Competitive Lotka-Volterra equations

The competitive Lotka–Volterra equations are a simple model of the population dynamics of species competing for some common resource. They can be further generalised to include trophic interactions.

The form is similar to the Lotka–Volterra equations for predation in that the equation for each species has one term for self-interaction and one term for the interaction with other species. In the equations for predation, the base population model is exponential. For the competition equations, the logistic equation is the basis.

The logistic population model, when used by ecologists often takes the following form:

Here x is the size of the population at a given time, r is inherent per-capita growth rate, and K is the carrying capacity.

Given two populations, x₁ and x₂, with logistic dynamics, the Lotka–Volterra formulation adds an additional term to account for the species' interactions. Thus the competitive Lotka–Volterra equations are:

Here, α₁₂ represents the effect species 2 has on the population of species 1 and α₂₁ represents the effect species 1 has on the population of species 2. These values do not have to be equal. Because this is the competitive version of the model, all interactions must be harmful (competition) and therefore all α-values are positive. Also, note that each species can have its own growth rate and carrying capacity. A complete classification of this dynamics, even for all sign patterns of above coefficients, is available, which is based upon equivalence to the 3-type replicator equation.

This model can be generalized to any number of species competing against each other. One can think of the populations and growth rates as vectors and the interaction α's as a matrix. Then the equation for any species i becomes

or, if the carrying capacity is pulled into the interaction matrix (this doesn't actually change the equations, only how the interaction matrix is defined),

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