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Continuous and Discontinuous Change
in a Southwestern Woodland
Donald A. Jameson 1
Abstract - The traditional view of ecosystems stressed by man's activities
is that a given stress will result in a given plant community equilibrium, i.e.,
there is a single equilibrium state for any stress regime, regardless of the
previous state. This is the essence of a single steady state system, which
allows for only "smooth" changes. Much has appeared in some ecological
literature lately about alternative models that allow discontinuous changes.
The simplest such model is a fold catastrophe. In this model, the response
to some extreme value of a control variable is a single stable steady state.
The response at the opposite extreme has another stable steady state. In
between there is a zone where either state can be considered a stable
steady state (along with some unstable steady states); the particular
equilibrium state depends on the originating extreme state. The next most
complex multiple steady state model is the cusp catastrophe. Whereas the
fold model requires a "jump", i.e., an outside influence, to move from one
steady state to another, the cusp model also allows a "smooth return" along
one control axis that does not require an outside influence. The
Southwestern pinyon-juniper woodland has many examples that appear to
fit this last model - with both discontinuous and continuous change
introduced by combinations of Climate, fire, grazing and wood harvesting.
The cusp model also may alleviate concerns of those who are unwilling to
depart from the earlier paradigm that time and succession will cure all ills.
The relationship of the appropriate equilibrium states to climatic stress and
activities of mankind are a fruitful area of study.
SINGLE VERSUS MULTIPLE
EQUILIBRIUM STATES
appropriate, random fluctuations may alter the current state but
the system will return to the predetermined state for any given
stress level.
The migration to a stress-free steady state, once the stress is
removed, may take a longer time than can be allowed for study.
Alternative study techniques may involve a space-for-time
exchange, i.e., using ecological relationships displayed in space
as a surrogate for ecological relationships that change in time.
In fact, it may not be discemable whether a phenomena is
occurring in space or in time. "When a moving ecological
community reaches an obselVer" might be the same question as
"when a moving obselVer reaches the ecological community" .
Although there are dangers in making assumptions about
ecological patterns when the underlying processes are not
obselVable or controllable (Cale et al. 1989), it is certainly
tempting to do so when the time required to change exceeds the
life of the obselVer.
A traditional view of ecology is that a given stress to an
ecological community will result in a given community
equilibrium. In range management, for example, it has been
assumed that there is a single equilibrium state (called range
condition) for any grazing prescription, regardless of whether
the previous condition was higher or lower. This is the essence
of the concept of a single steady state system. When any outside
stress is removed, the system migrntes to a single stress-free
condition, which may be known by some term such as "potential
natural vegetation". If the single steady state concept is
1 Donald A. Jameson is Professor Emeritus, Colorado State
University, and USDA Forest Service, retired. He currently resides
at Sedona, AZ.
137
ELEMENTARY CATASTROPHE THEORY
In contrast to the traditional view of single steady states,
multiple steady state concepts allow more than one state of the
system to result from a given input or stress (Zeeman 1976).
The resultant steady state for any stress may be determined by
a previous state, rather than being independent of any previous
state as in the single steady state concept. A classic example in
range management is the California annual grasslandlStipa
pulchra. Range managers have been forced to recognize that the
Stipa pulchra communities of pre-European settlement will not
be replaced through natural succession, and that the potential
steady state, even if grazing is eliminated, is now an annual
community dominated by introduced Mediterranean annuals.
However, this has been commonly taught in range management
classes as perhaps the single ~ception to a single steady state
concept of range condition
Recently there has been considerable attention in the literature
(reviewed by Laycock 1991) of many interpretations of multiple
steady states observed in nattIml vegetation systems, but explicit
system models that are subjectJo control and statistical analyses
are rare. However, Jameson (1991) reported experimental results
that contained at least a partial test of a hypothesis that a cooland wann-season grass community possessed multiple steady
state properties.
