Community matrix explained
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
\begin{array}{rcl}\dfrac{dx}{dt}&=&x(\alpha-\betay)\\
\dfrac{dy}{dt}&=&-y(\gamma-\deltax),
\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}
\end{bmatrix}=A\begin{bmatrix}u\ v\end{bmatrix},
where
u =
x −
x* and
v =
y −
y*. In mathematical biology, the
Jacobian matrix
evaluated at the equilibrium point (
x*,
y*) is called the community matrix.
[2] By the
stable manifold theorem, if one or both eigenvalues of
have positive real part then the equilibrium is unstable, but if all eigenvalues have negative real part then it is stable.
See also
References
Notes and References
- Berlow. E. L.. Neutel. A.-M. . Cohen. J. E.. De Ruiter . P. C. . Ebenman . B.. Emmerson . M. . Fox . J. W.. Jansen. V. A. A.. Jones . J. I. . Kokkoris . G. D. . Logofet . D. O. . McKane . A. J. . Montoya . J. M . Petchey . O.. Interaction Strengths in Food Webs: Issues and Opportunities . Journal of Animal Ecology. 73. 5. 585–598. 2004. 10.1111/j.0021-8790.2004.00833.x. 3505669. free.
- Book: Kot, Mark . Elements of Mathematical Ecology . Cambridge University Press . 2001 . 0-521-00150-1 . 144 .
