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| The '''competitive Lotka–Volterra equations''' are a simple model of the [[population dynamics]] of species competing for some common resource. They can be further [[Generalized Lotka–Volterra equation|generalised]] to include trophic interactions. | | The writer's title is Milford Cuccia but it's not the most masucline title out there. My husband and I reside in Maryland. Invoicing is her day occupation now but she plans on altering it. One of the extremely very best issues in the world for him is to perform croquet but he can't make it his occupation. He's been working on his website for some time now. Verify it out right here: http://eu4wiki.de/index.php?title=4_Stunning_Examples_Of_Beautiful_Nya_Online_Svenska_Casino<br><br> |
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| ==Overview==
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| The form is similar to the [[Lotka–Volterra equation]]s for predation in that the equation for each species has one term for self-interaction and one term for the interaction with other species. In the equations for predation, the base population model is [[exponential function|exponential]]. For the competition equations, the [[logistic equation]] is the basis.
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| The logistic population model, when used by [[ecology|ecologists]] often takes the following form:
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| :<math>{dx \over dt} = rx\left(1-{x \over K}\right).</math>
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| Here ''x'' is the size of the population at a given time, ''r'' is inherent per-capita growth rate, and ''K'' is the [[carrying capacity]].
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| ===Two species===
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| Given two populations, ''x''<sub>1</sub> and ''x''<sub>2</sub>, with logistic dynamics, the Lotka–Volterra formulation adds an additional term to account for the species' interactions. Thus the competitive Lotka–Volterra equations are:
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| :<math>{dx_1 \over dt} = r_1x_1\left(1-\left({x_1+\alpha_{12}x_2 \over K_1}\right) \right)</math>
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| :<math>{dx_2 \over dt} = r_2x_2\left(1-\left({x_2+\alpha_{21}x_1 \over K_2}\right) \right).</math>
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| Here, ''α''<sub>12</sub> represents the effect species 2 has on the population of species 1 and ''α''<sub>21</sub> represents the effect species 1 has on the population of species 2. These values do not have to be equal. Because this is the competitive version of the model, all interactions must be harmful (competition) and therefore all ''α''-values are positive. Also, note that each species can have its own growth rate and carrying capacity. A complete classification of this dynamics, even for all sign patterns of above coefficients, is available,<ref>[[Immanuel Bomze|I. Bomze]], Lotka–Volterra equation and replicator dynamics: a two-dimensional classification. ''Biological Cybernetics'' 48, 201–211 (1983); [[Immanuel Bomze|I. Bomze]], Lotka–Volterra equation and replicator dynamics: new issues in classification. ''Biological Cybernetics'' 72, 447–453 (1995).</ref> which is based upon equivalence to the 3-type [[replicator equation]]. | |
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| ===''N'' species===
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| This model can be generalized to any number of species competing against each other. One can think of the populations and growth rates as [[vector (geometric)|vectors]] and the interaction ''α'''s as a [[matrix (mathematics)|matrix]]. Then the equation for any species ''i'' becomes
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| :<math>\frac{dx_i}{dt} = r_i x_i \left(1- \frac{\sum_{j=1}^N \alpha_{ij}x_j}{K_i} \right) </math> | |
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| or, if the carrying capacity is pulled into the interaction matrix (this doesn't actually change the equations, only how the interaction matrix is defined),
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| :<math>\frac{dx_i}{dt} = r_i x_i \left( 1 - \sum_{j=1}^N \alpha_{ij}x_j \right) </math>
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| where ''N'' is the total number of interacting species. For simplicity all self-interacting terms ''α''<sub>ii</sub> are often set to 1.
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| ===Possible dynamics===
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| The definition of a competitive Lotka-Volterra system assumes that all values in the interaction matrix are positive or 0 (''α<sub>ij</sub>'' ≥ 0 for all ''i,j''). If it is also assumed that the population of any species will increase in the absence of competition unless the population is already at the carrying capacity (''r<sub>i</sub>'' > 0 for all ''i''), then some definite statements can be made about the behavior of the system.
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| # The populations of all species will be bounded between 0 and 1 at all times (0 ≤ ''x<sub>i</sub>'' ≤ 1, for all ''i'') as long as the populations started out positive.
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| # Smale<ref>S. Smale, J. Math. Biol. '''3''', 5., 1976</ref> showed that Lotka-Volterra systems that meet the above conditions and have five or more species (''N'' ≥ 5) can exhibit any [[Asymptote|asymptotic]] behavior, including a [[Fixed point (mathematics)|fixed point]], a [[limit cycle]], an [[Torus|''n''-torus]], or [[Chaotic attractor|attractors]].
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| # Hirsch<ref>M. Hirsch, SIAM J. Math. Anal. '''16''', 423., 1985</ref><ref>M. Hirsch, Nonlinearity '''1''', 51., 1988</ref><ref>M. Hirsch, SIAM J. Math. Anal. '''21''', 1225., 1990</ref> proved that all of the dynamics of the attractor occur on a [[manifold]] of dimension ''N''-1. This basically says that the attractor cannot have [[dimension]] greater than ''N''-1. Why is this important? A limit cycle cannot exist in fewer than two dimensions. An ''n''-torus cannot exist in less than ''n'' dimensions, and finally, chaos cannot occur in less than three dimensions. So, Hirsch proved that competitive Lotka–Volterra systems cannot exhibit a limit cycle for ''N'' < 3, or any [[torus]] or chaos for ''N'' < 4. This is still in agreement with Smale that any dynamics can occur for ''N'' ≥ 5.
