A slow pushed front in a Lotka–Volterra competition model

Abstract

We study invasion speeds in the Lotka–Volterra competition model when the rate of diffusion of one species is small. Our main result is the construction of the selected front and a rigorous asymptotic approximation of its propagation speed, valid to second order. We use techniques from geometric singular perturbation theory and geometric desingularization. The main challenge arises from the slow passage through a saddle-node bifurcation. From a perspective of linear versus nonlinear speed selection, this front provides an interesting example as the propagation speed is slower than the linear spreading speed. However, our front shares many characteristics with pushed fronts that arise when the influence of nonlinearity leads to faster than linear speeds of propagation. We show that this is a result of the linear spreading speed arising as a simple pole of the resolvent instead of as a branch pole. Using the pointwise Green's function, we show that this pole poses no a priori obstacle to marginal stability of the nonlinear travelling front, thus explaining how nonlinear systems can exhibit slower spreading that their linearization in a robust fashion.