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Individual-based modeling
Spatially-explicit population models
Over the last 10
years or so, we are witnessing a conceptual revolution in the field of
population and community biology, and a paradigm shift in ecological modeling.
These changes were stimulated by recent increases in computational capabilities
and the subsequent rapid development of individual-based models or IBM's and
spatially explicit population models or SEPMs.
These types of simulation models allow for taking a first principle
approach in which natural history and field observations are used to deduce the
causal relationships among components of the natural systems and the resulting
system dynamics. Individual-based models assign the ecological objects of the
model (e.g., the individuals) a set of rules, which can be fairly literal
translations of their behavior found under field conditions. These rules are not
arbitrary functions but should be based on natural history knowledge or are
hypothesis on to test with the model.
The causal relationships among objects and the resulting dynamics emerge as a
consequence of individuals acting according to behavioral rules, but the causal
relationships are not assumed a priory as necessary in more traditional top-down
modeling approaches.
I use
individual-based models because they allow the efficient and lucid integration
of life history information into demographic models, and they consider important
processes like demographic stochasticity in a natural way (because the unit of
the model, the individual, is also the biological unit). In addition, for
empiricists, individual-based models are intuitive in a way that matrices and
differential equations are not. The fact that the unit of the model, the
individual, is also the biological unit makes individual-based models especially
suitable for
pattern-oriented modeling.
Typically, large
volumes of data are generated via simulation experiments that modify the
behavioral rules, and the management or climatic scenarios applied. This allows
to make deductions about the causal relationships among ecological objects,
spatial and temporal dynamics, scales, and mechanisms in the field system.
Because of the potentially high realism of simulation models, these deductions
can be tested under field conditions.
The initial
enthusiasm about the IBMs and SESMs, however, has been dampened by critical
voices outlining the data
requirements, problems associated with parameter estimation and possible
magnifications of parameter errors, which may make the development of such
models an onerous process (Wiegand
et al.
2004). IBMs and
SEPMs are able to include many biological details and they may contain many
parameters. This can be a problem, because of not only error propagation or
lacking direct estimates of model parameters,
but also because the inherent complexity may prevent an exhaustive model
analysis. Thus, while the new bottom-up approach is tempting in theory, there
are mayor obstacles in practice. To overcome them I am
working on the development of constructive approaches
and new modeling techniques that (1) allow
using the available empirical data more efficiently to face problems
associated with parameter uncertainty and error propagation, (2) facilitate
model calibration and evaluation to assure that the models indeed capture
relevant aspects of reality, and (3) provide guidance in analyzing the complex
model dynamic. For details see
Wiegand et al. 1999,
2003, 2004
a,
2004
b,
Grimm et
al. 2005.
Spatial pattern analysis
Over the last decade, there has been increasing interest
in the study of spatial patterns in ecology. Ecologists study spatial pattern to
infer the existence of underlying processes and identify the scale at which
those processes are operating. One approach for inferring underlying processes
from spatial patterns is to characterize spatial point patterns using statistics
such as Ripley’s K or the pair-correlation function and contrast the observed
patterns to patterns produced by null models representing specific hypotheses
about the underlying processes. The use of point pattern analysis in ecology has
been increasing exponentially over the last decade and half.
Although statisticians have developed increasingly advanced statistical methods
for point pattern analysis relevant for ecologists, ecological applications of
spatial point pattern analysis generally remain far below their potential. Although
appropriate statistical methods have been available for decades, appropriate
software that can deal with “real world” data sets and is flexible enough to
accommodate the appropriate null models for hypothesis testing does not exist (but
see Baddeley and Turner 2005). For example, many study areas have irregular
boundaries and contain variable environmental conditions influencing the
distribution of study organisms, yet available software packages can, in general,
only be applied to relatively “tame” situations, e.g., rectangular study areas
containing spatially homogeneous patterns. This prohibits the inclusive study of
more complex situations. Most available software also assumes that ecological
objects can be approximated as points, suppressing information about the size
and shape of the objects being studied, and offers only a very limited number of
null models (such as total spatial randomness, independence, or random labelling).
Based on many years of teaching experience
and
collaborative research in the
field of ecological point-pattern analysis, the software
Programita (Wiegand
and Moloney 2004) was developed to accommodate the needs of “real world”
applications in ecology. The software grew in response to requests of colleagues
and students who approached the authors with their specific research problems
rather than from a motivation to include all existing methods (as e.g. the
statistical package by Baddeley and Turner 2005).
Programita has been designed to provide a more general analysis of point
patterns, allowing for irregular study boundaries, analysis of inhomogeneous
point patterns, consideration of size and shape of ecological objects and
incorporation of a broad range of null models for describing spatial structures
of point patterns and hypothesis testing.
