|
|
| Line 1: |
Line 1: |
| {{Santa Fe CSSS 2007}}
| |
|
| |
|
| The '''competitive Lotka-Volterra equations''' are a simple [[model (abstract)|model]] of the [[population dynamics]] of species [[competition|competing]] for some common [[Natural resource|resource]].
| |
|
| |
| ==Overview==
| |
|
| |
| The form is similar to the [[Lotka-Volterra equation]]s for [[predator|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.
| |
|
| |
| The logistic population model, when used by [[ecology|ecologists]] often takes the following form:
| |
|
| |
| :<math>{dx \over dt} = rx\left({K-x \over K}\right)</math>
| |
|
| |
| 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]].
| |
|
| |
| ===Two species===
| |
|
| |
| Given 2 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' [[biological interaction|interactions]]. Thus the competitive Lotka-Volterra equations are:
| |
|
| |
| :<math>{dx_1 \over dt} = r_1x_1\left({K_1-x_1-\alpha_{12}x_2 \over K_1}\right)</math>
| |
| :<math>{dx_2 \over dt} = r_2x_2\left({K_2-x_2-\alpha_{21}x_1 \over K_2}\right)</math>
| |
|
| |
| 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]].
| |
|
| |
| ===''N'' species===
| |
|
| |
| 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 (spatial)|vectors]] and the interaction ''α'''s as a [[matrix (mathematics)|matrix]]. Then the equation for any species ''i'' becomes
| |
|
| |
| :<math>\frac{dx_i}{dt} = r_i x_i \left( \frac{K_i - \sum_{j=1}^N \alpha_{ij}x_j}{K_i} \right) </math>
| |
|
| |
| 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),
| |
|
| |
| :<math>\frac{dx_i}{dt} = r_i x_i \left( 1 - \sum_{j=1}^N \alpha_{ij}x_j \right) </math>
| |
|
| |
| where ''N'' is the total number of interacting species. For simplicity all self-interacting terms ''α''<sub>ii</sub> are often set to 1.
| |
|
| |
| ===Possible dynamics===
| |
|
| |
| 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.
| |
|
| |
| # 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.
| |
| # 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]] [[Chaotic attractor|attractors]].
| |
| # 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. A [[Torus|''n''-torus]] cannot exist in less than ''n'' dimensions, and finally, [[Chaotic attractor|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 [[Chaotic attractor|chaos]] for ''N'' < 4. This is still in agreement with Smale that any dynamics can occur for ''N'' ≥ 5.
| |
| #*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 [[Asymptote|asymptotic]] dynamics of the system.
| |
|
| |
| ==4-dimensional example==
| |
|
| |
| [[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.]]
| |
|
| |
| 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
| |
|
| |
| :<math>r_i = \begin{bmatrix} 1 \\ 0.72 \\ 1.53 \\ 1.27 \end{bmatrix} \ \ \ \ \ \ \alpha_{ij} = \begin{bmatrix} 1 & 1.09 & 1.52 & 0 \\ 0 & 1 & 0.44 & 1.36 \\ 2.33 & 0 & 1 & 1.47 \\ 1.21 & 0.51 & 0.35 & 1 \end{bmatrix}</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 [[dimension|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 [[derivative]]s 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>\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>
| |
|
| |
| Note that there are always 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 was 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]].
| |
|
| |
| ==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'' effects ''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 ''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
| |
|
| |
| :<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>
| |
|
| |
| 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
| |
|
| |
| :<math>\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 ''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 (dynamics)|orbit]] is a stable [[attractor]], but it must be either a [[limit-cycle]] or [[torus|''n''-torus]] - but not a [[strange attractor]] (this is because the largest [[Lyapunov exponent]] of a [[limit-cycle]] and [[torus|''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 (mathematics)|fixed point]] [[attractor]]. When searching a [[dynamical system]] for non-[[fixed point (mathematics)|fixed point]] [[attractor]]s, 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>Wildenberg, J.C., Vano, J.A., Sprott, J.C. Complex spatiotemporal dynamics in Lotka–Volterra ring systems. Ecological Complexity <b>3<b>, 140. 2006</ref> If all species are identical in their spatial interactions, then the interaction matrix is [[circulant matrix|circulant]]. The [[Eigenvalue, eigenvector and eigenspace|eigenvalues]] of a [[circulant matrix]] are given by<ref>Hofbauer, J., Sigmund, K., 1988. The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge, U.K, p. 352.</ref>
| |
|
| |
| :<math>\lambda_k = \sum_{j=0}^{N-1} c_j\gamma^{kj}</math>
| |
|
| |
| 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]].
| |
|
| |
| The [[Lyapunov function]] exists if the real part of the [[Eigenvalue, eigenvector and eigenspace|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
| |
|
| |
| :<math>Re(\lambda_k) = 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>
| |
|
| |
| for k = 0…''N''-1. Now, instead of having to [[Numerical integration|integrate]] the system over thousands of time steps to see if any dynamics other than a [[Fixed point (mathematics)|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 ''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 (mathematics)|fixed point]]. If ''α''<sub>1</sub> = 0.852 then the real part of one of the complex [[Eigenvalue, eigenvector and eigenspace|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>\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 function]]s that have not been discovered.
| |
|
| |
| The [[Eigenvalue, eigenvector and eigenspace|eigenvalues]] of the circle system plotted in the [[complex plane]] form a [[trefoil]] shape. The [[Eigenvalue, eigenvector and eigenspace|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.
| |
|
| |
| ==References==
| |
| <div class="references-small">
| |
| <references/>
| |
| </div>
| |
