Community matrix

In mathematical biology, the community matrix is the linearization of the Lotka–Volterra equation at an equilibrium point. The eigenvalues of the community matrix determine the stability of the equilibrium point.

The Lotka–Volterra predator-prey model is

\( \begin{array}{rcl} \frac{dx}{dt} &= x(\alpha - \beta y) \\ \frac{dy}{dt} &= - y(\gamma - \delta x), \end{array} \)

where x(t) denotes the number of prey, y(t) the number of predators, and α, β, γ and δ are constants. The linearization of these differential equations at an equilibrium point (x*, y*) has the form

\( \begin{bmatrix} \frac{du}{dt} \\ \frac{dv}{dt} \end{bmatrix} = A \begin{bmatrix} u \\ v \end{bmatrix}, \)

where u = x − x* and v = y − y*. The matrix A is called the community matrix. If A has an eigenvalue with positive real part then the equilibrium is unstable, but if all eigenvalues have negative real part then it is stable.

References

Murray, James D. (2002), Mathematical Biology I. An Introduction, Interdisciplinary Applied Mathematics 17 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95223-9.