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.

Anomalous spreading in a system of coupled Fisher–KPP equations

“…This is the case for scalar problems, but for systems this bound no longer holds. This fact was recently shown in an example of a Lotka-Volterra competition model, [17],…”

Criteria for Pointwise Growth and Their Role in Invasion Processes

“…Like any other algebraic pointwise growth mode, these double double roots give linear predictions for the selected speed of the nonlinear system. In the context of a Lotka-Volterra competition model, a double double root was found that overestimates the invasion speed of the nonlinear system, see [22]. Examples in [21] show that double double roots sometimes give correct predictions for spreading speeds.…”

“…u κ 2 , κ > 1, and try to derive nonlinear predictions for spreading speeds in the leading edge. Such nonlinear coupling generated slow pushed fronts in the Lotka-Volterra equation; see [22].…”

A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations

“…This is more complicated and it is not the case that the observed speed in the nonlinear system is simply the fastest of these linear speeds. This issue has been explored in depth in [11,12,13] where a distinction is made between linear spreading speeds based upon the analyticity (or lack thereof) of the pointwise Green's function in a neighborhood of the singularities enforcing these linear speeds. In [11], the double roots leading to anomalous spreading in the linear system were called relevant if anomalous spreading speeds were also observed numerically in the nonlinear regime and irrelevant if the invasion speed was slower than the anomalous one in the nonlinear system despite the existence of faster spreading speeds in the linear system.…”