Pascual and Levin (1999) examined the time series of the densities of fish in boxes of various sizes and use ideas from dynamical systems to search for what they call the ``intermediate scale of nontrivial determinism." For the parameters simulated on the previous page, their length is 64.
Keeling et al (1997) have taken a simpler approach. Let SL be the number of fish in an L by L box in equilibrium. Letting var(SL) be the variance of SL; they look at v(L)=var(SL)/L2 and define (see page 1591 of Keeling et al (1997)) the coherence length Lc to be ``the point where v(L) asymptotes to a constant value." For the parameter values under consideration, their length is 125.
Our new approach is to let c(z) be the covariance of two sites separated by a displacement of z. Elementary formulas from statistics imply that var(SL) is a weighted sum of the c(z) Let the correlation length be the point where the sum over a box of side L first exceeds 75% of its limiting value. As the next graph shows the new correlation length is only 11.
Here we are not trying to argue that our new formula gives the ``right answer," just giving a definition that is simpler than Pascual and Levin's and more stable than asking about when an asymptotic limit is achieved. Indeed, statistical mechanics tells us that the role of the correlation length is to indicate the order of magnitude of the place where correlation lengths start to decay exponentially. The exact units, e.g., feet or meters, in which this is measured is not important.