Abstract
In this paper, we study the traveling front solutions of the Lotka-Volterra competition-diffusion system with bistable nonlinearity. It is well-known that the wave speed of traveling front is unique. Although little is known for the sign of the wave speed. In this paper, we first study the standing wave which gives some criteria when the speed is zero. Then, by the monotone dependence on parameters, we obtain some criteria about the sign of the wave speed under some parameter restrictions.
Highlights
In this paper, we study the following Lotka-Volterra competitiondiffusion system ut = uxx + u(1 − u − kv), vt = dvxx + av(1 − v − hu), (1.1)where u = u(x, t) and v = v(x, t) represent population densities of two competing species, and a, h, k, d are positive constants with certain ecological meanings
Throughout this paper, we only focus on the bistable nonlinearity
In [1] or [2], they proved the existence of traveling front solutions
Summary
The parameters h and k satisfy the bistability condition min{h, k} > 1. We say that (u, v)(x, t) is a traveling front solution of (1.1) with speed s if (u, v)(x, t) = (U, V )(ξ), where ξ = x − st for some functions U, V (called wave profiles), such that (U, V )(±∞) ∈ {(1, 0), (0, 1)} and (U, V )(∞) = (U, V )(−∞). For given a and d, we have the following relations between parameters (h, k) and (b, c): (h, k) = (b/a, ac), (b, c) = (ah, k/a). In [1] or [2], they proved the existence of traveling front solutions.
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