BCB 322:Landscape Ecology

Lecture 2: Theories & ModelsHierarchy theory, diffusion theory &

percolation theory

Introduction• Landscape heterogeneity, complexity of

the ecosystem components, resource restraints & population behaviour all affect organisms in a landscape.

• The interaction of these components is estimated through several models and theories.

• Most of these theories evolved in different contexts

• All aim to interpret landscape complexity (systems & structures), and together

Principal theories• We shall be looking at four theories in greater

detail:– Island biogeography theory– Hierarchy theory– Diffusion theory– Percolation theory

• We’ll also consider two models:– Metapopulation model– Source-sink systems model

• Between them these models cover a lot of the conceptual ground of landscape ecology

• There is considerable variation in the details of some of the models

Hierarchy theory• Landscapes are intrinsically complex, with

variation in resources at all scales• Hierarchy theory attempts to explain how

scale-specific components of the landscape are in contact with components visible at other resolutions

• HT considers that any system is a component of other systems at a larger scale, and is itself comprised of sub-systems.

• eg: Landscape classification, with the micro-, meso-, macro- and megachores each comprising combinations of the finer-scale classifications

Hierarchy theory• (eg): River watersheds• River basin comprises sub-

basins, each comprised of smaller basins

• Similarly, different mammals are associated with stream order.

• River basins (geological), stream order (physical) and animal size (biological) all interact.

• Clearly, landscapes are very complex systems Harris, 1984 (reprinted

in Farina, 1998)

Hierarchy theory• To understand complex systems, one needs

to focus on organizational level. • This means choosing a relevant

spatiotemporal scale to study the system (hence the components of the system)

• The horizontal structure of a hierarchical system comprises subsystems or holons

• Each holon is an aggregate of lower-level holons, and is part of a higher one

• The borders of a holon may be easily visible (the edge of a forest) or invisible (the edge of a frog’s distribution)

Hierarchy theory: holon borders• eg: the structure of a grassland (higher holon)

depends on the processes of grazing and woodland encroachment acting on local scales

• Finer-scale holons tend to have a faster behaviour rate than larger ones (grazing behaviour at a low level,

• Outputs from one level to another are aggregates of the component processes

• Consequently, holon levels and borders effectively act as filters for behaviour. Boundaries exist where there is a discontinuity in the rate of change of variables.

• This is the basis of the hierarchical understanding of systems

Hierarchy theory: signal filters• Burning in a field occurs rarely

(low frequency) and changes the structure of a grassland (visible at higher levels of organisation)

• Browsing of woodland margins by migratory animals occurs more frequently (annually), with reduced effect on the matrix (visible to a lesser extent at higher levels)

• Localised seed-gathering by mice in a savannah frequently causes local shortages of seed (high frequency), but from a higher level the effect may be seen as constant O’Neill et al, 1986 (reprinted in

Farina, 1998)

Hierarchy theory: Incorporation• Incorporation is the process by which

perturbation is absorbed by a level of the system

• Low frequency fires in a savannah tend to increase soil fertility, reduce woodland encroachment & provide high-quality fodder

• Consequently, they increase biodiversity & complexity

• Frequent (human-induced) fires can destroy the seed bank & reduce biodiversity, when the system can no longer incorporate the event

• The system becomes less complex, turning from a woodland-grassland matrix to simple grassland, then to arid semi-desert.

Diffusion theory• This theory describes

the movement of organisms through a landscape.

• Describes plants & animals, although they obviously operate on different timescales

• The principle is based on the diffusion of particles in a liquid.

NDNftN 2)(

• N=population size• f(N) = population growth

function• D = diffusion coefficient

(describes spatial movement rate)

• = diffusion operator (describes the rate of change of N with distance – the density gradient)

2

Eqn (1)

Turner et al, 2001

Diffusion theory• When invading a uniform landscape, the rate of spread

(V) will reach asymptotes equal to

where r is the intrinsic growth rate & D is the diffusion coefficient

• Equation tested by Andow et al (1990), and was found to work well for – Invasion of muskrats in Europe– Invasion of cabbage white butterfly in North America

• However, in the case of the cereal leaf beetle, movement patterns were considered on a finer scale, and it appeared that D was underestimated.

