Английская Википедия:Community matrix
Шаблон:Short description Шаблон:Use dmy dates 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
- <math> \begin{array}{rcl}
\dfrac{dx}{dt} &=& x(\alpha - \beta y) \\ \dfrac{dy}{dt} &=& - y(\gamma - \delta x), \end{array} </math> 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
- <math> \begin{bmatrix} \frac{du}{dt} \\ \frac{dv}{dt} \end{bmatrix} = \mathbf{A} \begin{bmatrix} u \\ v \end{bmatrix}, </math>
where u = x − x* and v = y − y*. In mathematical biology, the Jacobian matrix <math>\mathbf{A}</math> evaluated at the equilibrium point (x*, y*) is called the community matrix.[2] By the stable manifold theorem, if one or both eigenvalues of <math>\mathbf{A}</math> 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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