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Competition

In mathematical terms, the competition between two populations or species can be expressed as follows:

dN_t / dt = r₁N₁ (K₁ - N₁ - alpha N₂) / K₁

and

dN₂ / dt = r₂N₂ (K₂ - N₂ - beta N₁) / K₂

N₁ and N₂ are the numbers of individuals of the populations (species) 1 and 2, respectively, alpha is the competition coefficient (with an inhibiting effect) population 2 exerts on population 1, beta is the competition coefficient giving the inhibiting effect population 1 has on population 2. Just to remember: r₁ and r₂ are the growth rates of population 1 and population 2, K₁ is the capacity of the habitat for population 1, K₂ its capacity for population 2.

If

alpha > K₁ / K₂ und beta > K₂ / K₁

only one of the two populations/ species will survive. Which of them is dependent on the starting conditions, but if

alpha > K₁ / K₂ und beta < K₂ / K₁

then population/ species 1 will be eliminated. The elimination of species 2 occurs accordingly if

alpha < K₁ / K₂ und beta > K₂ / K₁

Coexistence is only possible, if

alpha, beta < K₁ / K₂ und alpha, beta < K₂ / K₁

Under these conditions, both species are in equilibrium.

The two clover species Trifolium fragiferum and Trifolium repens are able to exist in mixed culture and are thus an example for coexistence. The coexistence of two species with equal requirements does nevertheless lead to a reduction of their performances. In this case, the leaf index, i.e. the ratio of leaf surface to covered soil surface is lower for both species than it is in pure culture.

Competition: two clover species in pure and in mixed culture. Blue: Trifolium fragiferum (pure culture), red: Trifolium repens (pure culture), yellow: Trifolium repens (cultivated in mixed culture together with Trifolium fragiferum), turquoise: Trifolium fragiferum (in mixed culture with Trifolium repens). Ordinate: Leaf surface index ( = leaf surface/ degree of covered soil surface), Abscissa: weeks after sowing (according to J. L. HARPER and J. N. CLATWORTHY, 1963).