mathematical biology Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development a ...

, the community matrix is the

linearization In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, lineariz ...

of the Lotka–Volterra equation at an

equilibrium point In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation. Formal definition The point \tilde\in \mathbb^n is an equilibrium point for the differential equation :\frac = \ma ...

. The

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

s of the community matrix determine the

stability Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems ** Asymptotic stability ** Linear stability ** Lyapunov stability ** Orbital stability ** Structural sta ...

of the equilibrium point. The Lotka–Volterra predator–prey model is :

\begin 
\dfrac &=& x(\alpha - \beta y) \\
\dfrac &=& - y(\gamma - \delta x),
\end

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 In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that lineari ...

the non-linear system is

topologically equivalent In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated fun ...

to a linearization of the system about an equilibrium point (''x''*, ''y''*), which has the form :

\begin \frac \\ \frac \end = \mathbf \begin u \\ v \end,

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

Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...

\mathbf

evaluated at the equilibrium point (''x''*, ''y''*) is called the community matrix. By the stable manifold theorem, if one or both eigenvalues of

\mathbf

have positive real part then the equilibrium is unstable, but if all eigenvalues have negative real part then it is stable.

References

* . Mathematical and theoretical biology Population ecology Dynamical systems {{mathapplied-stub

See also

References