Zoology of a non-local cross-diffusion model for two species

We study a nonlocal two species cross-interaction model with cross-diffusion. We propose a positivity preserving finite volume scheme based on the numerical method introduced in [J. A. Carrillo, A. Chertock, and Y. Huang, Commun. Comput. Phys., 17 (2015), pp. 233–258] and explore this new model numerically in terms of its long-time behaviors. Using the so-gained insights, we compute analytical stationary states and travelling pulse solutions for a particular model in the case of attractive-attractive/attractive-repulsive cross-interactions. We show that, as the strength of the cross-diffusivity decreases, there is a transition from adjacent solutions to completely segregated densities, and we compute the threshold analytically for attractive-repulsive cross-interactions. Other bifurcating stationary states with various coexistence components of the support are analyzed in the attractive-attractive case. We find a strong agreement between the numerically and the analytically computed steady states in these particular cases, whose main qualitative features are also present for more general potentials.