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

d x d t = x ( α − β y ) d y d t = − y ( γ − δ x ) , {\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

[ d u d t d v d t ] = A [ u v ] , {\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 A {\displaystyle \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 A {\displaystyle \mathbf {A} } have positive real part then the equilibrium is unstable, but if all eigenvalues have negative real part then it is stable.

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