Ripley’s K-function and the O-ring
statistic
The function K(r)
is the expected number of points in a circle of radius r centered at an
arbitrary point (which is not counted), divided by the intensity l of the
pattern. The alternative pair correlation function g(r), which
arises if the circles of Ripley’s K-function are replaced by rings, gives
the expected number of points at distance r from an arbitrary point,
divided by the intensity of the pattern. Of special interest is to determine
whether a pattern is random, clumped, or regular. Significance is usually
evaluated by comparing the observed data with Monte Carlo envelopes from the
analysis of multiple simulations of a null model. The common null model is
complete spatial randomness (CSR), but other null models may be appropriate
depending on the biological question asked. Points in a point-pattern may
contain information in addition to position, often referred to as marks (e.g.,
a species identifier, a life stage identifier, or whether the individual
survived or died), and many biological questions concern the relationship
between points with different marks (e.g., facilitation or competition among
adult trees and seedlings, or shrubs and grasses). Bivariate extensions of
Ripley’s K and the pair-correlation function provide appropriate methods
to address such questions. However, the analysis of bivariate point patterns is
more complicated than that of univariate patterns because various other null
models in addition to CSR become possible. The appropriate null model for
bivariate analysis must be selected carefully based on the biological hypothesis
to be tested.
Using rings
instead of circles (Fig. 1) has the advantage that one can isolate specific
distance classes, whereas the cumulative K-function confounds effects at
larger distances with effects at shorter distances. Note that the K-function
and the O-ring statistic respond to slightly different biological
questions. The accumulative K-function can detect aggregation or
dispersion up to a given distance r and is therefore appropriate if the
process in question (e.g., the negative effect of competition) may work only up
to a certain distance, whereas the O-ring statistic can detect
aggregation or dispersion at a given distance r. The O-ring
statistic has the additional advantage that it is a probability density function
(or a conditioned probability spectrum) with the interpretation of a
neighborhood density, which is more intuitive than an accumulative measure.
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Fig. 1.
Numerical implementation of the K- function and
the O-ring statistic for an irregularly shaped study region encircled
by the dashed line. Points of pattern 2 are represented by closed circles,
the focal point i of pattern 1 as open circle. Note that we approximate
circles and rings with the underlying grid structure. (A) For numerical
implementation of Ripley’s bivariate K-function we count the number
of points of pattern 2 inside the part of the circles around point i of
pattern 1 which falls inside the study region (i.e., the gray shaded area),
and the number of cells within this area. (B) For implementation of the
bivariate O-function we count the number of points of pattern 2
inside the part of the ring around point i of pattern 1 which falls inside
the study region (i.e., the gray shaded area), and the number of cells
within this area.
Pattern-oriented modeling
Only recently, pattern-oriented modeling has been proposed as framework for
exploiting the available data at all steps of the bottom-up modeling process:
from the initial model construction to parameter estimation and to detection of
deficiencies in the model structure and knowledge (Grimm 1994;
Grimm et al. 1996;
Wiegand et al. 2003,
2004, Grimm and Railsback
2005;
Grimm et al. 2005).
Grimm et al. (1996) defined a “pattern” as a characteristic, clearly
identifiable structure in nature itself or in the data extracted from nature. A
pattern is anything that goes beyond random variation and thus indicates an
underlying process that generates this pattern. For example, in semiarid
grasslands such patterns could be fence line effects, data on phytomass
production, basal cover, and species composition, or size class distributions of
individual tufts under different grazing regimes. All these patterns are the
outcome of the interplay between demographic processes, competitive interactions
of individual grass tufts, and constraining factors of climate and management.
In general, such patterns represent high-level manifestations of population
dynamic processes and are the outcome of interplay between demographic
processes, dispersal characteristics and various constraining factors (e.g.,
management actions, or a climatic pattern). Therefore, empirically observed
patterns contain a great deal of information and memory about the history of the
system.
The
information that is hidden in the observed patterns can be used in various ways
(Wiegand et al. 2003,
2004). First,
focusing on observed patterns guides the construction of a structurally
realistic model that matches the predominant scales of the real system to
facilitates a direct comparison of model relations with observations, and which
contains a ”correct” representation of the basic processes and interactions that
determine the dynamics of the system on the given scales. A structurally
realistic model should only be as complex as is required to reproduce the
patterns and to fulfill the objectives. Second, comparison of observed and
predicted patterns, in the context of systematically varying the processes that
may be involved in the formation of the pattern, can help to provide a better
understanding of the functioning of the system and to detect gaps in the current
knowledge and deficits of the model structure. Third, indirect parameter
estimation through comparison of observed and predicted patterns can be used to
determine the values of unknown or uncertain model parameters with higher
precision that would be possible otherwise. Indirect parameter estimation also
avoids error propagation. Finally, in structurally realistic models one can
compare internal model relations (secondary predictions) with empirical findings
that adds further possibilities for model validation and for evaluating
the model performance.
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Pattern-oriented modeling. The hypotheses on parameters and processes are
constrained by the observed patterns, and comparison between the observed
patterns and the patterns produced by the model restricts the range of uncertain
parameters and can be used for detecting the underlying processes that produce
the observed patterns. The observed patterns are compared with patterns produced
by several alternative hypotheses on processes and/or a large number of
plausible parameter combinations. Because the high-level output of the model (the
patterns), which contains relationships between the model parameters, must match
the observed pattern, error propagation does not occur. A helpful analysis of
the model can be obtained by analyzing internal model relations (secondary
predictions), and the model can be validated by comparing secondary predictions
with independent field data.
