Propagating Direction Near the Strong-Competition Borderline in the Two-Species Lotka-Volterra Model

Abstract

This study investigates the propagating direction of bistable traveling waves in the two-species Lotka-Volterra competition-diffusion model under strong competition. From an ecological perspective, the sign of the wave speed is critical, as it dictates which species eventually prevails. We focus on a near-symmetric scenario where intrinsic growth rates and inter-specific competition coefficients are identical, leaving diffusion rates as the sole source of asymmetry. This framework is motivated by the conjecture "Unity is not strength" as described by Alzahrani et al., Girardin and Nadin, and Girardin, which proposes that the species with the higher diffusion rate gains a competitive advantage, directly dictating the wave speed's sign. While extensive literature, including the significant recent progress by Nakamura and Ogiwara, has validated this conjecture under specific assumptions, a comprehensive proof remains elusive. In this paper, we explore the subtle regime where inter-specific competition weakens, approaching the strong-competition borderline. Leveraging our previous finding, a minimax formulation for the zero-wave-speed condition, we successfully analyze the asymptotic behavior of the wave and construct a sharp test function to determine the sign of the wave speed. Consequently, we verify "Unity is not strength" conjecture within this new parameter regime and derive explicit bounds that characterize how the zero-wave-speed condition is influenced by the interplay between competition strength and diffusion rates.