Generalized Lotka–Volterra equation

The generalized Lotka–Volterra equations are a set of equations which are more general than either the competitive or predator–prey examples of Lotka–Volterra types. They can be used to model direct competition and trophic relationships between an arbitrary number of species. Their dynamics can be analysed analytically to some extent. This makes them useful as a theoretical tool for modeling food webs. However, they lack features of other ecological models such as predator preference and nonlinear functional responses, and they cannot be used to model mutualism without allowing indefinite population growth.

The generalised Lotka-Volterra equations model the dynamics of the populations ${\displaystyle x_{1},x_{2},\dots }$ of ${\displaystyle n}$ biological species. Together, these populations can be considered as a vector ${\displaystyle \mathbf {x} }$. They are a set of ordinary differential equations given by

${\displaystyle {\frac {dx_{i}}{dt}}=x_{i}f_{i}(\mathbf {x} ),}$

where the vector ${\displaystyle \mathbf {f} }$ is given by

${\displaystyle \mathbf {f} =\mathbf {r} +A\mathbf {x} ,}$

where ${\displaystyle \mathbf {r} }$ is a vector and ${\displaystyle A}$ is a matrix known as the interaction matrix.