12. Individual-Based Models and Assessment of Population Variability
189
The temporal dynamics of the state variable models are described by differential, partial differential, or difference equations. Usually, the parameters of these
models are assumed to be constant. However, noise can be introduced into the
parameters, representing fluctuations in the environment, to produce stochastic
difference or differential equation models (May and MacArthur 1972; Turelli
1977). Analysis of such models is difficult for all but the simplest models. Because a principal advantage of the state variable approach is that models may be
analytically tractable, such stochastic state variable models are seldom used.
State variable models can be made spatially explicit to model populations in
spatially heterogeneous regions by dividing the region into a number of subregions, connected by dispersal or migration, with variables representing the
population sizes in the various subregions. This allows the model to incorporate
some of the effects on variation in the total population that may result from the
environmental conditions in different regions of space acting in phase or out of
phase.
Thus, state variable models have the capability of representing complex population situations in which demographic stochasticity is not too important. However, such models become less mathematically tractable as they are made larger to
deal with more complex situations. In small populations in which demographic
stochasticity is expected to be important, birth-and-death models can be used
(e.g., Pielou 1969), but these models are mathematically intractable in all but the
simplest cases, and most of the advantages of an analytic formulation are lost.
In addition to the difficulties imposed by demographic stochasticity when using
a state variable approach, we argue here that both the internal complexity of a
population and the way in which environmental changes influence population
dynamics often cannot be encompassed even by very complex state variable
models. For example, the variability of a small population may depend critically
on the detailed structure of the population through time, such as age and size
structure, sex ratio, physiological conditions of the organisms in the population,
genetic variability, and so forth. State variable models can take into account some
of these detailed characteristics, but this again adds to their complexity, increasing
the number of variables that must be followed.
Spatial heterogeneity further complicates the situation. In particular, resources,
predators, refuge areas, disturbances that alter habitats, and other factors affecting
populations, are often patchily distributed, and the relationships between the
populations and these patchily distributed factors may be too complex to be
represented by most conceivable spatially explicit state variable approaches. For
example, important effects on population dynamics, and therefore variability,
may result from (1) the details of the spatial relationships of patches (Cain 1991;
Fahrig 1991), (2) the spatial pattern of disturbances that create new open patches
available for colonization (Wu and Levin 1994; Moloney and Levin 1996), (3) the
size distribution of patches (e.g., Hyman at al. 1991; Turner and Gardner 1991),
and patterns of temporal changes in patches of resource availability (Fleming et al.
1994; Wolff 1994).
189
The temporal dynamics of the state variable models are described by differential, partial differential, or difference equations. Usually, the parameters of these
models are assumed to be constant. However, noise can be introduced into the
parameters, representing fluctuations in the environment, to produce stochastic
difference or differential equation models (May and MacArthur 1972; Turelli
1977). Analysis of such models is difficult for all but the simplest models. Because a principal advantage of the state variable approach is that models may be
analytically tractable, such stochastic state variable models are seldom used.
State variable models can be made spatially explicit to model populations in
spatially heterogeneous regions by dividing the region into a number of subregions, connected by dispersal or migration, with variables representing the
population sizes in the various subregions. This allows the model to incorporate
some of the effects on variation in the total population that may result from the
environmental conditions in different regions of space acting in phase or out of
phase.
Thus, state variable models have the capability of representing complex population situations in which demographic stochasticity is not too important. However, such models become less mathematically tractable as they are made larger to
deal with more complex situations. In small populations in which demographic
stochasticity is expected to be important, birth-and-death models can be used
(e.g., Pielou 1969), but these models are mathematically intractable in all but the
simplest cases, and most of the advantages of an analytic formulation are lost.
In addition to the difficulties imposed by demographic stochasticity when using
a state variable approach, we argue here that both the internal complexity of a
population and the way in which environmental changes influence population
dynamics often cannot be encompassed even by very complex state variable
models. For example, the variability of a small population may depend critically
on the detailed structure of the population through time, such as age and size
structure, sex ratio, physiological conditions of the organisms in the population,
genetic variability, and so forth. State variable models can take into account some
of these detailed characteristics, but this again adds to their complexity, increasing
the number of variables that must be followed.
Spatial heterogeneity further complicates the situation. In particular, resources,
predators, refuge areas, disturbances that alter habitats, and other factors affecting
populations, are often patchily distributed, and the relationships between the
populations and these patchily distributed factors may be too complex to be
represented by most conceivable spatially explicit state variable approaches. For
example, important effects on population dynamics, and therefore variability,
may result from (1) the details of the spatial relationships of patches (Cain 1991;
Fahrig 1991), (2) the spatial pattern of disturbances that create new open patches
available for colonization (Wu and Levin 1994; Moloney and Levin 1996), (3) the
size distribution of patches (e.g., Hyman at al. 1991; Turner and Gardner 1991),
and patterns of temporal changes in patches of resource availability (Fleming et al.
1994; Wolff 1994).