Lotka-Volterra models consider a system of n species each with density x_i. The following finite difference equation is used for the dynamics of the system:
This equation is the finite difference equation analogue of the more common Lotka-Volterra differential equation given by
The finite difference equation should not be viewed as an approximation of the differential equation. Instead it describes a different situation, where growth occurs in distinct generations, rather than in a continous fashion.
We will now give an interpretation of the models parameters, and describe a procedure for random generation of them.
The parameter a_i is the intrinsic growth rate of species i, while b_ij describes how the density of species j affects the growth rate of species i. Completely arbitrary choices of a_i and b_ij is likely to cause the dynamic to be divergent, and therefore we will enforce some restrictions upon a_i and b_ij. Let a_i>0 (since a_i should be interpreted as a growth rate) and b_ii = -a_i. Then delta x_i = 0 when x_i = 1 in the absence of non-zero non-diagonal elements b_ij. The self-limiting condition b_ii = - a_i means that the model will be similar to the logistic map if non-diagonal elements are ignored. The different growth rates a_i could be chosen randomly in some fashion, but no natural choice of distribution spring to mind, and we have chosen to set all a_i=a. The non-diagonal elements b_ij are 0 with probability (1-C) and with probability C an independent random number drawn from a gaussian distribution with mean m and standard deviation s. If m>0 interaction is on average cooperative, while if m<0 interaction is on average competitive.
The movie shows changes in the attractor of a size 30 system when the coupling strength is increased. Delay embedding is used to obtain a visual representation of the attractor at each coupling strength. The end of the movie is very close to the boundary where the map starts diverging. Measured Lyapunov exponents are shown at the top of each frame. Click on the screenshots to view the movie.
Click on the thumbnail to se a larger version. Coordinates for the plotted points were formed by taking the length of the state vector in three consequtive steps.
A continuous time version of the model. The plot is a phase diagram of the two dominant components. Click on the thumbnail to se a larger version.