Spreading speed and linear determinacy for two-species competition models

Abstract

One crucial measure of a species' invasiveness is the rate at which it spreads into a competitor's environment. A heuristic spread rate formula for a spatially explicit, two-species competition model relies on 'linear determinacy' which equates spread rate in the full nonlinear model with spread rate in the system linearized about the leading edge of the invasion. However, linear determinacy is not always valid for two-species competition; it has been shown numerically that the formula only works for certain values of model parameters when the model is diffusive Lotka-Volterra competition [2]. This paper derives a set of sufficient conditions for linear determinacy in spatially explicit two-species competition models. These conditions can be interpreted as requiring sufficiently large dispersal of the invader relative to dispersal of the out-competed resident and sufficiently weak interactions between the resident and the invader. When these conditions are not satisfied, spread rate may exceed linearly determined predictions. The mathematical methods rely on the application of results established in a companion paper [11].

Climate and competition: the effect of moving range boundaries on habitat invasibility

“…Species 1 propagates to the right and replaces species 2, while the latter retreats. This problem has been thoroughly studied in Lewis et al (2002), and it has been shown that if parameters of the model satisfy two inequalities,…”

How Predation can Slow, Stop or Reverse a Prey Invasion

“…Even though the minimum possible speed is often the one selected in multispecies waves [see for example, Murray (1989)], it should be noted that this is not always the case (Hosono, 1998). For further discussion of this see Lewis et al (2000) and Weinberger et al (2000).…”

Invasion waves in the presence of a mutualist

“…In contrast to the case of a mutualist-invader, this speed is the same as for the 2-species competition model of [7] with no mutualist present. An intuitive explanation is that in our case the mutualist does not directly interact with the invader.…”

“…For the mutualist-invader we transform (2) by extending the corresponding substitution for the 2-species model from [7]:…”

“…as the linearization at the edge of the spreading wavefront, provided that the initial density vector is equal to (0, 1) outside a bounded subset. The analysis of the system (1) in [7] uses the substitution q 1 = p 1 , q 2 = 1 − p 2 , which converts it into a cooperative parabolic system of reaction-diffusion type.…”