A slow pushed front in a Lotka-Volterra competition model
This paper studies a two-species Lotka-Volterra competition model, in which the two species diffuse on the real line, NEWLINE\[NEWLINEu_t=\epsilon^2 u_{xx}+(1-u-a_1 v)u,NEWLINE\]NEWLINE NEWLINE\[NEWLINEv_t=v_{xx} +r(1-a_2 u-v)v.NEWLINE\]NEWLINE Parameters are chosen so that \(a_1<1<a_2\) and, in the corresponding space-homogeneous model, the stable state is the \(u=1\), \(v=0\). It is also assumed that the diffusion rate \(\epsilon^2\) of the species \(u\) is sufficiently small. The question of interest is to understand the process of invasion of the species \(u\) into a homogeneous population of species \(v\), by studying travelling front connecting the unstable state \(u=0\), \(v=1\) to the stable state \(u=1\), \(v=0\). Using geometric singular perturbation theory, it is proved (for \(\epsilon\) sufficiently small) that there is a slowest monotone travelling front, which is marginally stable, and is thus the selected front. Of central interest here is the fact that the speed at which this front travels is slower than the linear spreading speed, that is the speed of waves obtained from the linearization of the equations around the unstable equilibrium. This contrasts with the more familiar phenomenon for scalar equations, whereby the speed of the selected front is higher than the linear spreading speed. An analysis of the resolvent of the linearized system shows that this effect is connected to the fact that the linear spreading speed is a simple pole of the resolvent rather than the a branch pole.
scientific article; zbMATH DE number 6063068