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
- [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \mathbf{A} }[/math] have positive real part then the equilibrium is unstable, but if all eigenvalues have negative real part then it is stable.
See also
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. et al. (2004). "Interaction Strengths in Food Webs: Issues and Opportunities". Journal of Animal Ecology 73 (5): 585–598. doi:10.1111/j.0021-8790.2004.00833.x.
- ↑ Kot, Mark (2001). Elements of Mathematical Ecology. Cambridge University Press. p. 144. ISBN 0-521-00150-1. https://books.google.com/books?id=7_IRlnNON7oC&pg=PA144.
- Murray, James D. (2002), Mathematical Biology I. An Introduction, Interdisciplinary Applied Mathematics, 17 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95223-9.
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