The effect of "very fast" dispersal on two species competition with drift

Classical theory predicts that for two competing populations subject to a constant downstream drift, the faster disperser will competitively exclude the slower disperser. In the current work, we consider a novel model of a "much faster" dispersing species, modeled via a $p$-Laplacian operator, competing with a slower disperser. We prove global existence of weak solutions to this model for any positive initial condition, in the regime $\frac{3}{2} < p <2$. Counterintuitively, we show that while the faster disperser always wins - the "much faster" disperser could actually lose, for certain initial data. Several numerical simulations are conducted to confirm our analytical findings. Our results have implications for biodiversity, refuge design, and improved biological control, driven by habitat fragmentation and climate change.