Community matrix

Inmathematical biology,thecommunity matrixis thelinearizationof ageneralized Lotka–Volterra equationat anequilibrium point.^[1]Theeigenvaluesof the community matrix determine thestabilityof the equilibrium point.

For example, theLotka–Volterra predator–prey modelis

{\begin{array}{rcl}{\dfrac {dx}{dt}}&=&x(\ Alpha -\beta y)\\{\dfrac {dy}{dt}}&=&-y(\gamma -\delta x),\end{array}}

wherex(t) denotes the number of prey,y(t) the number of predators, andα,β,γandδare constants. By theHartman–Grobman theoremthe non-linear system istopologically equivalentto a linearization of the system about an equilibrium point (x*,y*), which has the form

{\begin{bmatrix}{\frac {du}{dt}}\\{\frac {dv}{dt}}\end{bmatrix}}=\mathbf {A} {\begin{bmatrix}u\\v\end{bmatrix}},

whereu=x−x* andv=y−y*. In mathematical biology, theJacobian matrix $\mathbf {A}$ evaluated at the equilibrium point (x*,y*) is called the community matrix.^[2]By thestable manifold theorem,if one or both eigenvalues of $\mathbf {A}$ 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.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.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.

[1]

[2]

Community matrix

See also

References