25. Ecosystem Modeling
of system behavior. The basic concept of detennining how a system changes in response to changes
in the parameters is intuitively appealing. It mimics
the sort of experimental manipulations that one
would like to be able to perform to better understand how a system works.
Stability Analysis
The fundamental concern in analyzing the stability
of a system is the long-term response of the system
to an external change or a perturbation. Clearly, this
is a topic of interest in considering the response of
ecosystems to changes in the environment. There
are two rather different mathematical traditions and
associated methods for detennining stability of dynamic systems. The first is appropriate only for linear systems and originates from some of the greats
of physics and mathematics (Newton, Euler, Laplace, Lagrange, Maxwell). The second applies to
linear and nonlinear systems and was the doctoral
dissertation of a Russian mathematician, A.M. Lyapunov (Timothy and Bona 1968).
Linear stability analyses is involved with the response of the system to perturbations or changes in
inputs. In linear systems, a system is stable if its
response to an impulse function approaches zero as
time approaches infinity (DiSephano et al. 1967).
There are a number of methods that can be used to
detennine the stability of linear systems. Linear
compartment models in which the material being
transferred from one compartment to another are
conserved and are stable because the structure of
the matrix of parameters for the models.
One method that has been applied in several ecosystems to detennine stability of systems is the procedure known as "loop analysis" (Levins 1974;
Yodzis 1989). Loop analysis is derived from criteria
for detennining the stability of linear systems called
the Routh-Hurwitz criteria (Timothy and Bona
1968). The appealing aspect ofloop analysis is that
it allows one to detennine from the structure of interactions whether a linear system will be unstable.
This finding is independent of the particular parameter values (other than their signs) that are involved.
One cannot use the analysis to prove that a system
is stable (which can depend on the magnitudes as
well as the signs of the parameters). Applications
of loop analysis to nonlinear systems are usually
couched on the assumption that the system behaves
383
in a linear manner, at least under the conditions
being considered.
The second approach to the problem of systems
stability was that of A.M. Lyapunov. In this case,
stability concepts could be applied to nonlinear (as
well as linear) systems. Lyapunov recognized two
sorts of stability. The first sort of stability in a Lyapunov sense concerned whether or not the systems
would return to the vicinity its former state following a small perturbation. In a system that is stable,
any (small) perturbation on a system within E of the
state of the system at some time (to) produces a
response from the system (xr(t» that is bounded
within a distance 0 of the unperturbed response
(x(t» of the system. In a system that is asymptotically stable, the difference between the perturbed
system dynamic and the unperturbed system dynamic goes to zero after a sufficiently long time.
Future Directions:
Multiple Commodity Models
and Individual-Based Models
In a field as creative as ecosystem modeling, it is
sometimes difficult to predict future directions of
development. Two areas that seem to be developing
both theoretical and practical innovations are both
products of increased computational power. These
are complex models simulating multiple commodities (light, water, nutrients) and models that simulate ecosystem structural change by simulating
each individual of the dominant organisms in the
community.
Multiple Commodity Models
Multiple commodity models are a logical outgrowth of compartment models described above.
The models used in ecosystem studies often have
similar internal assumptions that are a part of the
rationale for scaling the models to larger space and
longer time scales. These assumptions are also a
basis for the organization of available data on ecosystem performance into dynamic models. Important among these assumptions are:
1. Generality: The concept that ecosystems function in accordance to some overarching rules
that control structure and/or function. Examples
of system behavior. The basic concept of detennining how a system changes in response to changes
in the parameters is intuitively appealing. It mimics
the sort of experimental manipulations that one
would like to be able to perform to better understand how a system works.
Stability Analysis
The fundamental concern in analyzing the stability
of a system is the long-term response of the system
to an external change or a perturbation. Clearly, this
is a topic of interest in considering the response of
ecosystems to changes in the environment. There
are two rather different mathematical traditions and
associated methods for detennining stability of dynamic systems. The first is appropriate only for linear systems and originates from some of the greats
of physics and mathematics (Newton, Euler, Laplace, Lagrange, Maxwell). The second applies to
linear and nonlinear systems and was the doctoral
dissertation of a Russian mathematician, A.M. Lyapunov (Timothy and Bona 1968).
Linear stability analyses is involved with the response of the system to perturbations or changes in
inputs. In linear systems, a system is stable if its
response to an impulse function approaches zero as
time approaches infinity (DiSephano et al. 1967).
There are a number of methods that can be used to
detennine the stability of linear systems. Linear
compartment models in which the material being
transferred from one compartment to another are
conserved and are stable because the structure of
the matrix of parameters for the models.
One method that has been applied in several ecosystems to detennine stability of systems is the procedure known as "loop analysis" (Levins 1974;
Yodzis 1989). Loop analysis is derived from criteria
for detennining the stability of linear systems called
the Routh-Hurwitz criteria (Timothy and Bona
1968). The appealing aspect ofloop analysis is that
it allows one to detennine from the structure of interactions whether a linear system will be unstable.
This finding is independent of the particular parameter values (other than their signs) that are involved.
One cannot use the analysis to prove that a system
is stable (which can depend on the magnitudes as
well as the signs of the parameters). Applications
of loop analysis to nonlinear systems are usually
couched on the assumption that the system behaves
383
in a linear manner, at least under the conditions
being considered.
The second approach to the problem of systems
stability was that of A.M. Lyapunov. In this case,
stability concepts could be applied to nonlinear (as
well as linear) systems. Lyapunov recognized two
sorts of stability. The first sort of stability in a Lyapunov sense concerned whether or not the systems
would return to the vicinity its former state following a small perturbation. In a system that is stable,
any (small) perturbation on a system within E of the
state of the system at some time (to) produces a
response from the system (xr(t» that is bounded
within a distance 0 of the unperturbed response
(x(t» of the system. In a system that is asymptotically stable, the difference between the perturbed
system dynamic and the unperturbed system dynamic goes to zero after a sufficiently long time.
Future Directions:
Multiple Commodity Models
and Individual-Based Models
In a field as creative as ecosystem modeling, it is
sometimes difficult to predict future directions of
development. Two areas that seem to be developing
both theoretical and practical innovations are both
products of increased computational power. These
are complex models simulating multiple commodities (light, water, nutrients) and models that simulate ecosystem structural change by simulating
each individual of the dominant organisms in the
community.
Multiple Commodity Models
Multiple commodity models are a logical outgrowth of compartment models described above.
The models used in ecosystem studies often have
similar internal assumptions that are a part of the
rationale for scaling the models to larger space and
longer time scales. These assumptions are also a
basis for the organization of available data on ecosystem performance into dynamic models. Important among these assumptions are:
1. Generality: The concept that ecosystems function in accordance to some overarching rules
that control structure and/or function. Examples