Competition of Multiple Species in Advective Environments

Abstract

We study the effect of changes in flow speed on competition of an arbitrary number of species living in advective environments, such as streams and rivers. We begin with a spatial Lotka–Volterra model which is described by n reaction–diffusion–advection equations with Danckwerts boundary conditions. Using the dominant eigenvalue $\lambda \le 0$ of the diffusion–advection operator subject to boundary conditions, we reduce the model to a system of ordinary differential equations. We impose a “transitive arrangement” of the competitors in terms of their interspecific coefficients and growth rates, which means that in the absence of advection, we have the following situation: for all $1\le i<j\le n$, species i out-competes species j, while species j has higher intrinsic growth rate than species i. Changing advection speed in the original spatial model corresponds to changing the value of $\lambda$ in the spatially implicit model. Considering the cases of the odd and even n separately, we obtain explicit intervals of the values of $\lambda$ that allow all n species to be present in the habitat (coexistence interval). Stability of this equilibrium is shown for $n\le 4$.