Download - BCB 322: Landscape Ecology
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.