2.4 System Dynamics
equilibrium point or cycle characteristic of simple
dynamical systems such as we discussed above.
Chaotic systems fluctuate in a complex and nonperiodic manner that can be described by fractal
geometry. The prevalence of chaos in real-world
ecosystems is still a matter of debate, although
some data strongly suggest a chaotic component in
the dynamics, for example, of insect population dynamics and childhood diseases (Schaffer and Kot,
1986a, b). Chaotic dynamics are apparent in certain nonlinear mathematical models of ecological
systems (e.g., May, 1974; May and Oster, 1976;
Gilpin, 1979; Hastings et al., 1993). These models
suggest limitations to our ability to predict ecosystem dynamics. Chaotic systems display limited predictability due to extreme sensitivity to initial conditions. In other words, small uncertainties in the
state of a system at time t (or small perturbations)
can become amplified over time such that predictions of the expected state at future times become
increasingly less accurate. On the other hand, although long-term prediction may be compromised,
nonlinear forecasting of shorter-term events might
well have applications in the assessment and management of ecological systems, although such applications have not been widely explored (Sugihara
and May, 1990; Casdagli and Eubank, 1992). The
major limitation is that these techniques generally
require large quantities of precise data, which are
difficult to obtain for most ecological variables.
2.4.3 Self-organized Criticality
A potentially valuable perspective is the emerging
description of ecological dynamics in terms of selforganized criticality, in which ecosystems are
poised at the edge of chaos, not displaying simple
equilibrial or periodic behavior, but not displaying
fully developed chaotic dynamics either (Bak et al.,
1988; Bak and Chen, 1991; Kauffman, 1993). The
systems are self-organized in the sense that the dynamics of the system itself drive it to the edge of
chaos. For example, a forest contains trees that
started to grow there (initial condition) and, over
time, a fuel load is building up, so fire may remove
the trees and create a clearing, from which the
process begins anew. A characteristic feature of
systems poised on the edge of chaos is fluctuations
that vary in magnitude, displaying a power law relationship (linear in a log-log plot) between the
magnitude of the fluctuation and its frequency of
occurrence. Such power law relationships are evident in nature, such as the frequency-magnitude relation for earthquakes or the frequency-discharge
relationship for floods (Malamud et al., 1996) and
forest fires (Malamud et al., 1998). Analysis of ex35
tinction dynamics in the introduced avifauna of the
Hawaiian islands provides evidence that ecological
dynamics may also show such power law scaling
(Keitt and Marquet, 1996). Certain models of evolutionary dynamics display self-organized critical behavior (Bak and Sneppen, 1993; Kauffman, 1993).
Edge-of-chaos systems are sometimes described as
transient dynamic systems (see Section 2.4.4) because they do not converge toward any attractor. If
self-organized criticality proves to be a common feature of ecosystems, power laws should find general
applicability in quantifying the frequency-magnitude
relationships of various events. This would allow better prediction of the expected frequency of extreme
events (floods, fires, droughts, etc.) and enhance our
understanding of their role in the long-term dynamics of ecosystems.
2.4.4 Transient Dynamics
Mathematically, equilibria, periodic cycles, and
chaos are all types of attractors that describe the ultimate behavior of the system. The concept of transient dynamics, on the other hand, focuses on the
behavior of the system before an attractor is
reached. If a forest is clear-cut, the transient dynamics are what occurs before the area settles into
being a functioning forest again. Transient dynamics may be important in the analysis of ecosystems,
because disturbances or changes in the constraints
(e.g., fire frequency, climate) may prevent the system from reaching the attractor on the time scale
of interest. Some systems may never converge to
any attractor. Such systems may be said to have
long or infinite transients, and they may display
more open-ended behavior, showing greater adaptability in the face of perturbations or environmental fluctuations. History plays a significant role in
these systems; their current response to a perturbation may be contingent on past events. Evolution
may be an appropriate metaphor for systems that
are constantly changing. Evolutionary innovation
results in continuous change of species and morphologies through time. Technological systems
may be another example--cultural use of resources
keeps the system in flux. For instance, rubber was
extracted from plant sources before World War II,
but the pressure on these resources shifted with the
development of synthetic rubber. People in tropical regions that depended on natural rubber for their
livelihood had to shift to extraction of other resources. Assessment and analysis techniques have
not yet been developed for long-transient dynamic
systems, but we should keep these concepts in our
consideration because they may offer a means to
understand the continuing processes of change that
equilibrium point or cycle characteristic of simple
dynamical systems such as we discussed above.
