Abstract1: We consider some mathematical models in one-dimensional advective environments. Individuals are exposed to unidirectional flow, with the possibility of being lost through the boundary. Our analysis suggests that, in contrast to the case of no advection, slow dispersal is generally selected against in advective environments. When the diffusion and advection rates are small and comparable,we determine some criterion for the existence and multiplicity of evolutionarily stable strategies. We also study whether that these strategies are convergent stable.

Abstract2:We consider the global boundedness and stability to the predator-prey system with prey-taxis. By Moser iteration and $L^p$-estimates, we show that the intrinsic interaction between predators and preys in the predator-prey system is sufficient to prevent the population overcrowding without imposing any technical assumption on prey-taxis as done in the existing results. Furthermore, by constructing appropriate Lyapunoval functional, we show that prey-only steady state is globally asymptotically stable if the predation is weak, and the co-existence steady state is globally asymptotically stable under some conditions (like the prey-taxis is weak or prey diffuse fast) if the predation is strong. This is a joint work with Haiyang Jin.