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.
Contents
The Generalised Lotka-Volterra equations model the dynamics of the populations
where the vector
where
Meaning of parameters
The generalised Lotka-Volterra equations can represent competition and predation, depending on the values of the parameters, as described below. They are less suitable for describing mutualism.
The values of
The values of the matrix A represent the relationships between the species. The value of
Positive values for both
Indirect negative and positive effects are also possible. For example, if two predators eat the same prey then they compete indirectly, even though they might not have a direct competition term in the community matrix.
The diagonal terms
Dynamics and solutions
The Generalised Lotka-Volterra equations are capable of a wide variety of dynamics, including limit cycles and chaos as well as point attractors (see Hofbauer and Sigmund). As with any set of ODEs, fixed points can be found by setting
This may or may not have positive values for all the
Alternative views
A credible, simple alternative to the Lotka-Volterra predator–prey model and their common prey dependent generalizations is the ratio dependent or Arditi-Ginzburg model. The two are the extremes of the spectrum of predator interference models. According to the authors of the alternative view, the data show that true interactions in nature are so far from the Lotka-Volterra extreme on the interference spectrum that the model can simply be discounted as wrong. They are much closer to the ratio dependent extreme, so if a simple model is needed one can use the Arditi-Ginzburg model as the first approximation.