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bibkey: “visomirskigriffin2026spatial” authors: “Matthew Visomirski; Christopher Griffin” year: 2026 title: “Coherent Structures and Travelling Waves in Spatial Replicators from a Biased Volterra Lattice” doi: “10.1016/j.chaos.2026.118260” url: “https://arxiv.org/abs/2601.06314v1” claim: “Conjecture 2 asserts that the alternating sine initial data on every biased even cycle evolve to a frozen wave maintaining ecological niches.” strata_touched:

• “D5/S3/FluidDynamics/ReactionDiffusion/VisomirskiGriffinFrozenWaveRefutation” license: “citation-only” triage: “anchor”

Verified locator

DOI: 10.1016/j.chaos.2026.118260

Source: https://arxiv.org/abs/2601.06314v1

Crossref resolves the DOI to the article by Matthew Visomirski and Christopher Griffin in Chaos, Solitons & Fractals 208, article 118260. Section 2, page 4, defines the biased lattice with predecessor coefficient $1+a$ and successor coefficient $-1$, and indices modulo the cycle length. Section 3, page 10, equation (6), gives the spatial replicator with common diffusion. Section 4.2, pages 17–19, gives the alternating sine data and the simulated parameters $a=-1/2$, $D=1/100$; Conjecture 2 is on page 19.

Biased lattice and spatial equation

Section 2, page 4: “If we replace each instance of $+1$ with $1+a$ in the interaction matrix $\mathbf{A}$, we obtain a biased Volterra lattice defined by the system of differential equations,” and “Again, index addition and subtraction is considered modulo $n$. We assume $-1<a<\infty$ to maintain the cyclic relation given in the graph.”

$${\dot{u}}_i=u_i(u_{i-1}-u_{i+1})-a{u}_i[C_n(\mathbf{u})-u_{i-1}],C_n(\mathbf{u})=\sum _{i=0}^{n-1}{u}_i{u}_{i+1}.$$

Section 3, page 10: “Let $\mathbf{A}$ be the biased-cycle interaction matrix (for an arbitrary cycle) assuming that diffusion is constant over species. In one-dimension, Vicker’s equation becomes,”

$$\frac{\partial u_i}{\partial t}=u_i({\mathbf{e}}_i^T\mathbf{Au}-\mathbf{uAu})+D\frac{{\partial }^2{u}_i}{\partial x^2}.$$

Source statement

“Consider the one-dimensional spatial replicator, i.e., Eq. (6) with reaction term generated by a $2n$-cycle with $n\ge 2$. Assuming periodic boundary conditions and initial conditions of the form,”

$$u_i(x,0)=\{\begin{array}{ll}\frac{1}{2n}(1+\sin (x)),& \text{if }i\text{ is odd},\\ \frac{1}{2n}(1-\sin (x)),& \text{if }i\text{ is even},\end{array}$$

“the dynamics will evolve to a frozen wave stationary solution. That is, ecological niches will be maintained.”

The formal encoding uses Fin (2*n), with index zero representing species one. Its limit predicate asks for continuous periodic profiles different from the homogeneous equilibrium and convergence in squared $L^2$ distance. It omits the stationary equation, so it is implied by the source’s stronger stationary-wave assertion. The refutation at $n=25$ applies to every classical solution satisfying the prescribed equation and data; it supplies no existence theorem. The six- and eight-cycle simulations and the source’s separate travelling-wave results fall outside that refutation.