Some Comments on Spatial and
Temporal Scale in Ecological
Models
Don DeAngelis, Mark Grismer, and Laurel
Saito
Acknowledgments
Examples used here are due to
• Lou Gross and his group (U. of Tennessee)
• Wolf Mooij (Netherlands Institute of Ecology)
• and others.
Books on the Importance of Scale in Ecological
Systems
Giller, P. S., A. G. Hildrew, and D. G. Raffaelli (eds.) Aquatic Ecology: Scale, Pattern, and Process.
Blackwell Scientific 1994
Harris, G. P., Phytoplankton Ecology: Structure, Function, and Fluctuation. Chapman and
Hall 1986
Karlson, R. H. Dynamics of Coral Communities. Kluwer Academic 1999
MacCall, A. D. Dynamic Geography of Marine Fish Populations. Washington Sea Grant Program 1990
Mullin, M. M. Webs and Scales. Washington Sea Grant Program 1993
Okubo, A. Diffusion and Ecological Problems: Mathematical Models. Springer Verlag 1980
Seuront, L., and P. G. Strutton. Handbook of Scaling Methods in Aquatic Ecology. CRC Press 2004
Steele, J. H. (ed.) Spatial Pattern in Plankton Communities. Plenum Press 1978
What Ecological Factors Determine Natural
Spatial Scales?
We can differentiate spatial extent and spatial
resolution.
Spatial extent largely depends on the amount of area
needed to encompass the ecological system (e.g.,
population or community) under consideration.
This is the
approximate range of
the snail kite in Florida
Reality
Range of
Florida
panther
population
Extent of Geographic Range of Cape
Sable Seaside Sparrow
A much smaller area is needed to
model the entire population of Cape
Sable seaside sparrows.
Even smaller areas encompass major
subpopulations (outlined in red),
which may be considered as
individual ‘systems’ over short time
periods.
What Ecological Factors Determine Natural
Spatial Scales?
Spatial resolution necessary to accurately describe a
process is determined by the scale over which some
process or pattern in space varies according to some
criterion (Levin and Pacala 1997). So we choose a
scale of resolution small enough to capture that
variation.
What Ecological Factors Determine Natural
Spatial Scales?
Some of the main causes of such variation in a pattern
or process that must be considered in modeling are:
• Scale of variation in the abiotic conditions, including abiotc
disturbances
• Scale of variation in underlying biotic conditions
• Individual organism size
• Scale of ecological patterns emerging from model interactions
Variation in Abiotic Conditions
Example:
We are attempting to model the spatial dynamics of
numbers and biomass of small fishes across the
entire Everglades.
It is necessary to choose an appropriate spatial scale,
or spatial ‘cell’ size to represent the elevation
gradient across the Everglades, which will determine
the spatio-temporal pattern of flooding and drying.
The Modeled Functional Group
The small fishes (Gambusia, mollies, killifishes, etc.) constitute
one of the most important components of the Everglades
system. This 'functional group' provides much of the prey-base
for wading birds and other higher trophic level species.
Sailfin molly
(Poecilia latipinna)
Least killifish
(Heterandria formosa)
Variation in Abiotic Conditions
How fine do we need to go in spatial scale?
The South Florida Water Management Model (SFWMM)
produces output of daily water depths at a 2 x 2 mile
scale. However, it is possible, through algorithms
incorporating additional knowledge of vegetation, etc.
(see atlss.org website), to refine this to a finer scale.
Tree Island : Skinner’s Camp
Variation in Abiotic Conditions
How fine do we need to go in spatial scale?
A resolution of 500 x 500 m may capture much of the
biologically important change in mean elevation (and
thus water depth) across large-scale gradients of the
Everglades landscape. (We have confidence in
refinement to this scale, as we can use USGS
elevation data collected roughly at this scale.)
Slough to prairie transition
Variation in water depths
Wet
0
Dry
SFWMM
HRH
Variation in Abiotic Conditions
However, the 500 x 500 m resolution fails to capture
essential fine-scale heterogeneity within the cell.
This heterogeneity is in the form of alligator ponds,
solution holes, etc., that may form dry-season
refuges for fish, aquatic macroinvertebrates
(crayfish), etc.
Unit Cell Layout
Pond areas assumed permanently wet, marsh
areas periodically dry
Variation in Abiotic Conditions
How do we deal with that heterogeneity?
It is impractical to try to model explicitly at a finer spatial
scale of resolution. But is it really necessary to do
so?
Probably not. The internal pattern can be dealt with in a
spatially implicit manner, by quantifying the fractions
of cell area that are alligator holes, solution holes,
etc., and that therefore can serve as dry season
refuges. Their explicit locations are unimportant.
Variation in Abiotic Conditions
So we can ‘scale down’ by including finer scale
information in a spatially implicit way. How do we
‘scale up’ the dynamics of the fish population to the
whole landscape?
We take into account processes of fish movement
among cells.
