Dirección de la comunicación

`Man masters nature not by force but by understanding' - Jacob Bronowski 1956

(a) Introduction.

Ecological processes in a nite region take place on many dierent spatial scales (section 1.1). The construction of discrete explicitly spatial models brings the issues of scaling to the forefront, as various scales are necessarily imposed on the system by the implementation of the model. Three imposed length scales arise from a discrete lattice. The smallest of these is thecell size, which may be related to the size, or area of direct inuence of a sessile organism, or the space covered by a motile individual in a small time interval. Secondly, theneighbourhood size, usually dened in terms of the cell size, is the range over which biological interactions occur this often consists of four or eight cells. The largest imposed scale is thesystem length, N, so that the total number of cells is N2.

In addition to the imposed scales, there is an emergent length scale, at which the dynamics aris- ing from the system progress. This is usually approximated by the classicalcorrelation length, which is the separation distance at which any two sites are uncorrelated. Several techniques exist for determining correlation lengths, but these tend to be dicult and denite results are

28Mead, 1967 Diggle, 1976 Auld & Coote, 1980 Karlson & Jackson, 1981 Weiner & Conte, 1981 Gates,

1982 Hobbs & Hobbs, 1987 Green, 1989 Moloney et al., 1992 Pacala & Tilman, 1994.

29Hassell et al., 1991 Sole & Valls, 1991 Comins et al., 1992 Sole & Valls, 1992a Sole et al., 1992a 1992c

Sole & Bascompte, 1993 Hassell et al., 1994 Keeling, 1995.

30Karlson & Jackson, 1981 Hassell et al., 1991 Sole et al., 1992a Sole & Bascompte, 1993 Bascompte &

Sole, 1994 Hassell et al., 1994.

31Othmer & Scriven, 1971 1974 Kaneko, 1986 1989 Hassell et al., 1991 Sole & Valls, 1991 Moloney et al.,

not always forthcoming. Here a new length called thecoherence length scale, nc, is introduced.

Within regions (or subgrids) of length signicantly below nc, the states of individual cells are

strongly correlated. Disjoint regions of a size much larger than nc are statistically independent.

It is usually desirable to perform computations using grids of around the coherence length for two reasons. Firstly, if the system is smaller than nc, strong coupling obscures the true

dynamics, particularly in a toroidal system if a larger system is used, any dynamics will be averaged out and so more dicult to detect. Secondly very large grids will also inevitably be accompanied by excessive computational times. The length scale problem has been addressed by a few authors32 but a widely applicable and robust technique has yet to be developed.

A method is presented here for determining the coherence length scale of a lattice-based model, using the analysis of errors arising from locally-coupled interactions.

(b) Mathematical Theory.

The identication of the coherence length scale, nc, can be approached by analysing the errors

arising from spatial aspects of a model. It is assumed here that the model is such that the dynamics tend towards a statistically stationary distribution. This means that on any subgrid of size n there is a time independent probability () that the conguration will occur, after time t0 when transients have passed. The state xi() of cell i can be mapped by an observable

function F to a real number F(xi()). For any conguration the spatial average of the

observable on the subgrid, Fn(), is calculated at time t as:

Fn((t)) = 1n 2 X cellsi F(xi((t))):

Assuming statistical stationarity, the long term time averagehFi is given by: 32Wiens, 1989 DeRoos et al. 1991 Wissel 1991 Levin 1992 Rand & Wilson 1995.

hFi = X Fn()() = lim T!1 1 T t0 +T X t0+1 Fn((t)):

The innite limit must, of course, be approximated, but a value of several thousand is usually sucient for T. If an insucient number of iterations is used then the error function, dened below, will not tend to zero as n tends to innity. The error function, En, is dened in terms

of the uctuations of FnabouthFi(equation (1)).

E 2 n = 1T t 0+T X t=t 0 +1 ; Fn((t));hFi 2 (1) It is postulated thatEnsatises thecentral limit theorem(CLT) for large n. The CLT (Rosanov,

1977) is a fundamental theorem of probability theory and may be expressed as follows. Given ` random variables i (i = 12`) with nite means and variances, the sum:

S` = X`

i=1

i:

may be normalised to give the sum in equation (2), whereE() and var() are respectively the

mean and variance of random variable . S

` = S`;E(S`) p

var(S`) (2)

Then the random variables satisfy the CLT if: lim

`!1

S

` N(01)

that is, S

` approaches a standard normal distribution with mean 0 and variance 1 as ` becomes

Sn2 = n2 X i=1 F(xi) n2 :

The meanE(Sn2) of the sum Sn2 may be approximated byhFi . Thus:

E 2 n = D (Sn2;E(Sn2)) 2 E which from equation (2) gives:

E 2 n ' D var(Sn2) S n2 2 E :

Two simple lemmas give that var(a) = a2var() for constant a and var( + ) = var() +

var() for independent random variables and . From these results: var(Sn2) = var 0 @ n2 X i=1 F(xi) n2 1 A = 1n4var 0 @ n2 X i=1 F(xi) 1 A:

If the observables,F(xi), are taken to be identical independent random variables with variance 2, then: var(Sn2) = 1n 4 n2 X i=1 var(F(xi)) = 1n 4n 2 2 1 n2:

This arrives at the conclusion that:

E 2 n ' 2 n2S n2 2 and hence:

En

1

nN(01):

Given that the expected deviation from 0 of a N(01) normal distribution is 1, the error varies as the inverse of the length scale, n (the size of the subgrid). This result can only be derived from the CLT by assuming n (and hence N) is large. However, the proof of the CLT depends onStirling's formula:

z!

p

2zzze;z as z !1:

Through the examination of the higher order terms of Stirling's series for the factorial function (Arfken, 1970) the degree of inaccuracy for smaller z may be estimated (equation (3)). The derivation of the variation ofEnuses the CLT in terms of n

2. Putting z = n2 in equation (3)

shows that the inaccuracy is well below 1% for a subgrid as small as 55. Hence the error can

be assumed to vary as 1

n at all but the very smallest scales. This theoretical error for random

variables with given variance may now be denoted byE 0 n. z! = p 2zzz_e;z 1 + 1 12z +288z1 2 + (3) The error,E 0

n, applies to random variables with a distribution given by an innite lattice. On

small subgrids (0 < n < nc), the absolute error En will be smaller or larger than E 0

n, but at

large scales (n > nc) the error will approach E 0

n asymptotically. Thus, by tting a kn curve to

En (where k is a constant) for nN, the coherence length scale nc can be estimated as the

value of n whereEn rst meets the tted curve.

If En > E 0

n for n < nc there is a greater error that that expected for the distribution on

the innite lattice, so there is aggregation of mass or individuals - positive spatial coherence. If En < E

0

n there is less aggregation this situation is termed negative spatial coherence here.

as fast with n as doesE 0 n, that is: 0 > @En @n > @E 0 n @n = ;1

then there is aaggregatingtendency, as mass or individuals accumulate in clumps. In contrast, if the interactions are such that mass or similar individuals tend to move apart (disaggregation), then: ;1 = @ E 0 n @n > @En @n :

Thus positive or negative coherence may be identied at scales below nc, as well as aggregating

or disaggregating tendencies as the scale of observation is varied. The error may alternatively be represented by a plot of nEn against n (gures 23, 34 and 55). In these cases aggregating

and disaggregating tendencies are represented respectively by: @

@n(nEn) > 0

and

@

@n(nEn) < 0

and the tted line will simply be a constant k. Detailed proofs of these results may be found in Keeling et al. (1995).