The critical concept of scale “…the problem of relating phenomena across scales is the central problem in biology and in all of science”. Simon Levin,

The critical concept of scale

“…the problem of relating phenomena across scales is the central problem in biology and in all of science”. Simon Levin, 1992.

Scales over which humans observe the world

Why be concerned about scale?

• Characterization of biogeographic pattern involves:– Scale of the phenomena– Scale of its detection through observation– Scale at which a pattern is statistically analyzed

and communicated.

Cartographic scale

From Lewis Carroll’s Alice in Wonderland

Cartographic scale

1:250,000 1:50,000 1:2500

Representative fractions

Cartographic scale

• Assuming that you have two maps of the same paper size, which of the two representative fractions shows the most detail and the smaller surface area?

1:50,000 1:10,000

Large versus small cartographic scale

Large cartographic scale Small cartographic scale

Absolute and relative scaling

• Absolute scale– Distance is physical, measureable

• Relative scale– Distance is not a physical distance but a more

intangible construct, often similarity

Operational scale

• Refers to the spatial and temporal dimensions of an object or a process

• Operational scale may not be independent of our observations, and in fact may be very dependent upon them.

• Operational scale can be constrained by our technology and our senses.

Operational scale

And so these men of IndostanDisputed loud and long,Each in his own opinionExceeding stiff and strong,Though each was partly in the right,And all were in the wrong!

John Godfrey Saxe1878

Decreasing (finer) grain, extent is constant

Increasing extent, grain is held constant

• Extent: spatial (or temporal) dimension of an object or process observed or analyzed

• Grain: level of spatial (or temporal) resolution at which an object or process has been measured or observed.

Ecological scaling

Which map shows more grain?

How much grain is shown can be a subjective decision. Cartographic scale and ecological scale are often intertwined.

Haggett’s scale coverage problem

• Nature has an immense extent and a fine grain

Haggett’s scale coverage problem

• To make even small descriptions of it, we have to sample

• Sampling requires sacrificing grain for extent or extent for grain.

Fallacies of scale

• Individualistic fallacy: extrapolating to the broad scale based on observations conducted at small, local scales

• Ecological fallacy: making local-scale characterizations based on broad-scale observations.

Modifiable areal unit problem

• Two components – Aggregation problem– Zoning problem

Hierarchy theory

• Landscapes organized into spatial and temporal domains of shaping processes

• Scale hierarchy is a system of vertical interconnections

• Higher levels constrain the lower levels

Hierarchy theory

Non-hierarchical controls also relevant

Examples: Tilt, insolation, climate, photosynthesis: hierarchical

Invasive species, dispersal, natural selection: non- hierarchical

Cartesian scale• Hierarchy theory

exemplifies a Cartesian scaling of the world

• Scales are imposed• Not necessarily “true”

scales• Often tied to XYZ

coordinate system• Space as a container

Constructivist scale

• Scale emerges out the interactions of entities

• Scales are produced or constructed

• There is no fixed and unchanging hierarchy of scales.

Constructivist scale• Boundaries in space and

frequencies of environmental exposures shaped by organisms not just our observation of them.

Evolutionary scaling

• Scaling is a practice related to adaptation

– Scale and scaling are practiced by all forms of life• Are our observational

scales relevant to the organism of interest?

• How do other organisms scale the world and how does it influence what they (and we) can know and act upon?

Rules for ecological scale for humans

1. Patterns are dependent upon the scale of observation

2. The important explanatory variables change with scale.

3. Statistical relationships may change as scale changes.

4. Patterns are generated by processes acting over various temporal and spatial scales.

Rules for ecological scale

5. Scale can be used to justify or refute certain management practices and ideas about nature

Example: Successional response to clearcut logging

• Grain and extent of post-logging sampling determine criteria for judging response– Large extent, fine grain: criteria for

recovery not likely to be met – Small extent, coarse grain: criteria for

recovery easier to meet

Rules for ecological scale 6. The scales experienced by an organism define what

it sees and responds to. For example, what might constitute a patchy resource to an insect, could be perceived by a larger vertebrate as homogeneous.

How to work with scale

• There is no single correct scale or level at which to describe a system.

• This does not mean that all scales serve equally well or that there are not rules or guidelines.

• Be aware of the different types of scaling and their relatedness

• Don’t be too anthropocentric • Employ sampling designs and methods that

are sensitive to multiple scales– Nested observations– Power laws– Fractals– Networks

How to work with scale

Power laws• Summarize how relationships

change with changes in scale• Often expressed on a log-log plot. • Y = constant (X)n

• Similar slopes are thought to have similar structuring processes (n = slope)

• Examples• Species-area relationships (left)

• Animal metabolic rates and body mass (next slide)

– However, a valid criticism of power laws is that they often lack an explanatory process

Metabolic rate and body mass

Fractals• A fractal pattern appears the

same across all scales. It is scale invariant.

• The relationship between size of box and pattern in it is constant.

• Fractals follow their own power law relating how number of boxes needed to cover a shape change in relation to their size.

Networks

• Can represent relationships at a variety of scales at once.

• Structural properties of networks provide means of understanding how they work.– Nodes and links– Degree centrality and

betweenness– Weak versus strong links– Directional versus non-

directional graphs

Random and scale free networks

Spatial autocorrelation

• A method to summarize how patterns change with scale