Landscape Layout and Movement
White - movement from low water to high water
areas
Red - movement from high fish density to low
density areas
Variation in Abiotic Conditions
Then the model for fish biomass and number dynamics
can produce spatially explicit output on a 500 x 500
m scale across the landscape, on 5-day time steps
over 31-year simulations (in this case comparing two
scenarios).
Average Fish density on
February 15 from 1965 - 1995
Variation in Abiotic Conditions
From this output, combined with other factors, such as
the water depths that wading birds can utilize, we can
estimate the habitat suitability (for reproduction) for
long-legged and short-legged wading bird colonies
across the Everglades, or for specific sub-regions for
given years.
SESI Output for Long-Legged Wading Birds in N. Taylor Slough: For 1993
In computing this, we introduced a further spatial scale, the wading birds’
foraging area. Wading Bird Nesting Colony Suitability is determined by
foraging suitability of many 500 x 500 m pixels surrounding the colony
The Wading Bird
Foraging Index for a
given pixel (potential site
of a nesting colony) is
determined by the
‘collective foraging
suitability’ of the 500-m
pixels in the ‘core’ area
surrounding the colony
pixel.
The core foraging radius
for long-legged and
short-legged wading
birds, ForagRadius, is
currently 1.5 and 3.0 km,
respectively.
Black denotes Wading Bird colony
Variation in Abiotic Conditions
A scale of resolution of 500 x 500 m seems to be
reasonable for describing the response of many
Everglades biota to abiotic conditions, finer scale is
used for the American crocodile, which responds to
mangrove estuaries and short scale variations in
depth.
Fig 1
Open water
Mangrove
Dwarf Mangrove
Marsh
Nest sites
R1
Fringe
Road
Small
Landscape
Non-habitat
R2
R3
R4
R5
Variation in Underlying Biotic Conditions
Biotic conditions, such as vegetation
type (denotes by different colors in map),
may also affect the dynamics of a
population. In this case vegetation type
is known to 30 x 30 m resolution, and so
can be used explicitly in the model, even
though water depth continues to be
modeled at a resolution of 500 x 500.
We have used this additional scale of
resolution for vegetation in some specific
cases, though not in general. So more
than one spatial scale of variation can be
used, depending on the availability of
data at various scales of resolution and
the importance of including it.
Effect of Individual Body Size on Needed Scale of
Resolution
• In modeling competition of sessile benthic organisms
(e.g. Robles and Descharnais, Ecology 2002) and
terrestrial plants, organism size may dictate the scale
of resolution.
• This may apply as well to animal territories or home
ranges
Effect of Emergent Patterns on Needed Scale of
Resolution
Depending on the type of model, some spatial features
of certain scales may emerge as a result of the
interactions in a model. Two examples are:
• Emergence of tree islands from models that combine
hydrology, erosion/deposition, tree colonization, nutrient
accumulation, to form tree islands.
• Competition between species along an ecological gradient.
Tree Islands in the Everglades Landscape
Our model for tree island formation has a spatial scale of 10 x10 m
Emergent Spatial Structure of Fish Community in
the Everglades
We are now attempting to model the spatial structure of
communities of small fishes across elevation gradients. Fishes
differ in their life history traits (fecundity/survival and movement
ability are possible trade-offs). This suggests that there is some
partitioning of space, at least on a temporary basis.
An abstract model of fish competition in a spatio-temporally varying
one-dimensional landscape attempts to capture that (DeAngelis,
Trexler, and Loftus, CJFAS 2005)
It is essential to choose a spatial resolution (200 m in this case)
that allows emergence of patter in space.
Elevation and Hydroperiod along Spatial Gradient
CHARACTERISTICS OF TRANSECT
ELEVATION AND HYDROPERIOD
2.5
2
ELEVATION: ARBITRARY
UNITS
1.5
1
HYDROPERIOD: FRACTION OF YEAR
FLOODED
0.5
0
1
3
5
7
9
11
13
15
17
19
SPATIAL CELL NUMBER
21
23
25
27
29
31
Hydrologic Driving Function in ODFISH
HYDROPERIODS OF SPATIAL CELLS
DEPTH OF WATER IN METERS
2
1.5
1
0.5
0
0
50
100
150
200
-0.5
-1
-1.5
-2
TIME IN DAYS
250
300
350
400
Biomass Distributions of Hypothetical Small Fish
Species Populations on a Given Day Across an
Elevation Gradient
BIOMASSES (G/M2)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1
6
11
16
21
26
31
SPATIAL CELL NUMBER
Spatial gradient represented by 31 cells, each 200 m in length
Temporal Scales of Variation
Temporal resolution in the model must be chosen
on a fine enough scale to capture short time scale
events that are important to population dynamics.
In particular, rapid changes in water depth must
be represented.
Local drying events
can happen within a
few weeks, forcing snail
kites to move.
Wetland Dry
Sudden Increases in Water Depth Can Interrupt
Breeding Cycle of Cape Sable Seaside Sparrow
Increase in water
depth to 15 cm will
cause any nesting
to cease. It can
commence if water
depths decrease.
