Community matrix

Short description: 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

dxdt=x(α−βy)dydt=−y(γ−δx),

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

[dudtdvdt]=𝐀[uv],

where u = x − x* and v = y − y*. In mathematical biology, the Jacobian matrix 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 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

1. ↑ 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.
2. ↑ 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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