A nonlocal advection system for two competing species with resources recovering time

Competitive interactions between multiple cell populations are crucial for modeling biological processes such as tumor growth and tissue regeneration. In this study, we investigate the nonlocal advection model for two-species competition, which characterizes cell growth and dispersion phenomena in co-culture experiments. To capture realistic phenomena in biology, we introduce the time delay representing population migration and resources recovering time. The primary objectives are to investigate the impact of time delays on competitive dynamics under various parameter settings and to develop numerical methods that ensure biological feasibility of solutions. Accordingly, we design a positivity-preserving finite volume scheme based on an upwind flux approach, guaranteeing non-negative population densities and discrete conservation properties. We examine the convergence orders of the scheme through the numerical experiments and explore the effects of time delays on species competition dynamics under different parameter settings.