The Math Behind Persistence and Extinction in Freshwater Ecosystems

The spatial dynamics of freshwater organisms in rivers with directed flow are examined in this research using a nonlocal reaction-diffusion-advection model. We derive sharp criteria for the persistence or extinction of species by examining the primary eigenvalue of the related integro-differential operator and proving global well-posedness. The research indicates that long-term behaviors are greatly impacted by rising advection rates, with extinction being the unavoidable result of sufficiently high directional flow. We also investigate the stability of stationary solutions and limiting profiles under different advection rates. Numerical simulations are used to verify theoretical results. These findings shed fresh light on how diffusion, advection, and survival interact in aquatic environments as well as the drift dilemma.