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

\begin{matrix} \frac{d x}{d t} & = & x (α - β y) \\ \frac{d y}{d t} & = & - y (γ - δ x), \end{matrix}

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{matrix} \frac{d u}{d t} \\ \frac{d v}{d t} \end{matrix}] = 𝐀 [\begin{matrix} u \\ v \end{matrix}],

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. 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.

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. (2004). "Interaction Strengths in Food Webs: Issues and Opportunities". Journal of Animal Ecology. 73 (5): 585–598. Bibcode:2004JAnEc..73..585B. doi:10.1111/j.0021-8790.2004.00833.x. JSTOR 3505669.
↑ Kot, Mark (2001). Elements of Mathematical Ecology. Cambridge University Press. p. 144. ISBN 0-521-00150-1.

Murray, James D. (2002), Mathematical Biology I. An Introduction, Interdisciplinary Applied Mathematics, vol. 17 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95223-9.

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[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.; Jones, J. I.; Kokkoris, G. D.; Logofet, D. O.; McKane, A. J.; Montoya, J. M; Petchey, O. (2004). "Interaction Strengths in Food Webs: Issues and Opportunities". Journal of Animal Ecology. 73 (5): 585–598. Bibcode:2004JAnEc..73..585B. doi:10.1111/j.0021-8790.2004.00833.x. JSTOR 3505669.

[2] Kot, Mark (2001). Elements of Mathematical Ecology. Cambridge University Press. p. 144. ISBN 0-521-00150-1.

Community matrix

See also

References