Incoprea Research · Supplied Round 3 · 30 October 2026

A supplied Fisher–KPP check: wave speed depends on a product, not a ratio alone

Evidence status: Passed exact algebra within the homogeneous Fisher–KPP model; supplied numerical front speed is an approximation.

In the standard homogeneous Fisher–KPP equation, the minimal traveling-wave speed is 2√(rD). Its squared value depends on the product of population growth rate r and diffusion coefficient D. The supplied calculation gives a clear counterexample to treating r/D alone as the speed-setting quantity within that model. It does not establish a universal formula for every biological invasion.

The hypothesis and the model boundary

The posted invasion hypothesis attributes spread to a ratio of growth rate to dispersal distance and says a fast grower with short dispersal can outrun a slow grower with long dispersal. The supplied test replaces movement with a diffusion coefficient D. D has units of length²/time; it is not a dispersal distance, which has units of length.

The model is ∂u/∂t=D∂²u/∂x²+ru(1−u), for density or frequency u scaled between zero and one. Growth is logistic, movement is local diffusion, and parameters are constant in a one-dimensional homogeneous environment. For this equation, the minimal admissible wave speed is c*=2√(rD); sufficiently steep invasion profiles select that minimal speed asymptotically.

Why the product matters

Growth rate has units 1/time, so rD has units length²/time², and its square root has the units of speed. The ratio r/D instead has units 1/length². A dimensional ratio of that form cannot by itself specify a dimensional speed without another scale.

The supplied exact comparison makes the insufficiency concrete. Its 100-point scan keeps rD=0.5 while varying r/D from 0.005 to 50, a factor of 10,000. Every calculated minimal speed remains √2≈1.414214 in the chosen length/time units. Conversely, doubling both r and D preserves their ratio but doubles 2√(rD).

Those two examples have different growth/dispersal balances but equal speeds because their products match. This does not refute the word “can” in the original comparison: a fast grower can outrun another invader when its product is larger. One equal-speed pair shows that the ordering is not guaranteed by the qualitative labels or by the ratio alone.

The supplied numerical check is approximate

The code also integrates a finite-difference version of the equation at r=0.5 and D=1. It uses spatial spacing 0.5, time step 0.1, a domain length of 1,500 and a maintained occupied left boundary. It measures displacement of the u>0.5 front between two saved times separated by 300 time units.

The recorded numerical speed is 1.35, compared with the exact minimal-speed value 1.414214: approximately 4.54% lower. Finite grid spacing, finite time and the front-location rule limit this comparison. No grid-refinement or asymptotic-convergence analysis is supplied. It is a rough consistency check, not numerical confirmation to the displayed precision. The product-versus-ratio result comes from the exact formula and controlled parameter comparisons.

Established result, limited ecological reach

Fisher’s primary 1937 wave analysis is the established source for the growth–diffusion speed relation; it is not a new Incoprea discovery. Real populations can involve heterogeneous habitat, long-distance movement, density-dependent dispersal, interactions and Allee effects. Such systems need not obey this particular minimal-speed law, and a diffusion coefficient must be justified rather than equated with a single movement distance.

The supplied scripts evaluate formulas and a deterministic model. Their grids are not observed invasions, and equal lower/upper entries in the source’s JSON are not statistical confidence intervals.

Provenance

The analysis and original code come from the supplied DeepSeek-assisted Round 3 package. Codex checked units and interpretation, reproduced the unchanged script in isolation, and edited this publication. The result file and numeric series reproduce byte-for-byte. No new ecological dataset, experiment or follow-up simulation was created. The original hypothesis remains unchanged through Incoprea Research. This is an internal computational check, not external peer review.