In population dynamics, cross-diffusion terms describe the attempt of individuals of one species to avoid individuals of another species. Even though their particular form can seem artificial, it can be easily seen, at least at a formal level, that the cross-diffusion terms appear in the fast-reaction limit of a “microscopic” (in terms of time scales) presenting only standard diffusion and fast-reaction terms. The limiting cross-diffusion system naturally incorporates the processes happening at different time scales. This approach led to the justification of the cross-diffusion SKT model for competing species, proposed to account for stable inhomogeneous steady states (patterns) exhibiting spatial segregation.

In this talk, I will introduce the fast-reaction system for competing species leading to the SKT model. By combining linearised analysis and numerical continuation techniques, we will understand why cross-diffusion is the key ingredient for pattern formation and we will see the possible emerging patterns in the bifurcation diagram.

The same approach can be also exploited in other contexts, for instance predator-prey models and plant ecology.