Measures of Spatial Pattern.

`spatial pattern is a conspicuous characteristic of any ecosystem'- Garcia-Moliner et al. 1993

(a) The Clumping Index.

A vital aspect of spatial heterogeneity is pattern, which can be dened as a departure from randomness (Galiano, 1982 Addicott et al., 1987). The character of the distribution of species and habitats is of fundamental ecological and sociobiological importance. The geometric shape of the constituents of an ecosystem, may, for example, aect the response to disease or para- sites. For example a complex of small habitats will provide a physical barrier to the spread of an epidemic (Jetschke, 1992) and the destruction of ne-scale vegetational mosaics by re manage- ment programs has created large patches through which res spread rapidly (Minnich, 1983). Larger patches are, however, benecial under other circumstances: predators are able to remove aphid clusters if they can move over suciently large areas, whereas a patchy environment in- hibits predator movements and leads to pest outbreaks (Kareiva & Andersen, 1988). Thus the levels ofclumping or aggregation of species, resources or features in the physical environment are important for understanding the functioning of systems and the response of individuals. Many measures of aggregation have been developed in the biological and physical sciences (Leg- endre & Fortin, 1989). Biometricians have concentrated on statistical tests for distinguishing aggregated, random and regular patterns (Thomas, 1951 Diggle, 1977 Galiano, 1982 Perry, 1995). Many spatial models clearly produce clumped patterns, so that a statistical test to prove the presence of aggregation does not provide signicant new information. More useful is the dynamical approach to the variation in the level of clumping which is developed here for discrete spatial models, which is also highly relevant to other lattice-based data, such as satellite images.

A simple parameter, the clumping index is based on the traditional joint-count statistics of Moran and others (Moran, 1948 Krishna Iyer, 1949 1950 Ford & Diggle, 1981). The index Ci is dened for state i in a rectilinear grid by equation (4). nij is the number of interfaces

between a cell of type i and a cell of type j. Here the state variable i is discrete, such as a CA state. However, the state may be an aggregation of automata states (chapter 7), or a nite or innite set of CML states (chapter 4).

Ci = Pnii

j6=inij

(4) The signicance of the value of Ciis now discussed. If the distribution of state i is random with

density , then the probability of two neighbouring cells being in state i is 2. The probability

that a state i cell has a neighbour in a dierent state is 2(1;). Thus the clumping index for

a random distribution of density is:

C = ₂₍₁

;):

Arelative clumping index, CRican now be dened (equation (5)). This is greater or less than 1 according to whether the pattern is more or less clumped than a random distribution of density . If Ni is the number of cells of type i and N is the grid size, then the density is Ni

N. Thus

the clumping index for a distribution, relative to a random distribution of the same density, is given by equation (6). CRi = 2(1;)nii P j6=inij (5) CRi = 2(N2 ;Ni)nii NiP j6=inij (6) The standard and relative indices both have their uses. The clumping index can be plotted as a path through time as a function of density, with the random clumping curve, C, displayed on

the same graph for comparison. In this way, increases and decreases in the degree of clumping over time can be displayed. Alternatively, the relative clumping index can be plotted as a

function of time. The eect of dierent mechanisms on spatial structure can be observed using either of the indices the standard index plot allows density uctuations to be observed at the same time. A second use of the indices is in the identication of transience, that is the dynamical behaviour of a system before it has `settled down' (chapter 7).

Figure 3 shows the results of computing the clumping index Ci for random distributions of a

particular cell state on a coupled map lattice, at dierent densities between 0 and 1, tted to the curve C. These results verify the assumptions behind the derivation of the relative clumping

index C_Ri. Additionally, the method can be used to indicate the eectiveness of the random number generator used, as the clumping indices should lie as near as possible to the curve C.

Regular mm square clumps will have a clumping index of m ;1

2 and an n

m rectangular

clump will have index n(m;1)+m(n;1)

2(m+n) . Therefore an index of value C is equivalent to regular

square blocks of size (2C + 1)(2C + 1). This does not, however, indicate how much space

there is between the clumps and illustrates the importance of the relativeindex CR_.

(b) Multifractal Theory Applied to Lattice Models.