Occam's razor teaches that the simplest usable model should
be used to explain observed results, and the simplest model of
multiple steady state systems is the fold catastrophe (Zeeman
1976). A drawing of a fold is simple, it is only the proof that
it is the simplest multiple steady state model that is difficult.
Even simplistic graphics programs can be induced to draw an
appropriate fold. In the fold model, the response to some extreme
value of a control variable has a single stable steady state. The
response at the opposite extreme has another stable steady state.
In between there is a zone where either state can be considered
a stable steady state (along with some unstable steady states).
If we should sample for some measurement variable along
the control axis (such as percent of vegetation made up of
pre-European species), we would expect to fInd a low variance
at the two extremes of the fold and a high variance in the middle
zone, resulting in a cloud of observations in the multiple
equilibrium region of the system In fact, the high variance might
obscure the fact that there are steady states of any kind, and we
might assume that we have chaos.
In the annual grassland example, the percentage of vegetation
made up of pre-European species is a simple index of
community composition. In other ecosystems, other indices may
be more appropriate. Some nominees for approaches to the index
question include species richness indices, patch connectivity,
diversity indices, fractal dimensions, phase transition parameters,
discontinuity detection algorithms, edge detection algorithms,
changes in spatial autocorrelations, and others.
An important consideration of the study of such systems is
their equilibria, i.e., to what states does the system migrate
because of its own properties.
Some extremely helpful aids in addressing the concept of
equilibria are found in the area of catastrophe theory within the
definition:
Elementary catastrophe theory is the study of how the
equilibria of a dynamic system changes as the control
parameters change.
We will now review various catastrophe manifolds. These
manifolds are system response surfaces that show response of
state variables x resulting from application of various controls
u. Although the ~stem dynamics as it returns to equilibrium
has a time dimension, the time trace of the system is not
explicitly shown However, if the system is near the equilibria
and the changes in controls are small, the trace of system
dynamics is nearly the same as the equilibria manifold. The
basic concepts of both adaptive management and mathematical
catastrophes were developed in the 1960' s. However, it was not
until a decade later that ~se two concepts were sufficiently
digested to be incorporated into studies of ecological systems
(Zeeman 1976, Jones 1977, Bar-Shalom and Tse 1976), and
were combined even later (Casti 1980).
The theoretical properties of systems that can be managed
incrementally are well known (Bar-Shalom and Tse 1976) and
have been discussed in tenns of biological and ecological
systems (Jameson 1986). It has been well documented that fIxed
schedule plans for management of ecological systems are not
satisfactory, but again it was not until recently that there was
sufficient understanding so that basic causes of failure of fIxed
schedule systems could be reasonably well discussed (Walters
1986). Traditional approaches to analyses of uncertain systems
emphasize the importance of using a stochastic model. However,
a stochastic model is not always necessary unless model
equilibria demonstrate properties of certain of the catastrophe
classes.
Several environmental systems have been modeled as fold
catastrophes (McMurtrie and Wolf 1983, Noy-Meir 1982,
Walker and Noy-Meir 1982, Walker et al. 1981). Jones (1977)
presented a model of spruce budworm outbreaks as a cusp
catastrophe. Loehle (1985) published a theoretical paper on
application of catastrophe theory to grazing, but no concrete
examples were included. Johnson and Parsons (1985) studied
an example pasture system to collect data on a fold catastrophe
response. Many earlier authors seemed tentative in suggesting
the occurrence of a cusp, but Lockwood and Lockwood (1993)
explicitly applied catastrophe theory to weather-driven
grasshopper population dynamics with a detailed analysis of
historical data that demonstrated the properties of a cusp. These
published analyses have been a posteriori; except for the limited
analyses of Jameson (1991) there has not yet been a published
result that has examined the biological response of an
environmental system to an applied treatment to examine a cusp
catastrophe hypothesis.
