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.

• The paper proves that as competition nears the threshold, the total species density converges uniformly to the carrying capacity, forming a sharp transition interface.
• Using a minimax variational characterization with a tailored profile ansatz, the study derives explicit bounds on the critical propagation threshold and wave speed.
• Verification of the 'Unity is not strength' conjecture confirms that, under nearly symmetric conditions, the fast-diffusing species invades the slower one.

Propagation Direction near the Strong-Competition Borderline in Two-Species Lotka–Volterra Systems

Introduction and Problem Formulation

The paper studies the directionality of bistable traveling waves in the classical two-species Lotka–Volterra competition-diffusion system, specifically analyzing the regime near the strong-competition threshold. The system under consideration is: $$\begin{cases} u_t = u_{xx} + u(1 - u - cv), \ v_t = d v_{xx} + v(a - b u - v), \end{cases}$$ with strong interspecific competition $(1/c < a < b)$ and positive parameters. The key ecological quantity is the sign of the unique traveling wave speed $s$: its sign determines which species (with wavefront $(u,v)(\xi)$, $\xi=x-st$) advances, resulting in competitive exclusion.

The analysis focuses on a symmetric scenario with equal intrinsic growth rates and competition coefficients, leaving only the diffusion rates (notably $d > 1$ or $d < 1$) as the source of asymmetry. The central conjecture ("Unity is not strength"), previously proposed by Alzahrani et al., Girardin, and others, suggests that the faster-diffusing species inevitably invades the slower one in this setting—i.e., the sign of $s$ matches the sign of $d-1$.

Verification of this conjecture has proved elusive in the parameter regime near the strong-competition threshold ($k \to 1^+$), where explicit analytical descriptions of the wave and its speed become singular and technically challenging. This paper aims to rigorously establish the propagation direction in precisely this subtle regime, blending sharp variational characterizations, asymptotics, and carefully constructed test functions.

Summary of Main Results

Asymptotics Near the Strong-Competition Threshold

A salient contribution is the proof that, as the competition parameter $(1/c < a < b)$0,

$(1/c < a < b)$1

in traveling wave solutions with zero wave speed. This implies that, in the strong-competition limit, the total species density approximates the carrying capacity and the interface between species forms a tight transition region.

Variational Characterization of the Propagation Threshold

Building on prior work, the paper utilizes a minimax formulation: $(1/c < a < b)$2 Here, $(1/c < a < b)$3 is the class of admissible monotone functions representing the wave profile, and $(1/c < a < b)$4 is a functional encoding the core dynamics (see Theorem 2 in the paper).

This construct enables the derivation of explicit lower and upper bounds for the critical parameter $(1/c < a < b)$5 (the interface between wave propagation and stalling), as $(1/c < a < b)$6.

Verification of the "Unity is not Strength" Conjecture in the Borderline Regime

The main technical achievement is the explicit construction of an ansatz for the profile function,

$(1/c < a < b)$7

with a judiciously chosen perturbation $(1/c < a < b)$8 and small parameter $(1/c < a < b)$9 proportional to $s$0, allowing for precise analysis in the singular limit. Through a blend of maximum-principle techniques, careful asymptotics, and eigenfunction-based contradiction arguments, the paper establishes that:

• For all $s$1, there exists $s$2 such that

$s$3

i.e., the more diffusive species invades as $s$4, thus rigorously confirming the conjecture in a region previously inaccessible to analysis.

Moreover, explicit upper and lower bounds on the propagation threshold are derived: $s$5 demonstrating that for sufficiently small $s$6, the slow-diffuser cannot overcome the fast diffusing competitor, and quantifying the effect of both diffusion and competition parameters.

Technical Innovations

Key methodological components include:

• Reduction to a sharp transition layer: Rigorous compactness and maximum-principle arguments show that in the limit $s$7, the system "collapses" to a razor-thin interface, justifying the approximation $s$8.
• Profile ansatz and minimax test function: The explicitly constructed function $s$9, with sophisticated parameter dependence in $(u,v)(\xi)$0, both satisfies analytic boundary conditions and allows precise estimation of the critical functional $(u,v)(\xi)$1.
• Asymptotic expansion and error control: Detailed balance of higher-order correction terms ($(u,v)(\xi)$2) establish bounds on the critical value $(u,v)(\xi)$3 even as these terms become singular for vanishing $(u,v)(\xi)$4.
• Contradiction via eigenfunction analysis: The use of auxiliary test functions and integration identities provides a powerful tool for precluding the existence of anomalous behaviors, enabling sharp convergence results.

Numerical and Analytical Implications

The provided bounds are explicit in $(u,v)(\xi)$5 and $(u,v)(\xi)$6, revealing the delicate scaling relations governing strong competition. Notably, the results quantify that the advantage conferred by higher diffusivity becomes vanishingly sensitive to the deviation of competition strength from the threshold. This has corollaries for understanding competitive exclusion and pattern formation in ecological systems and for constructing numerical schemes that must resolve very steep population fronts.

The singularity at $(u,v)(\xi)$7 (where the system transitions from strong to weak competition, and the mathematical structure changes) is clarified, dispelling potential misconceptions about the robustness of classic Lotka–Volterra results to subtle parameter changes.

Theoretical and Practical Significance

The findings settle, in a substantial newly-addressed regime, the question of which structural property—diffusive mobility or competitive intensity—dominates propagation direction. The confirmation that increased dispersal always prevails (as $(u,v)(\xi)$8) is directly relevant for biological invasions and evolutionary strategies, particularly in nearly-symmetric contests where differences are muted except for mobility traits.

On a theoretical level, the extension of variational and maximum-principle techniques to this singular regime represents an advancement in the mathematical analysis of reaction-diffusion competition systems. The work also suggests promising generalizations: analogous threshold asymptotics may be tractable for multi-species or spatially heterogeneous systems by similar perturbative and functional analytic routes.

Conclusion

This paper rigorously demonstrates the dominance of higher dispersal near the strong-competition borderline in the two-species Lotka–Volterra model, confirming the “Unity is not strength” conjecture in a nuanced setting. The derivation of quantitative bounds and constructive minimax analysis provides both theoretical clarity and tools applicable to future work in competition-diffusion equations and related frameworks.