The figure shows 2
possible nesting
cyles of 45 days.
USGS
30
DEPTH OF WATER IN CELL
Nesting can start in a
cell as soon as
water depths
decrease to 5 cm.
Nesting can start
25
72 days
20
1 cycle
45 days
15
1 cycle
45 days
10
5
0
1
15 29 43 57 71 85 99 113 127 141 155 169
DAY OF REPRODUCTIVE SEASON
Water Depth May Rapidly Become Unfavorable for Wading
Birds
60
WATER DEPTH IN CELL
At any given time
during the breeding
season, only pixels
that are in certain
water depth ranges
are usable and
contribute to the
foraging suitability
of the core area of
the colony.
WaterDepth
50
40
30
Range 5 to 35 cm
20
10
Long-legged wading
birds:
MINDEPTH to
MAXDEPTH:
Currently 5 to 35 cm
DAY OF REPRODUCTIVE SEASON
Short-legged wading birds:
MINDEPTH to MAXDEPTH:
Currently 1 to 20 cm
148
141
134
127
120
113
106
99
92
85
78
71
64
57
50
43
36
29
22
15
8
1
0
Temporal Scales of Variation
Other important processes operate on much longer
time scales. This includes vegetation succession
processes, which are modeled in a rule-based
model.
e
Years Since
Last
Hydroperiod
Disturbance
Herbaceous/Forested Ve getation Succession
3
Dry Prairie [29, 39]
30–60
3
Dry Prairie [29]
30–60
Graminoid Dry
Prairie EC [39]
20–50
3
{.53} Mesic–Hydric
Forest [16]
{.22} Slash Pine
Woodland [25]
{.16} Gallberry/
Palmetto CG [30]
30–60
{.08} Ham mock [2]
10–45
{0.996} Muhly
Grass Marsh[45]
{0.004} Sparsely
Wooded Wet
Prairie CG [52]
60–120
{0.996} Muhly
Grass Marsh[45]
{0.004} Sparsely
Wooded Wet
Prairie CG [52]
60–120
{0.996} Muhly
Grass Marsh[45]
{0.004} Sparsely
Wooded Wet
Prairie CG [52]
60–120
{.25} Graminoid Marsh CG [42] 120–270
{.7} Cladium [43] 130–330
{.03} Eleocharis [44] 150–300
{.03} Typha [46] 180–280
Forb
Emergent
Marsh [56]
230–360
{.25} Graminoid Marsh CG [42] 120–270
{.7} Cladium [43] 130–330
{.03} Eleocharis [44] 150–300
{.03} Typha [46] 180–280
Forb
Emergent
Marsh [56]
230–360
{.25} Graminoid Marsh CG [42] 120–270
{.7} Cladium [43] 130–330
{.03} Eleocharis [44] 150–300
{.03} Typha [46] 180–280
Forb
Emergent
Marsh [56]
230–360
{.01} Live Oak [5]
0–60
Decid. Shrub [37] 110–320
10
[16, 25, 30]
30–60
Trop. Hammock [2]
10–45
Live Oak [5] 0–60
0
[16, 25, 30]
30–60
10
[2] 10–45
[5] 0–60
{.91} Swamp Forest CG [17] 120–290
{.09} Brd Lvd/Mixed Evergreen Shrub
[28] 120–150
Forb Emergent Marsh [56] 230–360
Float ing Leaved
Veg. [57] 330–360
{.77} Decid/Trop. Swamp Forest [3]
60–180
{.23} Brd Lvd/Mixed Evergreen Shrub
[28] 120–150
Decid. Shrub [37] 110–320
Swamp Forest CG [17] 120–290
Float ing Leaved Veg. [57] 330–360
0
[16, 25, 30]
30–60
[2] 10–45
20
Decid/Trop. Swamp Forest [3]
60–180
Decid. Shrub [37] 110–320
[5] 0–60
0
40
{.55} Trop.
Hammock [2] 10–45
{.04} Live Oak [5]
0–60
{.4} Xeric–Mesic
Live Oak
EC [4] 0–30
Float ing
Leaved
Veg. [57]
330–360
Swamp Forest CG [17] 120–290
{.35} Bay/Gum/Cypress EC [6]
60–160
{.65} Swamp Forest [3] 60–180
Float ing
Leaved
Veg. [57]
330–360
Decid. Shrub [37] 110–320
Swamp Forest CG [17] 120–290
Cypress Forest CG [18] 200–340
Decid/Trop. Swamp Forest [3]
60–180
50
0
30
60
90
120
No Data
150
180
210
240
270
300
330
Conclusions
The empirical study suggests that management models of fishes
must incorporate rapid recovery from drought events. Some
species may experience greater population growth rates in
short-hydroperiod habitats, although ultimate population sizes
may be limited by periodic dry down, site productivity, and other
factors.
Exploratory models, such as ODFISH, may help understand the
factors underlying rapid recovery and help improve
management models, such as ALFISH.