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. 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

${\begin{bmatrix}{\frac {du}{dt}}\\{\frac {dv}{dt}}\end{bmatrix}}=\mathbf {A} {\begin{bmatrix}u\\v\end{bmatrix}},$

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.