## Hierarchical Levels

Entities in a hierarchy are recognized as belonging to levels. Levels are sets, but the sets become levels because of robust asymmetry between them in a hierarchy. Mathematically, the asymmetry between levels makes hierarchies partially ordered sets. Hierarchy theory is the set theory that may precede network theory (see Ecological Network Analysis, Environ Analysis). A level of analysis assigns entities to levels, and is often explicit about their relationship to other entities assigned to other levels. There is a distinction between levels ofobservation and levels of organization. Levels of observation are ordered relative to each other on matters of size and scale. Meanwhile relationships between levels of organization follow from definitions chosen by observers, sometimes as a prelude to actual observation. For instance, organisms are subsumed by populations only by a definition. Hidden in the definition is a requirement for equivalence between population members. Meanwhile, a host and its parasite, while both organisms, are generally not assigned to the same population, in part because of inequivalent size. Host versus parasite is the basis of a hierarchy employing levels subtly different from those in the population/organism distinction. Hierarchy theory places entities in levels, taking care to be explicit about the definitions that lead to those levels, and the criteria that create order and linkage.

Scale versus definition has potential for generating different sorts of hierarchies. Some hierarchies focus on size and containment, while others are control hierarchies where upper level entities simply control lower levels. A Watt governor may be placed at a higher level in a control hierarchy, while being smaller than the whole steam engine it controls. Whether it is a scalar or a control hierarchy depends on the use for which the hierarchical conception is intended, something for which the observer must take responsibility. Time against space plots are popular in landscape ecology (see Landscape Ecology). But such hierarchies can miss out on the interesting situations where large space maps onto short time spans, or long time spans map onto small places (Figure 2). The globe is large enough to be the context of continental movement over hundreds of millions of years, but at the same time the rotation of the globe is also responsible for diurnal phenomena, at the fast end of ecological happenings. Surfaces arise when narrow space applies to large differences in time constants (strong temporal connection within, but weak connections across surfaces). Ecotones would be a case in point, because there is rapid exchange and fast process inside the abutting ecosystems or communities areas, while the exchanges across the narrow ecotone may be remarkably slow. Thus ecotones are spatially small, while representing slow exchanges that might cause the ecotone to move in a process of gradual encroachment. Conversely, in a communication channel, small differences in time constants apply along the long connection. Corridors would be an example here, where there is rapid movement along the extended length of the corridor. In ecology, these special places, such as ecotones and corridors, are at least as interesting as situations where time and space widen in concert. Complexity in hierarchies arises from the challenge of mapping between levels, as scale and definition entwine.

Fast Slow

Figure 2 A common graph appearing in landscape ecology plots increases in time against space, focusing on the quasilinear pattern of larger things seen as behaving over longer time periods. But such plots ignore potential control systems and their hierarchies. Local intransigence can control large entities (Wallace's realms where Australia's fauna is isolated from Asia by the narrow Torres Strait, separating millions of years of evolution). Barriers and surfaces occur at the lower right, while communication channels and corridors appear upper left.

### Fast Slow

Figure 2 A common graph appearing in landscape ecology plots increases in time against space, focusing on the quasilinear pattern of larger things seen as behaving over longer time periods. But such plots ignore potential control systems and their hierarchies. Local intransigence can control large entities (Wallace's realms where Australia's fauna is isolated from Asia by the narrow Torres Strait, separating millions of years of evolution). Barriers and surfaces occur at the lower right, while communication channels and corridors appear upper left.

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