Editing Competitive Lotka–Volterra equations (section)

==Spatial arrangements==
[[Image:Competitive LV Spatial Bee Example.JPG|thumb|right|300px|An illustration of spatial structure in nature. The strength of the interaction between bee colonies is a function of their proximity. Colonies ''A'' and ''B'' interact, as do colonies ''B'' and ''C''. ''A'' and ''C'' do not interact directly, but affect each other through colony ''B''.]]

===Background===
There are many situations where the strength of species' interactions depends on the physical distance of separation.  Imagine bee colonies in a field.  They will compete for food strongly with the colonies located near to them, weakly with further colonies, and not at all with colonies that are far away.  This doesn't mean, however, that those far colonies can be ignored.  There is a [[Transitive relation|transitive]] effect that permeates through the system.  If colony ''A'' interacts with colony ''B'', and ''B'' with ''C'', then ''C'' affects ''A'' through ''B''.  Therefore, if the competitive Lotka–Volterra equations are to be used for modeling such a system, they must incorporate this spatial structure.

===Matrix organization===
One possible way to incorporate this spatial structure is to modify the nature of the Lotka–Volterra equations to something like a 
[[reaction–diffusion system]].  It is much easier, however, to keep the format of the equations the same and instead modify the interaction matrix.  For simplicity, consider a five species example where all of the species are aligned on a circle, and each interacts only with the two neighbors on either side with strength {{math|''α''<sub>−1</sub>}} and {{math|''α''<sub>1</sub>}} respectively.  Thus, species 3 interacts only with species 2 and 4, species 1 interacts only with species 2 and 5, etc.  The interaction matrix will now be
<math display="block">\alpha_{ij} = \begin{bmatrix}1 & \alpha_1 & 0 & 0 & \alpha_{-1} \\ \alpha_{-1} & 1 & \alpha_1 & 0 & 0 \\ 0 & \alpha_{-1} & 1 & \alpha_1 & 0 \\ 0 & 0 & \alpha_{-1} & 1 & \alpha_1 \\ \alpha_1 & 0 & 0 & \alpha_{-1} & 1 \end{bmatrix}.</math>

If each species is identical in its interactions with neighboring species, then each row of the matrix is just a [[permutation]] of the first row.  A simple, but non-realistic, example of this type of system has been characterized by Sprott ''et al.''<ref>{{cite journal | last1=Sprott | first1=J.C. | last2=Wildenberg | first2=J.C. | last3=Azizi | first3=Yousef | title=A simple spatiotemporal chaotic Lotka–Volterra model | journal=Chaos, Solitons & Fractals | publisher=Elsevier BV | volume=26 | issue=4 | year=2005 | issn=0960-0779 | doi=10.1016/j.chaos.2005.02.015 | pages=1035–1043| bibcode=2005CSF....26.1035S }}</ref>  The coexisting [[equilibrium point]] for these systems has a very simple form given by the [[Inverse element|inverse]] of the sum of the row
<math display="block">\overline{x}_i = \frac{1}{\sum_{j=1}^N \alpha_{ij}} = \frac{1}{\alpha_{-1} + 1 + \alpha_1}.</math>

===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]].

===Line systems and eigenvalues===
[[Image:Competitive LV Spatial Eigenvalues.jpg|thumb|right|350px|The eigenvalues of a circle, short line, and long line plotted in the complex plane]]

It is also possible to arrange the species into a line.<ref name=Wildenberg/>  The interaction matrix for this system is very similar to that of a circle except the interaction terms in the lower left and upper right of the matrix are deleted (those that describe the interactions between species 1 and ''N'', etc.).

<math display="block">\alpha_{ij} = \begin{bmatrix}1 & \alpha_1 & 0 & 0 & 0 \\ \alpha_{-1} & 1 & \alpha_1 & 0 & 0 \\ 0 & \alpha_{-1} & 1 & \alpha_1 & 0 \\ 0 & 0 & \alpha_{-1} & 1 & \alpha_1 \\ 0 & 0 & 0 & \alpha_{-1} & 1 \end{bmatrix}</math>

This change eliminates the Lyapunov function described above for the system on a circle, but most likely there are other Lyapunov functions that have not been discovered.

The eigenvalues of the circle system plotted in the [[complex plane]] form a [[trefoil]] shape.  The eigenvalues from a short line form a sideways Y, but those of a long line begin to resemble the trefoil shape of the circle.  This could be due to the fact that a long line is indistinguishable from a circle to those species far from the ends.<ref name="Wildenberg" />