Random Lotka-Volterra maps

Micro controllers
Vector graphics
Joakim Linde and Mats G. Nordahl, Divergence and Chaos in Random Lotka-Volterra Maps, In preparation.

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:

x_i(t+1) = x_i(t) (1 + a_i + sum_j(b_ij x_j(t)) )

This equation is the finite difference equation analogue of the more common Lotka-Volterra differential equation given by

dx_i/dt = x_i(t) (a_i +  sum_j(b_ij x_j(t)))

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.