Nonlinear stability for diffusion models in biology

The reduction method for studying optimal nonlinear stability of constant solutions to some ecological models with diffusion, which include the Cantrell–Cosner and the May–Leonard systems, is given. A new canonical energy (Lyapunov function) is introduced, and it is proved that the regions of linear and nonlinear stability coincide with a known radius of attraction for the initial data. Attention is focused on a May–Leonard system with circular symmetry, an asymmetric May–Leonard system with diffusion, and a system for aggregation of glia in the brain.