Editing Competitive Lotka–Volterra equations (section)

==4-dimensional example==
[[Image:4D competitive LV color.png|thumb|right|350px|The competitive Lotka–Volterra system plotted in [[phase space]] with the {{math|''x''<sub>4</sub>}} value represented by the color]]

A simple 4-dimensional example of a competitive Lotka–Volterra system has been characterized by Vano ''et al.''<ref name="ReferenceA">{{cite journal | last1=Vano | first1=J A | last2=Wildenberg | first2=J C | last3=Anderson | first3=M B | last4=Noel | first4=J K | last5=Sprott | first5=J C | title=Chaos in low-dimensional Lotka–Volterra models of competition | journal=Nonlinearity | publisher=IOP Publishing | volume=19 | issue=10 | date=2006-09-15 | issn=0951-7715 | doi=10.1088/0951-7715/19/10/006 | pages=2391–2404| bibcode=2006Nonli..19.2391V | s2cid=9417299 }}</ref>  Here the growth rates and interaction matrix have been set to

<math display="block">r = \begin{bmatrix} 1 \\ 0.72 \\ 1.53 \\ 1.27 \end{bmatrix} \quad
\alpha = \begin{bmatrix} 1  & 1.09 & 1.52 & 0 \\ 0 & 1 & 0.44 & 1.36 \\ 2.33 & 0 & 1 & 0.47 \\ 1.21 & 0.51 & 0.35 & 1 \end{bmatrix}</math>

with <math>K_i=1</math> for all <math>i</math>. This system is chaotic and has a largest [[Lyapunov exponent]] of 0.0203.  From the theorems by Hirsch, it is one of the lowest-dimensional chaotic competitive Lotka–Volterra systems.  The Kaplan–Yorke dimension, a measure of the dimensionality of the attractor, is 2.074.  This value is not a whole number, indicative of the [[fractal]] structure inherent in a [[strange attractor]].  The coexisting [[equilibrium point]], the point at which all derivatives are equal to zero but that is not the [[Origin (mathematics)|origin]], can be found by [[Invertible matrix|inverting]] the interaction matrix and [[Matrix multiplication|multiplying]] by the unit [[column vector]], and is equal to

<math display="block">\overline{x} = \left ( \alpha \right )^{-1} \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0.3013 \\ 0.4586 \\ 0.1307 \\ 0.3557 \end{bmatrix}.</math>

Note that there are always {{math|2<sup>''N''</sup>}} equilibrium points, but all others have at least one species' population equal to zero.

The [[Eigenvalue, eigenvector and eigenspace|eigenvalues]] of the system at this point are 0.0414±0.1903''i'', −0.3342, and −1.0319.  This point is unstable due to the positive value of the real part of the [[Complex number|complex]] eigenvalue pair.  If the real part were negative, this point would be stable and the orbit would attract asymptotically.  The transition between these two states, where the real part of the complex eigenvalue pair is equal to zero, is called a [[Hopf bifurcation]].

A detailed study of the parameter dependence of the dynamics was performed by Roques and Chekroun in.<ref name="L.Roques 2011">{{cite journal | last1=Roques | first1=Lionel | last2=Chekroun | first2=Mickaël D. | title=Probing chaos and biodiversity in a simple competition model | journal=Ecological Complexity | publisher=Elsevier BV | volume=8 | issue=1 | year=2011 | issn=1476-945X | doi=10.1016/j.ecocom.2010.08.004 | pages=98–104| url=https://hal.archives-ouvertes.fr/hal-00511225/file/RoqChe_ecocplx_rev.pdf }}</ref>  
The authors observed that interaction and growth parameters leading respectively to extinction of three species, or coexistence of two, three or four species, are for the most part arranged in large regions with clear boundaries. As predicted by the theory, chaos was also found; taking place however over much smaller islands of the parameter space which causes difficulties in the identification of their location by a random search algorithm.<ref name="ReferenceA"/> These regions where chaos occurs are, in the three cases analyzed in,<ref name="L.Roques 2011"/> situated at the interface between a non-chaotic four species region and a region where extinction occurs. This implies a high sensitivity of biodiversity with respect to parameter variations in the chaotic regions. Additionally, in regions where extinction occurs which are adjacent to chaotic regions, the computation of local Lyapunov exponents <ref>{{cite journal | last=Nese | first=Jon M. | title=Quantifying local predictability in phase space | journal=Physica D: Nonlinear Phenomena | publisher=Elsevier BV | volume=35 | issue=1–2 | year=1989 | issn=0167-2789 | doi=10.1016/0167-2789(89)90105-x | pages=237–250| bibcode=1989PhyD...35..237N }}</ref> revealed that a possible cause of extinction is the overly strong fluctuations in [[species abundance]]s induced by local chaos.