Sign of the pulsating wave speed for the bistable competition–diffusion system in a periodic habitat

• Published in: Mathematische annalen Vol. 385; no. 3-4; pp. 1 - 36
• Main Authors: Ding, Weiwei, Liang, Xing
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2023; Springer Nature B.V
• Subjects: Competition; Diffusion rate; Ecological effects; Heterogeneity; Mathematics; Mathematics and Statistics; Species diffusion
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-022-02372-1

This paper is concerned with the speeds of pulsating waves for two-species competition–diffusion systems with bistable structures in periodically varying media. The existence and qualitative properties of pulsating waves have been established recently. In this work, assuming that the two species share the same diffusion rates, we study the sign of wave speeds by comparing the reactions and competitions. We first give a criterion when the speed is zero, and then provide some sufficient conditions ensuring the speed has a strict sign. We also show that the spatial heterogeneity has a significant consequence on the sign and gives rise to some new phenomena. More precisely, the pulsating waves in two opposite directions may have different speeds. Particularly, from the ecological point of view, our result indicates that the invasion of a new species may succeed in one direction but fail in the opposite one. Finally, we show the presence of multiple stationary waves which is in contrast with the uniqueness of non-stationary waves.