138
A FOLD CATASTROPHE EXAMPLE
A CUSP CATASTROPHE EXAMPLE
From elementaty algebra, the equation
The concept of cusp catastrophes in ecosystem management
will be introduced by an example of a woodland - grassland
transition. 1YPically, one would expect a shift toward grassland
species and away from woodland species with fire or wood
harvesting, and vice versa with heavy grazing. These reversals
seem entirely reasonable and have been frequently observed (see
Arnold, et al. (1964) and Jameson and Reid (1965) for fire and
post-Columbian grazing effects, Samuels and Betancourt (1982)
for prehistoric wood harvest effects).
Under xeric conditions, tree species may not become
established even with grazing and cessation of fire (Cinnamon
1988). More mesic climatic conditions may be required for this
shift to occur. On the other hand, the xeric climatic conditions
and limestone soils typical of the southern Little Colorado River
basin may not produce enough grass fuel for fire to be a factor
(Clary and Jameson 1981). However, in areas with more grass
fuel production, fire can be an effective deterrent to woody plant
reproduction (Jameson 1962). Another situation in which fire
suppression does not seem a likely factor in so-called invasions
is where the sprouting Juniperus deppeana is the dominant tree
species (Jameson and Johnsen 1964).
Betancourt (1987) stated "distributions of pinyon and juniper
species (and their associations) should be considered ephememl
over the past two million years ... traditionally attributed to
overgrazing and fire suppression, historic invasions could also
matk the current progress of continued migration, climatic
fluctuation, or recovery from historic and prehistoric
woodcutting." These statement are supported with analysis of
pack rat middens over the longer time periods (Van Devender
et al. 1987), and fire scar and tree growth chronologies over a
period of a few hundred years (Swetnam and Betancourt 1990).
Jameson (1969) analyzed the summer and winter precipitation
ratios across Arizona, but Webb and Betancourt (1992) have
indicated that even the summer-winter ratios may be
non-stationary because of variations in the Southern Oscillation
u =x + x + x (1)
is a cubic function that yields a single value of u for each
value of x. However, the equation
3
2
x3 + x2 + X =U (2)
yields 3 values of x for each value of u. This equation is the
simplest of the elementary catastrophes, or a fold catastrophe
(Fig. 1). The second derivative or inflection point of this
equation occurs at
6x + 2
x
=0
(3)
=-113
(4)
By subtracting the value of x of Equation (4) from the values
of x in Equation (2), the manifold equation can be simplified to
_x3 + X + U = 0; (5)
Equation (5) is the simplest eqUfttion that can represent the
discontinuous properties of a catastrophe. Setting the first
derivative of Equation (5) to 0 yields x = 1/3; these points
locate discontinuities that are one of the catastrophe properties.
Between these points, there are three solution values of x; the
upper and lower values represent stable equilibria and the
intermediate value represents an unstable equilibrium. As control
is increased or decreased across one of these discontinuity
points, the equilibrium "jumps" to the alternative equilibrium.
The studies reviewed by Laycock (1991) could be perceived
as a linked series of fold catastrophes, i.e., with discontinuities
between pairs of seveml states of the system However, the
behavior of the system is thereby constrained by these
discontinuities, and allows no "smooth return" to another state
as would be necessruy for many ecological situations.
(EI Nino).
)(
Stabiltty zone
1.4
The complex situation of fire, grazing, wood cutting, and
climatic shifts described in the preceding three paragraphs call
for a complex model to deal with the combinations. Organizing
information around any model has certain implications about
how phenomena are expected to behave, and Figure 2 is an
attempt to include all of these phenomena in a single model .
The basic "faSt" controls (Ul) considered will be fire, grazing
and wood harvest. As demonstrated in Figure 2, there is also a
second or "slow" control U2 dealing with climatic pattem
The front (folded) edge of Figure 2 represents four of five
properties of a catastrophe model listed by Lockwood and
Lockwood (1993):
II
en 1.0
c:
0
0.6
I...
0.2
Co
'0
CIJ
:::J
-0.2
-0.6
"i -1.0
>
-1.4
Stability mne
-2
-1
o
1
2
1.
