Community matrix

In mathematical biology, the community matrix is the linearization of a generalized Lotka–Volterra equation at an equilibrium point.^[1] The eigenvalues of the community matrix determine the stability of the equilibrium point.

For example, the Lotka–Volterra predator–prey model is

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

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

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

where u = x − x* and v = y − y*. In mathematical biology, the Jacobian matrix ${\displaystyle \mathbf {A} }$ evaluated at the equilibrium point (x*, y*) is called the community matrix.^[2] By the stable manifold theorem, if one or both eigenvalues of ${\displaystyle \mathbf {A} }$ 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.