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Pattern-oriented model selection and analysis
Because empirically observed patterns are high-level
manifestations of population dynamic processes and are the outcome of interplay
between demographic processes, dispersal characteristics and various
constraining factors (e.g., management actions, or a climatic pattern), they
contain a great deal of information and memory about the history of the system.
However, because the patterns describe features of the system at a higher
hierarchical level than are addressed by possible model rules (e.g.,
individual-based rules vs. population level data), the data of the pattern
cannot be included directly into the rule set of a model and are consequently
lost. Especially under the circumstances typical for conservation biology, where
management problems force hasty decisions to be taken in spite of scarce data,
we cannot afford such losses.
The pattern-oriented modeling strategy facilitates the inclusion of pattern data
at different hierarchical levels. In the process of pattern-oriented modeling,
one first constructs model rules on a low hierarchical level using data on the
behavior of the individuals (or model objects) as found under field conditions.
In a second step, one defines a set of patterns, which are basically the data on
hierarchical levels higher than the individual scale. In a third step one has to
implement the specific external driving forces and initial conditions (=
constraining factors) for the system dynamic under which the pattern were
created. For semiarid grasslands such constrains may be the long-term rainfall
pattern, a fence line with different grazing management at the two sides of the
fence, or an initial compositional state.
Once the model is constructed and the external driving forces and initial
conditions are implemented, the model is applied for a high number of model
parameterizations under selected alternative model process formulations.
Systematic comparison of the observed pattern with the patterns produced by the
model will filter implausible model parameterizations (e.g.,
Wiegand et al. 1998,
2004a,
2004b) or
model structures (Wiegand et al. 2003;
Kramer-Schadt 2004). Thus,
the data on higher hierarchical levels are not directly used for model
construction, but for selection of model parameterizations and processes. In
this way, the data from higher hierarchical levels enter the model via an
indirect process of model calibration. Clearly, preconditions of this approach
are that (1) the model is structurally realistic [i.e. a model that contains key
structural elements of the real system and produces predictions and internal
relations that can be compared to the pattern data (Wiegand
et al.
2003)],
(2) the model contains a "correct" representation of the basic processes and
interactions that determine the dynamics of the system on the given scales, (3)
the model includes mechanisms which are potentially able to drive the patterns,
(4) the patterns are "indicative" or "genuine" [i.e., they are able to constrain
the model dynamics and are not fulfilled for all model parameterizations and
process variants (Wiegand et al.
2003)].
Point (1) can be archived by choosing a model structure that in principle allows
reproduction of the observed patterns (Wiegand
et al. 2003, Grimm et al. 1996,
Grimm et al. 2005). Points (2) and (3) are not a problem
because a weak fulfillment of patterns will indicate deficiencies in the model
structure. However, even a "negative" result that a given set of mechanisms is
not able to produce one or more observed pattern can be valuable in stimulating
specific field investigations and subsequent modeling to identify the missing
mechanisms. Using multiple patterns can counteract point (4). While it might be
relatively simple to reproduce one feature of a system, the simultaneous
fulfillment of several patterns describing different features of the system is
by far non-trivial (Wiegand et al. 2003,
2004,
Grimm et al. 2005).
Related publications
DeAngelis
DL, Mooij WM (2003) In praise of mechanistically rich models. In: Models in
Ecosystem Science, C. D. Canham, J. J. Cole, and W. K. Lauenroth (eds).
Princeton University Press.
Grimm V
(1994) Mathematical models and understanding in ecology. Ecological
Modelling 75/76:641-51
Grimm V,
Frank K, Jeltsch F, Brandl R, Uchmanski J, Wissel C (1996) Pattern-oriented
modelling in population ecology. The Science of the Total
Environment 183: 151-166
Grimm V (2001) Den Wald vor
lauter Bäumen sehen: Musterorientiertes ökologisches Modellieren. In: Jopp F,
Weigmann G. (eds.): Rolle und Bedeutung von Modellen für den ökologischen
Erkenntnisprozeß. Theorie in der Ökologie, Band 4. Peter Lang Verlag,
Frankfurt/M., pp. 43-57
Grimm V,
Berger U (2002). Seeing the wood for the trees and vice versa:
pattern-oriented ecological modelling. In: Seuront L, Strutton PG (eds)
Scales in aquatic systems: measurement, analysis, simulation
Grimm, V., E. Revilla, U.
Berger, F. Jeltsch, W. Mooij, S. F. Railsback, H. Thulke, J. Weiner, T.
Wiegand, and D. L. DeAngelis. 2005 Pattern-oriented modeling of agent-based
complex Systems: lessons from ecology, Science 311:987-991.
Grimm V.,
and S. F.
Railsback (2005). Individual-based modeling and ecology. Princeton University
Press, Princeton, N.J.
Railsback SF
(2001) Getting "results": the pattern-oriented approach to analyzing natural
systems with individual-based models. Natural Resource Modeling 14:
465-474
Railsback
SF, Harvey BC (2002) Analysis of habitat selection rules using an
individual-based model. Ecology 83: 1817-1830.
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