The Mathematics of Life and Death

In order to examine the spatial dynamics of freshwater species in river habitats, this study creates a nonlocal reaction-diffusion-advection model. Using spectral analysis of an integro-differential operator, we incorporate directional flow (advection) and define conditions for species persistence or extinction. The findings show that advection rate has a significant impact on long-term survival, with considerable directional motion always resulting in extinction. With the help of numerical simulations, we further investigate the existence, uniqueness, and stability of stationary solutions. These results offer a robust mathematical framework for forecasting how ecosystems will react to river currents and comprehending the drift dilemma.