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May’s Criterion as K — ecosystem stability IS a coupling threshold
JIM’S OVERSIMPLIFICATION
An ecosystem is a web of things that depend on each other. Remove one species and the web deforms. Remove the wrong one and it collapses. There’s a threshold — when the number of connections times the coupling strength crosses it, the system crashes. Same math as a financial contagion.
K IN THIS DOMAIN
K here is species coupling. Remove one species, the web deforms. May's stability criterion: the ecosystem crashes when K × connectivity exceeds threshold.
An ecosystem is a web of things that eat each other. Remove one and the web wobbles. Remove the wrong one and it collapses. There’s a threshold, and Robert May proved it in 1972:
More species × stronger connections = less stable.
Wait, what? More species is less stable? That sounds wrong. Aren’t biodiverse ecosystems the healthy ones?
Yes — because they cheat. Tropical rainforests have thousands of species but weak individual connections. Everything depends on everything a little bit. Nobody depends on anything a lot. Many weak links beat few strong links. Weak coupling = resilience.
A monoculture is the opposite — one crop, one connection, total dependence. One disease wipes it out. That’s low coupling diversity. The Irish potato famine was a monoculture crash.
Predator-prey is literally two coupled oscillators. Rabbits go up, foxes follow. Too many foxes, rabbits crash. Fewer foxes, rabbits recover. The fox population trails the rabbit population by exactly a quarter cycle. This isn’t a metaphor. The differential equations ARE coupled oscillators.
Same math as financial contagion, neural synchronization, and protein stability. Different things that depend on each other, same coupling threshold where it all falls apart.
Honest limit: May’s criterion is known since 1972. We didn’t discover it. We’re showing it maps to the same coupling framework. No real ecosystem data was tested.
MAY'S STABILITY CRITERION
Robert May (Nature, 1972) proved that a random ecosystem with n species, connectance C, and interaction strength s is stable only when:
s√(nC) < 1
This is a coupling threshold. When the product of interaction strength, species count, and connectance exceeds 1, the system becomes unstable. The ecosystem collapses.
That threshold IS K. The stability boundary of an ecosystem is a coupling ceiling — the same kind of critical value that governs phase transitions in physics, synchronization in neuroscience, and melting in thermodynamics.
THE COMPUTATION
Interaction strength s=0.5, connectance C=0.3:
5 species: s√(nC) = 0.612 STABLE
10 species: s√(nC) = 0.866 STABLE
20 species: s√(nC) = 1.225 UNSTABLE
50 species: s√(nC) = 1.937 UNSTABLE
100 species: s√(nC) = 3.873 UNSTABLE
More species = more coupling = less stable.
The only way to add complexity without collapse is to REDUCE interaction strength.
RESTABILIZATION
Reducing interaction strength (s=0.3) restabilizes:
20 species, s=0.5: s√(nC) = 1.225 UNSTABLE
20 species, s=0.3: s√(nC) = 0.735 STABLE
This is how real ecosystems work:
Tropical forests have enormous species counts but WEAK individual interactions.
Many weak links beat few strong links. Weak coupling = resilience.
LOTKA-VOLTERRA AS COUPLED OSCILLATORS
The classic predator-prey equations are two coupled oscillators:
Lotka-Volterra simulation:
Coupling strength K = 1.45
Frequency ratio R = 0.91
Prey population oscillates.
Predator population oscillates with a quarter-cycle lag.
Predator-prey IS phase-locked oscillation.
The predator cycle trails the prey cycle by π/2.
More prey → more predators → fewer prey → fewer predators → repeat.
This is not a metaphor. The differential equations are coupled oscillators.
BIODIVERSITY AS R
Shannon evenness measures how evenly species are distributed. This IS the frequency ratio R — how synchronized the ecosystem’s energy distribution is:
Shannon evenness (5 species):
Equal distribution [200, 200, 200, 200, 200]:
Evenness = 1.000 — all species equally represented
Dominated ecosystem [980, 5, 5, 5, 5]:
Evenness = 0.076 — one species dominates everything
Low evenness = low R = fragile.
A monoculture is a single oscillator. No coupling. No resilience.
High evenness = many frequencies = robust coupling = stable.
THE CONNECTION
May’s criterion: s√(nC) < 1 — coupling threshold for ecosystem stability
Lotka-Volterra: K=1.45, R=0.91 — predator-prey as coupled oscillators
Shannon evenness: R of the ecosystem — biodiversity as frequency distribution
Three different ecological concepts. One coupling framework.
The same K/R/E/T that predicts protein folding, chip placement, and melting points.
HONEST LIMITS
What this is:
May’s criterion is a KNOWN result (1972). We did not discover it.
We are showing it maps to K/R/E/T. The reframing is ours, the math is May’s.
What the Lotka-Volterra numbers are:
K=1.45 and R=0.91 come from simulation, not field data.
No real ecosystem data was tested. These are textbook equations
run through the coupling framework.
What we haven’t done:
Real population time series (e.g., lynx-hare cycle data)
Multi-species food web analysis beyond pairwise Lotka-Volterra
Spatial ecology (metacommunities, island biogeography)
Empirical connectance matrices from field studies
The claim:
Not “we solved ecology.”
“The same K that predicts protein damage also describes ecosystem stability.”