Abstract
This work deals with the speed sign of travelling waves to the Lotka-Volterra model with diffusion and bistable nonlinearity. We obtain new conditions for the determinacy of the sign of the bistable wave speed by constructing upper or lower solutions with an inside parameter to be adjusted. The established conditions improve or supplement the results in the references and give insight into the combined effect of system parameters on the propagation direction of the bistable wave.
Highlights
We study the diffusive Lotka-Volterra competitive model φt = d1φxx + r1φ(1 − b1φ − a1φ), (1)
The purpose of this paper is to improve or supplement the results established in [17] by way of constructing novel upper or lower solutions and developing analytical techniques. Our presentation of this effort to determine the sign of the bistable wave speed can be summarized as follows: In Section 2, we introduce definitions, lemmas and decay rates at infinity of the bistable travelling wave which will be used later
We first establish conditions for the positive wave speed, under which the bistable travelling wave solution will spread to the left
Summary
The purpose of this paper is to improve or supplement the results established in [17] by way of constructing novel upper or lower solutions and developing analytical techniques Our presentation of this effort to determine the sign of the bistable wave speed can be summarized as follows: In Section 2, we introduce definitions, lemmas and decay rates at infinity of the bistable travelling wave which will be used later. The decay rate μ of the bistable travelling wave near β as z → ∞ satisfies the characteristic equation dμ2 + cμ + r(1 − a2) = 0, which has solutions μ3 = −c −. Μ1(c) is increasing and μ4(c) is decreasing in c ∈ R
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