Lotka-volterra diffusion-advection competition system with dynamical resources

Competition systems describing the competition between species for resources have been widely studied in the literature and wealthy results have been developed. Most of them (if not all) have essentially assumed that the resources are spatially varying without temporal dynamics. This is an idealized assumption since the most ecological environments and/or biospecies are dynamically changing. Hence the effect of temporal dynamics of resource ought to be taken into account to predict/interpret the competition outcomes more precisely. This constitutes the main motivation of this work and we consider a Lotka-Volterra reaction-diffusion-advection competition system with a dynamical resource whose dynamics is determined by an evolution equation, where the competing species have biased movement (advection) up the resource gradient. We first establish the global existence of classical solutions via Moser iteration and global stability of spatially homogeneous steady states for the constant resource growth rate by method of Lyapunov functionals. When the resource growth rate is spatially varying, we use numerical simulations to demonstrate the possible competition outcomes and find that the asymptotic dynamics of the competition system with dynamical resources is quite different from the case that resources have no dynamics. Moreover, we numerically observe that the advective strategy and/or the relative strength of advection sensory responses are key factors determining the competition outcomes and the asymptotic profiles of the solution.