The effect of competition on two populations was studied first by Volterra in 1927. Gause in 1932 reparametrized the model and studied the following system both experimentally and theoretically:
The system of equations admits three equilibrium points
N* = N2* = 0 (total extinction); Nj* = Kb N2* = 0 (extinction of N2), N* = 0, N2* = K2 (extinction of Nj); and N1* = (K1 — aK2)/(1 — ot(3), N2* = (K2 — fiKJ/ (1 — af3) (existence of both Nx* and N2*). In case
both populations will persist since the third equilibrium is stable. Now we can state the Gause competitive exclusion principle: in case condition  is not satisfied, one of the populations will drive the other into extinction.
In the above example, the competition is formulated as an additional factor to density dependence. In the following example, the competition for food is direct. Let us again use an analogy to a lake in which we have two species of algae A1 and A2 competing for the same source of nutrient N. For simplicity let us use the bilinear collision as a way of getting nutrient by the algae:
dN/dt = D(I - N) - aNAi - bNA2 dA1/dt = aNAi - DAi dA2/dt = aNA2 - DA2
We assume that a >0, b > 0, and a — b, which are plausible assumptions for any two species in nature.
There are three equilibria:
(N* = I, A1* — A2* — 0) ... extinction of both algae (N * = D/a, A* — I - D/a, A2* — 0) .. .extinction of algae A2 (N * — D/b, A* — 0, A2* — I - D/b) .. .extinction of algae A]
Note that an equilibrium in which both species persist (coexist) does not exist. The exception is when a — b, but then the two species are indistinguishable (the second and the third equations in  are identical).
The competitive exclusion principle may now be stated in a much stronger form: the two species cannot coexist if they compete for the same resource.
Of course, this has a consequence far beyond ecology. For example, it states that the two firms cannot coexist in a single market offering the same product: sooner or later the more efficient firm will drive the other into bankruptcy.
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