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

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

where x(t) denotes the number of prey, y(t) the number of predators, and α, β, γ and δ are constants. By the Hartman–Grobman theorem the non-linear system is topologically equivalent to a linearization of the system about an equilibrium point (x*, y*), which has the form

$\left[\begin{array}{c}\frac{du}{dt}\\ \frac{dv}{dt}\end{array}\right]=\mathbf{A}\left[\begin{array}{c}u\\ v\end{array}\right],$

where u = x − x* and v = y − y*. In mathematical biology, the Jacobian matrix $\mathbf{A}$ evaluated at the equilibrium point (x*, y*) is called the community matrix. By the stable manifold theorem, if one or both eigenvalues of $\mathbf{A}$ have positive real part then the equilibrium is unstable, but if all eigenvalues have negative real part then it is stable.

See also

• Paradox of enrichment

References

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