Two species after the same resource cannot share it — one wins, one goes extinct. Gause turned that sentence into arithmetic: two logistic populations, each pressing on the other through a competition coefficient, their zero-growth isoclines drawn on one plane. Where the isoclines cross decides the survivor. Down the center, data flows: the coefficients go in, the model integrates, the winner comes out. The blue team builds and defends it; the red team tries to break it. A model with named assumptions — not a law of nature, and not advice.
source Gause, The Struggle for Existence (Baltimore: Williams & Wilkins, 1934) — archive.org/details/struggleforexist00gauz. Link AMBER (scan of the original edition). Rendered, not quoted.
Two species, each logistic, each slowed by the other:
dN1/dt = r1N1(1 − (N1 + α₁₂N2)/K1)
dN2/dt = r2N2(1 − (N2 + α₂₁N1)/K2)
α₁₂ is how much one of species 2 weighs on species 1, measured in species-1 units. α = 1 means a competitor eats exactly as much of your niche as one of your own — a complete competitor.
Live state of the current setting:
| quantity | value |
|---|
Gause 1934: complete competitors cannot coexist. The isoclines decide who — this is the ecology sitting on top of the-lotka-volterra predator–prey machinery, where the same two-body coupling instead oscillates.
Read one register down and the winner-take-all of an occupied niche is the-replicator-dynamics in ecological dress: the fitter strategy fixes, the other is driven to zero. Each sphere is the next one's premise.
The blue team's live check: re-integrate the three canonical settings — complete competitors, weak competition, and strong mutual competition — and confirm each lands where the theory says. If red tampers, this badge is where it shows.
Four numbers set the whole contest: the carrying capacities K₁, K₂ (how many each species holds alone) and the competition coefficients α₁₂, α₂₁ (how hard each presses on the other). The growth rates r only set the speed — never the winner.
The two zero-growth isoclines — where each species stops changing — are straight lines:
| species | isocline | N-axis ∩ | other ∩ |
|---|---|---|---|
| 1 | N₁ + α₁₂N₂ = K₁ | K₁ | K₁/α₁₂ |
| 2 | N₂ + α₂₁N₁ = K₂ | K₂ | K₂/α₂₁ |
Feed these into the panel below. The crossing (or its absence) is the whole answer.
Move any slider — the isoclines redraw, the trajectory is integrated live (RK4, dt=0.05), and the outcome is classified from the Jacobian at the crossing. Never looked up.
What the machine proves, from the isocline geometry alone: coexistence is stable only when each species limits itself more than it limits the other (α₁₂·α₂₁ < 1). When α → 1 — a shared niche — the interior point degenerates and one species always excludes the other, reaching its own K while the loser hits zero. That is Gause's principle, computed.
The blue team's witness (left) re-integrates these settings live; the red team (right) tries to make the model lie.
So "complete competitors cannot coexist" is a theorem about this model, not a decree about ponds. Gause's own Paramecium data fit it — under his constant, well-mixed lab conditions, which is exactly the assumption at issue.
"One niche, one species — always, everywhere." Cut. Only under constant conditions and one limiting factor. Fluctuation and multiple resources break exclusion — the theorem's premises, not its conclusion, are what fail in the field.
"The species with the larger population wins." Cut. Under complete competition the higher K wins, not the larger start; under founder control (α>1) the one that starts higher wins. The engine shows both.
"α just means 'how aggressive' a species is." Kept, corrected. α is a conversion rate — one competitor measured in units of your own — which is why α=1 is the knife-edge of a shared niche.
The red team's move: quietly weaken the interspecific coefficient below 1 while the species are true complete competitors (α should be 1) — so the model falsely predicts stable coexistence. The blue team's witness (window 7) is watching.
Drop α from 1 to 0.5 for complete competitors and the loser survives at a false interior point — the witness re-integrates, finds two survivors where Gause demands one, and turns red. Nothing is faked; the attack is real and it is caught.