rDV 4

uniform landscape

Eqn (2)

Percolation theory• Real landscapes are only uniform when

considering very broad scales• At finer scales, percolation theory describes

organismal movement through the matrix.• Differs from diffusion theory in that it considers

the connectedness of the landscape• Also considers movement to be similar to a

that of a fluid• Below a critical threshold (pc), distribution is

patchy & separated into discrete regions• Above the threshold, movement through the

region is free• Experiments corroborate theory that the

percolation critical threshold (pc) is <0.5928

Percolation theory• The number & size of lattices are

related to P (probability of a cell being occupied by the target species)

• P = 0.4 (no percolation)– 49 clusters– Largest cluster = 18 cells

• P = 0.6 (some percolation)– 17 clusters– Largest cluster = 163 cells

• P = 0.8 (fully percolated)– 1 clusters– Largest cluster = 320 cells

• From this we can calculate landscape boundaries (total & inner edges) – useful for edge effect assessment in conservation.Gardner et al, 1992

Percolation theory: uses• The occupancy can signify any resource, and we can

thus estimate the likelihood of many events– resinous shrubs/trees: forest fires– carrier animals: disease spread– susceptible plants: pest outbreaks

• Also useful for resource usage studies in animals• If a landscape has a percolation value over to pc

(0.5928), it can move throughout the landscape to find resources

• Chance of finding no resources in n landscape units is

where P is the random distribution of the resource

nP)1( Eqn (3)

Farina, 1998

Percolation theory: uses• Therefore, the probability R or finding at least one

resource is

• We know that if R=0.5928 the animal can move through the landscape to find resources

• Substituting this into equation 4 gives us the relationship between n and P :

• This then tells us far the animal needs to travel to obtain sufficient resources.

nPR )1(1

)1ln(/89845.0 Pn

Eqn (4)

Eqn (5)

Percolation theory: resource use• Hence, when resources are well distributed (P<=pc), the

organism doesn’t have to move very far • Decreasing resource density will require an organism to

look further afield• When there are two or more available resources, n is

calculated using their combined potential• If a dominant organism consumes 90% of a resource,

the subdominant species has much lower resource availability, and must consequently search more land units

• The likelihood of finding subdominant species in a given land unit is hence much smaller than for dominant species, even in relation to their densities (sample is insufficient) (O’Neill et al, 1988)

• Furthermore, fragmented landscapes will reduce the viability of subdominant species first.

Summary• Hierarchy theory: all systems and processes in a

landscape are components of higher-level systems• Incorporation: the extent to which perturbation can

be absorbed by a system• Diffusion theory: in a homogeneous landscape,

population dispersion is related to the population growth rate and the rate at which it can move

• Percolation theory: in a fragmented landscape, movement rate is related to the integrity of the landscape. Over a critical threshold (pc = 0.5928) organisms can move freely through the landscape.

• Resource-gathering (& consequently home range) is related to resource density and landscape integrity

References• Andow, D.A., Karieva, P.M., Levin, S.A. & Okubo, A. (1990) Spread

of invading organisms. Landscape Ecology 4:177-188.• Farina, A. (1998) Principles and Methods in Landscape Ecology.

Chapman & Hall, London.• Harris, L.D. (1984) The fragmented forest. Island biograpgraphy

theory and the preservation of biotic diversity. University of Chicago Press, Chicago.

• Gardner, R.H., Turner, M.G., Dale, V.H. & O’Neill, R.V. (1992) A percolation model of ecological flows. In: Hansen, A.J. & di Castri, F. (eds.), Landscape boundaries. Consequences for biotic diversity and ecological flows. Springer-Verlag, New York, pp. 259-269.

• O’Neill, R.V., DeAngelis, D.L. Waide, J.B. & Allen, T.F.H. (1986) a hierarchical concept of ecosystems. Princeton University Press, Princeton, New Jersey.

• O’Neill, R.V., Milne, B.T., Turner, M.G. & Garnder, R.I.I. (1988) Resource utilization and landscape pattern. Landscape Ecology 2:63-69.

• Turner, M.G., Gardner, R.H. & O’Neill, R.V. (2001) Landscape Ecology in Theory and Practice: Pattern and Process. Springer-Verlag, New York 401pp.