Chaotic systems fluctuate in a complex and nonperiodic manner that can be described by fractal
geometry. The prevalence of chaos in real-world
ecosystems is still a matter of debate, although
some data strongly suggest a chaotic component in
the dynamics, for example, of insect population dynamics and childhood diseases (Schaffer and Kot,
1986a, b). Chaotic dynamics are apparent in certain nonlinear mathematical models of ecological
systems (e.g., May, 1974; May and Oster, 1976;
Gilpin, 1979; Hastings et al., 1993). These models
suggest limitations to our ability to predict ecosystem dynamics. Chaotic systems display limited predictability due to extreme sensitivity to initial conditions. In other words, small uncertainties in the
state of a system at time t (or small perturbations)
can become amplified over time such that predictions of the expected state at future times become
increasingly less accurate. On the other hand, although long-term prediction may be compromised,
nonlinear forecasting of shorter-term events might
well have applications in the assessment and management of ecological systems, although such applications have not been widely explored (Sugihara
and May, 1990; Casdagli and Eubank, 1992). The
major limitation is that these techniques generally
require large quantities of precise data, which are
difficult to obtain for most ecological variables.
2.4.3 Self-organized Criticality
A potentially valuable perspective is the emerging
description of ecological dynamics in terms of selforganized criticality, in which ecosystems are
poised at the edge of chaos, not displaying simple
equilibrial or periodic behavior, but not displaying
fully developed chaotic dynamics either (Bak et al.,
1988; Bak and Chen, 1991; Kauffman, 1993). The
systems are self-organized in the sense that the dynamics of the system itself drive it to the edge of
chaos. For example, a forest contains trees that
started to grow there (initial condition) and, over
time, a fuel load is building up, so fire may remove
the trees and create a clearing, from which the
process begins anew. A characteristic feature of
systems poised on the edge of chaos is fluctuations
that vary in magnitude, displaying a power law relationship (linear in a log-log plot) between the
magnitude of the fluctuation and its frequency of
occurrence. Such power law relationships are evident in nature, such as the frequency-magnitude relation for earthquakes or the frequency-discharge
relationship for floods (Malamud et al., 1996) and
forest fires (Malamud et al., 1998). Analysis of ex35
tinction dynamics in the introduced avifauna of the
Hawaiian islands provides evidence that ecological
dynamics may also show such power law scaling
(Keitt and Marquet, 1996). Certain models of evolutionary dynamics display self-organized critical behavior (Bak and Sneppen, 1993; Kauffman, 1993).
Edge-of-chaos systems are sometimes described as
transient dynamic systems (see Section 2.4.4) because they do not converge toward any attractor. If
self-organized criticality proves to be a common feature of ecosystems, power laws should find general
applicability in quantifying the frequency-magnitude
relationships of various events. This would allow better prediction of the expected frequency of extreme
events (floods, fires, droughts, etc.) and enhance our
understanding of their role in the long-term dynamics of ecosystems.
2.4.4 Transient Dynamics
Mathematically, equilibria, periodic cycles, and
chaos are all types of attractors that describe the ultimate behavior of the system. The concept of transient dynamics, on the other hand, focuses on the
behavior of the system before an attractor is
reached. If a forest is clear-cut, the transient dynamics are what occurs before the area settles into
being a functioning forest again. Transient dynamics may be important in the analysis of ecosystems,
because disturbances or changes in the constraints
(e.g., fire frequency, climate) may prevent the system from reaching the attractor on the time scale
of interest. Some systems may never converge to
any attractor. Such systems may be said to have
long or infinite transients, and they may display
more open-ended behavior, showing greater adaptability in the face of perturbations or environmental fluctuations. History plays a significant role in
these systems; their current response to a perturbation may be contingent on past events. Evolution
may be an appropriate metaphor for systems that
are constantly changing. Evolutionary innovation
results in continuous change of species and morphologies through time. Technological systems
may be another example--cultural use of resources
keeps the system in flux. For instance, rubber was
extracted from plant sources before World War II,
but the pressure on these resources shifted with the
development of synthetic rubber. People in tropical regions that depended on natural rubber for their
livelihood had to shift to extraction of other resources. Assessment and analysis techniques have
not yet been developed for long-transient dynamic
systems, but we should keep these concepts in our
consideration because they may offer a means to
understand the continuing processes of change that