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across scales. As Chave and Levin (2004, p. 32) note, scaling relationships - between
metabolic rate and body size, for example, or between area and species richness -
'are among the most robust empirical generalisations found' in ecological systems
- but they are not linear. They cite fi nancial crashes and traffi c jams as 'typical of
the dynamics found in complex adaptive systems': as the component parts interact
and adapt, positive feedback loops can trigger abrupt, extreme, unpredictable
change. Economies of scale are another example: how the division of labour and
the expansion of production result in non-linear increases in output and qualita-
tively new social phenomena can only be understood relationally.
Scale as relation requires a strong conceptual distinction from level. It is, so to
speak, an order removed from scale as level, defi ned by the spatial and temporal
relations among (processes at different) levels. To address scale as relation, then,
one must eschew the conventional synonymy of scale and level.
Among ecologists, scale as relation is part of a larger critique of equilibrium
models and assumptions. In a famous 1977 article, Robert May presented mathe-
matical models of systems with 'a multiplicity of stable states', inspired by empirical
cases of grazing ecosystems, fi sheries, insect outbreaks in forests, and host-parasite
systems. He likened ecosystem dynamics to a marble in a cup. If the cup formed 'a
single valley', then the system would always return to a single stable state following
disturbance, and historical effects would be unimportant. But if the cup were a
'dynamical landscape pockmarked with many different valleys, separated by hills
and watersheds', then 'the state into which the system settles depends on the initial
conditions: the system may return to this state following small perturbations, but
large disturbances are likely to carry it into some new region of the dynamical
landscape'. Scale is thus not only a spatial issue but also a temporal one. Any equi-
librium presupposes some period of time over which stability persists; it might turn
out to be unstable if evaluated at a different temporal scale. Moreover, 'if there are
many alternative locally stable states, historical accidents can be of overriding sig-
nifi cance' (May, 1977, p. 471).
Understood in this way, scale is central to current notions of sustainability and
resilience in complex adaptive systems involving humans and the environment. Once
one admits the possibility of multiple stable states, one cannot avoid the issue of
thresholds or 'breakpoints' between them. May (1977, p. 477) emphasised that
'continuous variation in a control variable can produce discontinuous effects' and
that 'increasingly severe nonlinearities can make the dynamical behaviour range
from a stable point, through a bifurcating hierarchy of stable cycles, into a regime
which is in many ways indistinguishable from random noise'. In the three decades
since, ecologists have struggled to model complex systems and quantify thresholds
of non-linear change. Predictive knowledge of thresholds has remained elusive, but
theory and conceptual models have advanced considerably and empirical observa-
tions are accumulating (Crumley, 1994; Westoby et al., 1989). There is also growing
interest in the hypothesis that unsustainable resource use results from 'mismatches
of scale' between human and natural processes (Lee, 1993; Cumming et al., 2006).
Determining the relevant processes involved, and their operational scales, thus
becomes a necessary prerequisite for advancing both research and management (for
an example involving fi sheries, see Perry and Ommer, 2003).
While ecologists turn to ever more sophisticated mathematics and models to
understand scale as relation, human geographers explore the matter through meta-
phors and theory. Howitt (1998) examines musical scales, pointing out that the
