Life seam / a probability distribution changes the answer
The Distance Hidden in the Tail
Post-glacial tree ranges seemed to move hundreds of metres each year while measured seeds mostly fell within tens of metres. The mismatch became Reid's paradox. Its cleanest lesson is not that the old parameters were slightly wrong. It is that a probability distribution can keep its mean and variance while changing the kind of motion it permits.
Clement Reid asked how oak crossed roughly 600 miles of Britain after the ice. His million-year figure was an order-of-magnitude estimate, not a measured migration rate. Later pollen chronologies suggested range movement across tree taxa on the order of hundreds to a thousand metres per year. Put local seed movement into the classical diffusion model and the front is far slower.
Begin with a deliberately generous translation: treat a low-tens-of-metres dispersal step as though it were available every year. This favours the classical model. It still misses.
Instrument I / the classical front
Give diffusion every advantage
For a Fisher-KPP population front, the linearized invasion speed is c = 2 sqrt(rD). The controls are logarithmic so you can push both parameters far beyond the representative starting point. The paleorecord band does not move.
Fisher-Skellam speed bench
Drag either input. The equation, shortfall, and parameter cost recompute together.
Default 0.1 per year. Clark reports simulated tree growth rates spanning roughly 0.03 to 0.5 per year.
Default RMS step computing. This yearly translation is more generous than a tree's paced reproduction.
Classical predicted front
computing
computed live as 2 sqrt(rD)
Gap to pollen band
computing
computing
The square root is the trap. If the front is short by a factor of ten, either r or D must grow by a factor of one hundred. A hundredfold speed gap costs ten thousandfold parameter inflation. Bigger numbers can force the equation to comply, but the required inputs cease to describe the measured system.
The deeper move is to change the object the equation treats as harmless: the dispersal kernel itself.
Instrument II / same bulk, different tail
Keep the mean. Keep the variance. Change the future.
Both kernels below are symmetric about zero. Both have variance sigma squared, with sigma = 40 m. Both feed the same Beverton-Holt growth map and the same integrodifference equation. Only the tail changes.
A matched-moment invasion
Computing the kernel moments.
Gaussian kernel
computing MGF
Its exponential tail gives a finite positive moment generating function, so the Kot speed formula has a finite minimum.
Student t, 3 degrees of freedom
computing MGF
Its density falls as an inverse fourth power. Mean and variance exist, but any nonzero exponential moment diverges.
The front is the rightmost cell with density at least 0.01. Tap the plot or press Enter to advance six generations.
Gaussian front
computing
Student t front
computing
Heavy-tail recent speed
computing
The check / every diagnostic number recomputed
The browser and the standalone verifier use the same equations and constants, but separate implementations. This table is filled by the browser from the live state.
| check | recomputed result | meaning |
|---|
Observed data
Local seed shadows of roughly 2 to 40 m; fitted red-oak scale 12.9 +/- 1.8 m; historical pollen estimates 100 to 1,000 m/yr. These are sourced ranges, not outputs of this page.
Diagnostic inference
The Fisher-Skellam shortfall, square-root inflation, matched moments, and MGF distinction follow from the stated equations.
Toy-model output
The two front series use a chosen growth map, 40 m kernel standard deviation, 10 m cells, 0.01 establishment threshold, and 36 generations. They are not an oak hindcast.
Independent run: node research/reids-paradox/verify-reids-paradox.mjs. It re-derives the base arithmetic, exact moments, MGF condition, all sampled front positions, window speeds, super-linearity, kernel mass, and boundary clearance.
The paradox also became smaller
A tail is not the only resolution
Pollen arrival is not a stopwatch on a continuous front. It depends on source area, production, transport, deposition, sampling, dating, and where the range began. Macrofossils and evidence for small northern populations changed that starting line.
Feurdean and colleagues estimated about 60 to 260 m/yr when northern refugia were allowed, compared with roughly 115 to 550 m/yr under a southern-refugia assumption. McLachlan, Clark, and Manos used chloroplast markers to argue for scattered northern populations and estimates below 100 m/yr. Rare long-distance dispersal remains a powerful mechanism. Cryptic refugia remain a live competing explanation. The historical paradox can be partly a tail problem and partly a starting-line problem.
What is exact, and what is chosen
Exact within the named models. The Fisher-KPP linear-front result is 2 sqrt(rD). The Gaussian and rescaled Student t kernels have mean zero and variance 1,600 square metres. The Gaussian MGF is finite for every real argument. The Student t MGF diverges away from zero because an exponential eventually outruns its inverse-fourth-power tail. Under the Kot integrodifference assumptions, that finite-versus-divergent MGF distinction separates finite constant-speed fronts from accelerating ones.
Chosen. Translating seed movement into an annual diffusion constant, the low-density multiplier, the Beverton-Holt growth map, lattice spacing, threshold, initial occupied width, and Student t family are free modelling choices. The continuum Student t tail is infinite; every computer grid truncates it. This finite run demonstrates acceleration over 36 generations, while the theorem supplies the asymptotic statement.
Not claimed. Not every leptokurtic kernel accelerates. A peaked kernel with an exponential bound retains a finite MGF and a finite asymptotic speed. Nor does this page establish which mechanism dominated any particular tree taxon. Real spread also depends on establishment, animals, climate, geography, competition, Allee effects, life stages, and the resolution of the fossil record.