the competing-species model

Now imagine two species that do not eat each other but want the SAME thing — two kinds of fish in one lake fighting over the same food, or two plants reaching for the same light. Each would do fine alone, but each crowds the other out. The competing-species model asks a sharp question: can they coexist, sharing the lake forever, or must one drive the other to extinction? The answer turns out to depend, beautifully, on just how hard each presses on the other.

Each species grows logistically on its own — toward its own carrying capacity — but is also held back by the rival. A common form is x' = x (1 - x - a y) and y' = y (1 - y - b x). The 1 - x part is the ordinary logistic self-limiting; the -a y and -b x parts say each species suffers more the more of the OTHER there is, with a and b measuring how strongly each feels the competition. Everything hinges on whether a and b are bigger or smaller than 1, that is, whether between-species crowding outweighs within-species crowding.

Trace the equilibria in the phase plane and four regimes appear. If competition is mild (a < 1 and b < 1) the two settle into stable coexistence — a shared steady state both populations approach. If competition is fierce (a > 1 and b > 1) coexistence becomes a saddle: an unstable knife-edge, and the eventual winner is decided by who started ahead — the principle of competitive exclusion, where the loser is wiped out. The two mixed cases let one species always win regardless of the start. So the same equations, with different strengths, predict harmony, winner-takes-all, or founder-controlled outcomes — which is exactly why ecologists reach for this model.

It is still a caricature: it freezes the carrying capacities, ignores space, age, and chance, and lumps all of 'competition' into two numbers. But it earns its place by turning a vague verbal debate — 'can two similar species share a niche?' — into a crisp geometric criterion you can test.