Community matrix

In mathematical biology, the community matrix is the linearization of a generalized [Lotka–Volterra equation] at an equilibrium point. The eigenvalues of the community matrix determine the stability of the equilibrium point.
For example, the Lotka–Volterra predator–prey model is
where x denotes the number of prey, y the number of predators, and α, β, γ and δ are constants. By the Hartman–Grobman theorem the non-linear system is topologically equivalent to a linearization of the system about an equilibrium point, which has the form
where u = x − x* and v = y − y*. In mathematical biology, the Jacobian matrix evaluated at the equilibrium point is called the community matrix. By the stable [manifold theorem], if one or both eigenvalues of have positive real part then the equilibrium is unstable, but if all eigenvalues have negative real part then it is stable.