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| #*More specifically, Hirsch showed there is an [[Invariant (mathematics)|invariant]] set ''C'' that is [[Homeomorphism|homeomorphic]] to the (''N''-1)-dimensional [[simplex]]<br /><math>\Delta_{N-1} = \left \{ x_i : x_i \ge 0, \sum x_i = 1 \right \}</math><br />and is a global attractor of every point excluding the origin. This carrying simplex contains all of the asymptotic dynamics of the system.
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| # To create a stable ecosystem the α<sub>ij</sub> matrix must have all positive eigenvalues. For large N systems Lotka-Volterra models are either unstable or have low connectivity. Kondoh <ref>Kondoh M Science 299, 1388 (2003)</ref> and Ackland and Gallagher <ref>Ackland GJ and Gallagher ID Phys. Rev. Lett. 93, 158701 (2004)</ref> have independently shown that large, stable Lotka-Volterra systems arise if the elements of α<sub>ij</sub> (i.e. the features of the species) can evolve in accordance with natural selection.
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| ==4-dimensional example==
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| [[Image:4D competitive LV color.png|thumb|right|350px|The competitive Lotka–Volterra system plotted in [[phase space]] with the ''x''<sub>4</sub> value represented by the color.]]
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| A simple 4-Dimensional example of a competitive Lotka–Volterra system has been characterized by Vano ''et al.''<ref>Vano, J.A., Wildenberg, J.C., Anderson, M.B., Noel, J.K., Sprott, J.C. Chaos in Low-Dimensional Lotka–Volterra Models of Competition. Submitted Nonlinearity. 2006</ref> Here the growth rates and interaction matrix have been set to
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| :<math>r_i = \begin{bmatrix} 1 \\ 0.72 \\ 1.53 \\ 1.27 \end{bmatrix} \quad \alpha_{ij} = \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>
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| 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
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| :<math>\overline{x} = \left ( \alpha_{ij} \right )^{-1} \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0.3013 \\ 0.4586 \\ 0.1307 \\ 0.3557 \end{bmatrix}.</math>
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| Note that there are always 2<sup>''N''</sup> equilibrium points, but all others have at least one species' population equal to zero.
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| 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]].
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| ==Spatial arrangements==
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| [[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''.]]
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| ===Background===
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| 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.
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| ===Matrix organization===
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| One possible way to incorporate this spatial structure is to modify the nature of the Lotka–Volterra equations to something like a
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| [[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 ''α''<sub>-1</sub> and ''α''<sub>1</sub> respectively. Thus, species 3 interacts only with species 2 and 4, species 1 interacts only with species 2 and 6, etc. The interaction matrix will now be
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| :<math>\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>
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| 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>Sprott, J.C., Wildenberg, J.C., Azizi, Y. A simple spatiotemporal chaotic Lotka–Volterra model. Chaos, Solitons & Fractals '''26''', 1035., 2005</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
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| :<math>\overline{x}_i = \frac{1}{\sum_{j=1}^N \alpha_{ij}} = \frac{1}{\alpha_{-1} + 1 + \alpha_1}.</math>
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| ===Lyapunov functions===
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| A [[Lyapunov function]] is a [[function (mathematics)|function]] of the system ''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.
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| The spatial system introduced above has a Lyapunov function that has been explored by Wildenberg ''et al.''<ref name = Wildenberg>Wildenberg, J.C., Vano, J.A., Sprott, J.C. Complex spatiotemporal dynamics in Lotka–Volterra ring systems. Ecological Complexity '''3''', 140. 2006</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>
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| :<math>\lambda_k = \sum_{j=0}^{N-1} c_j\gamma^{kj}</math> | |
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| for ''k'' = 0<sub>''N'' − 1</sub> and where <math>\gamma = e^{i2\pi/N}</math> the ''N''th [[root of unity]]. Here ''c<sub>j</sub>'' is the ''j''th value in the first row of the circulant matrix.
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| The Lyapunov function exists if the real part of the eigenvalues are positive (Re(''λ<sub>k</sub>'' > 0 for ''k'' = 0, …, ''N''/2). Consider the system where ''α''<sub>-2</sub> = ''a'', ''α''<sub>-1</sub> = ''b'', ''α''<sub>1</sub> = ''c'', and ''α''<sub>2</sub> = ''d''. The Lyapunov function exists if
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| :<math>\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 )</math> <math>= 1+(\alpha_{-2}+\alpha_2)\cos \left ( \frac{i4 \pi k}{N} \right ) + (\alpha_{-1}+\alpha_1)\cos \left ( \frac{i2 \pi k}{N} \right ) > 0 </math>
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| for 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).
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| Example: Let ''α''<sub>−2</sub> = 0.451, ''α''<sub>−1</sub> = 0.5, and ''α''<sub>2</sub> = 0.237. If ''α''<sub>1</sub> = 0.5 then all eigenvalues are negative and the only attractor is a fixed point. If ''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]].
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| ===Line systems and eigenvalues===
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| [[Image:Competitive LV Spatial Eigenvalues.jpg|thumb|right|350px|The eigenvalues of a circle, short line, and long line plotted in the complex plane]]
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| 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.).
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| :<math>\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>
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| 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.
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| 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.
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| ==Notes==
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| {{reflist}}
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| {{DEFAULTSORT:Competitive Lotka-Volterra equations}}
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| [[Category:Chaotic maps]]
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| [[Category:Equations]]
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| [[Category:Population ecology]]
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| [[Category:Community ecology]]
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| [[Category:Population models]]
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