Editing Competitive Lotka–Volterra equations (section)

===Lyapunov functions===
A [[Lyapunov function]] is a [[function (mathematics)|function]] of the system {{math|1=''f'' = ''f''(''x'')}} whose existence in a system demonstrates [[Lyapunov stability|stability]].  It is often useful to imagine a Lyapunov function as the energy of the system.  If the derivative of the function is equal to zero for some [[Orbit (dynamics)|orbit]] not including the [[equilibrium point]], then that orbit is a stable [[attractor]], but it must be either a limit cycle or ''n''-torus - but not a [[strange attractor]] (this is because the largest [[Lyapunov exponent]] of a limit cycle and ''n''-torus are zero while that of a strange attractor is positive).  If the derivative is less than zero everywhere except the equilibrium point, then the equilibrium point is a stable fixed point attractor.  When searching a [[dynamical system]] for non-fixed point attractors, the existence of a Lyapunov function can help eliminate regions of parameter space where these dynamics are impossible.

The spatial system introduced above has a Lyapunov function that has been explored by Wildenberg ''et al.''<ref name = Wildenberg>{{cite journal | last1=Wildenberg | first1=J.C. | last2=Vano | first2=J.A. | last3=Sprott | first3=J.C. | title=Complex spatiotemporal dynamics in Lotka–Volterra ring systems | journal=Ecological Complexity | publisher=Elsevier BV | volume=3 | issue=2 | year=2006 | issn=1476-945X | doi=10.1016/j.ecocom.2005.12.001 | pages=140–147}}</ref>  If all species are identical in their spatial interactions, then the interaction matrix is [[circulant matrix|circulant]].  The eigenvalues of a circulant matrix are given by<ref>Hofbauer, J., [[Karl Sigmund|Sigmund, K.]], 1988. The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge, U.K, p. 352.</ref>
<math display="block">\lambda_k = \sum_{j=0}^{N-1} c_j\gamma^{kj}</math>

for {{math|1=''k'' = 0<sub>''N'' − 1</sub>}} and where <math>\gamma = e^{i2\pi/N}</math> the ''N''th [[root of unity]].  Here {{math|''c<sub>j</sub>''}} is the ''j''th value in the first row of the circulant matrix.

The Lyapunov function exists if the real part of the eigenvalues are positive ({{math|Re(''λ<sub>k</sub>'') > 0}} for {{math|1=''k'' = 0, …, ''N''/2}}).  Consider the system where {{math|1=''α''<sub>−2</sub> = ''a''}}, {{math|1=''α''<sub>−1</sub> = ''b''}}, {{math|1=''α''<sub>1</sub> = ''c''}}, and {{math|1=''α''<sub>2</sub> = ''d''}}.  The Lyapunov function exists if
<math display="block">\begin{align}
\operatorname{Re}(\lambda_k) &= \operatorname{Re} \left ( 1+\alpha_{-2}e^{i2 \pi k(N-2)/N} + \alpha_{-1}e^{i2 \pi k(N-1)/N} + \alpha_1e^{i2 \pi k/N} + \alpha_2e^{i4 \pi k/N} \right ) \\
&= 1+(\alpha_{-2}+\alpha_2)\cos \left ( \frac{4 \pi k}{N} \right ) + (\alpha_{-1}+\alpha_1)\cos \left ( \frac{2 \pi k}{N} \right ) > 0
\end{align}</math>
for {{math|1=''k'' = 0, …, ''N'' − 1}}.  Now, instead of having to integrate the system over thousands of time steps to see if any dynamics other than a fixed point attractor exist, one need only determine if the Lyapunov function exists (note: the absence of the Lyapunov function doesn't guarantee a limit cycle, torus, or chaos).

Example: Let {{math|1=''α''<sub>−2</sub> = 0.451}}, {{math|1=''α''<sub>−1</sub> = 0.5}}, and {{math|1=''α''<sub>2</sub> = 0.237}}.  If {{math|1=''α''<sub>1</sub> = 0.5}} then all eigenvalues are negative and the only attractor is a fixed point.  If {{math|1=''α''<sub>1</sub> = 0.852}} then the real part of one of the complex eigenvalue pair becomes positive and there is a strange attractor.  The disappearance of this Lyapunov function coincides with a [[Hopf bifurcation]].