Abstract:

 

In this work we study a reaction-diffusion- advection model describing a predator-prey system in a river. We establish the existence of the minimal wave speed of traveling waves connecting the prey-only equilibrium (i.e., the invaded equilibrium) to a predator-present equilibrium (i.e., the predator-only equilibrium or the co-existence equilibrium). To this end, a positive traveling wave solution connecting the prey-only equilibrium is first obtained by constructing a pair of upper and lower solutions and applying the Schauder's fixed-point theorem. This traveling wave solution is then shown to connect a predator-present equilibrium by applying LaSalle's invariance principle. We investigate the effect of the flow and other factors on the minimal wave speed and hence on the spread of the prey and the invasion of the predator in the river.  We also explore the mechanics for linear and nonlinear determinacy by comparing the predator-prey Lotka-Volterra   system with the competitive Lotka-Volterra   system. It is shown that the linear determinacy follows from  the weak  interspecific competition or the predator-prey interaction and that the nonlinear determinacy is a result of the strong  interspecific competition.