The speeds of propagation for the monostable Lotka-Volterra competition-diffusion system in general unbounded domains

Abstract

This paper is concerned with the speeds of propagation for the monostable Lotka-Volterra competition-diffusion system in general unbounded domains of $\mathbb{R}^N$. We first establish various definitions of spreading speeds at large time in the situation where one species is an invader and the other is a resident. Then, we study fundamental properties of these new definitions, including their relationships and their dependence on the geometry of the domain and the initial values. Under the conditions that both species possess the same diffusion ability and that the interactions between them are sufficiently weak, we derive an upper bound for the spreading speeds in a large class of domains. Furthermore, we obtain general upper and lower bounds for spreading speeds in exterior domains, as well as a general lower bound in domains containing large half-cylinders. Finally, we construct some particular domains for which the spreading speeds can be zero or infinite.

Figures (5)

• Figure 1: the relationship between $R(e)$ and $R(e,z)$.
• Figure 2: the choice of $k$ points $y_1',\cdots,y_k'$, where the shaded area is $\Omega$.
• Figure 3: the line represents $\Gamma$ and the shaded area represents $\widehat{\Omega}$.
• Figure 4: the shaded area is $\Omega\backslash \overline {B_{2\pi}}$, which has the shape of a spiral.
• Figure 5: the subsets $\widetilde{\Omega}$ and $\widehat{\Omega}$ of $\mathbb{R}^2$, the shaded area $\widetilde{\Omega}$ is an infinite cusp.