems admitting invariant regions

Systems of nonlinear reaction-diffusion equations arise in various fields, including chemistry, population dynamics, pattern formation, phase transitions, and image processing. With the exception of few analytically solvable cases, they are treated by numerical methods carefully adjusted to capture the nonlinear phenomena exhibited by the solution. This article shows how to extend the notion of invariant regions generalizing the maximum principle for diffusion equations to the finite-difference method of lines, and how to consequently prove convergence of the underlying numerical scheme. We also provide two particular examples of reaction-diffusion systems in one-dimensional space with a diagonal diffusion operator, which are solved by the presented numerical method.