Community matrix

{{Short description|Community Matrix}}

{{Use dmy dates|date=June 2016}}

In mathematical biology, the community matrix is the linearization of a generalized Lotka–Volterra equation at an equilibrium point.{{cite journal|last1=Berlow|first1=E. L.

| last2 = Neutel| first2= A.-M. | last3 = Cohen| first3= J. E.| last4 =De Ruiter | first4=P. C. | last5 =Ebenman | first5= B.| last6 =Emmerson | first6=M. | last7 =Fox | first7= J. W.| last8 = Jansen| first8= V. A. A.| last9 =Jones | first9=J. I. | last10 =Kokkoris | first10=G. D. | last11 =Logofet | first11=D. O. | last12 =McKane | first12= A. J. | last13 = Montoya | first13=J. M | last14 =Petchey | first14= O.|title=Interaction Strengths in Food Webs: Issues and Opportunities |journal= Journal of Animal Ecology|volume=73|issue=5|pages=585–598|year=2004|doi=10.1111/j.0021-8790.2004.00833.x|jstor=3505669

| doi-access=free}} The eigenvalues of the community matrix determine the stability of the equilibrium point.

For example, the Lotka–Volterra predator–prey model is

:$\backslash begin\{array\}\{rcl\}$

\dfrac{dx}{dt} &=& x(\alpha - \beta y) \\

\dfrac{dy}{dt} &=& - y(\gamma - \delta x),

\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

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

where u = x − x* and v = y − y*. In mathematical biology, the Jacobian matrix $\backslash mathbf\{A\}$ evaluated at the equilibrium point (x*, y*) is called the community matrix.{{cite book |title=Elements of Mathematical Ecology |first=Mark |last=Kot |publisher=Cambridge University Press |year=2001 |isbn=0-521-00150-1 |page=144 |url=https://books.google.com/books?id=7_IRlnNON7oC&pg=PA144 }} By the stable manifold theorem, if one or both eigenvalues of $\backslash mathbf\{A\}$ have positive real part then the equilibrium is unstable, but if all eigenvalues have negative real part then it is stable.