Application of the theory of fractals provides another approach to the spatial structure of a lattice model. In particular, the recently-developed theory of multifractals can be used to consider the scales present in a spatial conguration. Multifractal theory was introduced in the basic form of an uncountably innite number ofgeneralised fractal dimensionsby Hentschel & Procaccia (1983) and Grassberger (1983) and extended by Halsey et al. (1986) and Pawelzik & Schuster (1987). In recent years the theory has been developed further and applied to many areas of physics. Applications include percolation and random networks33, clustering processes

(Coniglio et al., 1987 Coniglio & Zannetti, 1989 Vicsek, 1990), wave phenomena (Shalev et al., 1992 Du & Ott, 1993 Grussbach & Schreiber, 1993), mechanics (Silberschmidt, 1993) and turbulence (Mandelbrot, 1974 Levi, 1986 Argoul et al., 1989).

Although there has been simple fractal analysis of vegetation and other spatial distributions in

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 density clumping index

Figure 3: The clumping index for generated random distributions of cells at a range of densities (red). The tted curve () is C.

ecology (Burrough, 1981 Morse et al., 1985), multifractals have yet to be applied in ecology. Multifractal analysis investigates distributions of measures on a geometric support and it is therefore easily applied to the square grid of a CML or CA. (While the nite nature of the lattice means that it is not fractal, it can be considered to be a `truncated fractal'.)

The standard technique of box counting (Liebovitch & Toth, 1989 Block et al., 1990 Meisel et al., 1992) is used to determine a spectrum of fractal dimensions (Jensen et al., 1991), which provides a complete characterisation of the structure of the distribution. The grid is partitioned into boxes of length , where is here dened as the length divided by the grid size. Hence 1. The average mass in a -box denes a measure . A method of box counting is used which is weighted according to the measure of the boxes. A d-measure, Md(q), is dened as

in equation (7), where the variables are as follows: D is the set of boxes of size in the grid

i is the measure on the box i which is of size Z(q) is a partition function with moment

of order q. Md(q) = X i 2D qid= Z(q)d (7)

A set of mass exponents f(q)gis dened such that Md(q) remains nite and non-zero as

!0 where Z(q)

;(q). Hence (q) is given by equation (8). By analogy with statistical

physics, (q) is sometimes referred to in the literature as a fractal pressure and may be denoted by P(q). (q) = lim !0 logZ(q) ;log (8) The Lipschitz-Holder exponent is dened such that the measure in a box scales with the - power of the box size (equation (9)). Then the function f() is dened such that the number, N(), of -boxeswhere the measure scales in this way, scales with (equation (10)).

N()

;f() (10)

A relationship can now be found between (q(q)) and (f()). A detailed proof may be found in, for example, Feder (1988) or Falconer (1990). The result can be quickly justied as follows. If () is the density of -boxes, then ()d is the number of boxes with measures scaling between _and +d. Then, substituting

i = , the following expression for the

d-measure is obtained:

Md(q) =

Z

f()qd()d

which can be approximated by:

Md

~

q(~q);f((~q))+d

where ~q maximises q(q);f(). This occurs where:

d

dq (q(q);f()) = (q)

which is when:

df

d((q)) = q:

By comparison with the previous expressions for Md(q) (equation (7)) and Z(q), the pair

of equations (11) and (12) are obtained.

(q) = ;

d(q)

dq (12)

Thus values for (q) can be found for specic ranges of q 2 Rusing equation (8) and the

denition of Z(q). Then the exponents, (q), can be found from (12) and the spectrum f() from (11). Heuristically, f() can be considered to be the fractal dimension of fractal subsets Sof the geometric support S. These subsets can be thought of as having the same densities or

measures (of order _{) and they cover the support (S =}S). Thus the size of the measure

determines the scaling of the corresponding boxes.

A distribution is multifractal when there is a range of values of and f(). This can easily be seen from a graph of the pressure (q) against q (chapter 4). The gradient of the graph for any value of q is minus the value of at that point. Hence a multifractal will have a nonlinear plot of (q), whereas it will be a straight line for a simple fractal. It is also generally observed that approaches a nite constant as q!1and as q!;1, which leads to a nite range for

and for f(). f() is expected to be a concave curve within the nite range of (chapter 4). The maximum of this spectrum corresponds to a sharp peak in the number of boxes in the fractal set S. In this way the scaling structure of a distribution on a grid can be analysed in

a `fully quantitative fashion' (Stanley & Meakin, 1988).