How do the Lotka-Volterra equations model competition between two species?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

Common wildebeest — population ecology A herd of common wildebeest (*Connochaetes taurinus*) migrating across a dusty savanna in Africa. The animal is a keystone species (i.e., a species with a disproportionately large effect on its biological community) in plains and acacia savanna ecosystems from southeastern Africa to central Kenya.

The Lotka-Volterra equations model competition between two species by using a pair of linked differential equations that account for both self-regulation and interspecific interaction. These equations assess how the population growth of one species is influenced by its own carrying capacity and the presence of a competitor.1

The equations are expressed as:

$\frac{d N_{1}}{d t} = r_{1} N_{1} (1 - \frac{N_{1}}{K_{1}} - α_{1, 2} \frac{N_{2}}{K_{2}})$
$\frac{d N_{2}}{d t} = r_{2} N_{2} (1 - \frac{N_{2}}{K_{2}} - α_{2, 1} \frac{N_{1}}{K_{1}})$

In these formulas, $r$ represents the rate of increase, $N$ is the population size, $K$ is the carrying capacity, and $α$ represents the competitive effect of one species on the other. The term $α_{1, 2} N_{2} / K_{2}$ quantifies the reduction in the growth rate of species 1 caused by its interaction with species 2.1

Outcomes depend on the relative strength of self-regulation versus these interaction terms. If competitive effects are balanced, populations may stabilize; however, if one species exerts a stronger influence or has a higher initial abundance, it may drive the other to local extinction through competitive exclusion.1