Value of control u
Figur~ 1. - A fold catastrophe to depict discontinuous changes
In response x as a result of a management control u.
139
Modality: the system tends to be either in a tree
dominated state or in a grass dominated state;
intermediate values cannot be reached directly along
the grazing season axis and tend not to occur.
WOODLAND
solutions for a given input, it may not be possible to predict
what the response will be without knowing the ecological history
of the ecological system.
RISK AND CATASTROPHES
In Figure 3, the elementary fold catastrophe of Figure 1 is
tilted so that the greatest benefits are shown near the upper
discontinuity~ this figure represents those environmental
management situations where benefits can be increased, up to a
point, by inc~ing the input control. If the system were
deterministic, there would be no difficulty with such a
management strategy. However, if the system is stochastic, a
given control may result in either less than maximum benefits
or a shift beyond the upper discontinuity to the lower stability
zone. Once the system is in the lower stability zone, the original
benefits cannot be restored by reversing the control.
Figure 2. - A cusp catastrophe to depict a combined
discontinuous change in a pinyon-juniper woodland. Taken
from Jameson (1987).
2.
Inaccessibility: the infolded region of the figure
represents unstable equilibria which will not be
reached by successional activity.
3. Jumps (discontinuity or catastrophe): as the
controlling factor (fire versus grazing) moves
toward either extreme, a point is reached where
response can no longer move smoothly, the jump to
a different level at this discontinuity is what gives
catastrophe theory its name.
4. Hysteresis: The time path that the woodland!
grassland response must make as grazing control
moves to the right is different than the response as
control moves to the left.
The frfth property of a cusp catastrophe, divergence, is
represented in a movement from the front edge toward the back
edge of Figure 2.
It should be emphasized that models described here, although
postulated as reasonable for ecological obselVations, are only
conceptual. In fact, the surface for Figure 2 was generated from:
_(x3 +
XU1
+
U2 )
=0
~~_#;-->",..~):--,,-,'
,,
,/'
,,
""I _'/.1
-,........ ,-"..... ,/,-
/' ",/to
,,/'
, ~' , , / '
,
t //<::::~::~::::>/
.'
~",--,
,
,I'~~
,/'
CD
m
Control u
Figure 3. - A tilted fold catastrophe indicating the problems of
managing for a maximum benefit with a discontinuous
response surface. The heavy line represents the mean
response with a given control u; the dashed lines indicate
the response at individual sample years.
(6)
where x is the woodland/grassland response, UI is the "fast"
control of the fire/grazing axis, and U2 is the "slow" control
along the climatic axis.
This equation does not represent ecological mechanism, but
is the simplest mathematical form that will generate the desired
surface (Jones 1977, Zeeman 1976). It should be emphasized
that these model equations may contain parameters that are not
readily identifiable with any known biological states or
processes, and thus may not be experimentally detenninable.
The algebraic model also has some other interesting properties
in addition to those mentioned previously. Note that in the zone
between the edges of the fold in Figure 1, the equation gives
three solutions. However, for a given response and position
along the response axis, there is a single input or control. Thus
it may be possible to reconstruct from ecological evidence the
causes leading to a particular response, but, because of the three
Operationally, there are three strategies appropriate for
situations depicted by Figure 3:
(1) Choose a level of control U2 sufficiently left of the
optimal point such that benefits never fall to the
lower stability level, or
(2) initiate a recovery action to lift the system
performance from the lower stability zone to the
upper stability zone, or
(3) initiate an alternative control (UI in Fig. 2) to move
the system beyond the bifurcation point so that a
smooth return to the higher benefit level can be
restored.
140
The same history of thought also leads us to expect that we
can perceive problems and manage resources as though the
response sutfaces do not contain discontinuities. Even those who
insist that interesting models must be nonlinear seldom perceive
that discontinuity and bifurcation is a more serious problem than
the linear-in-a-neighborhood type of nonlinearity.
A new approach would require that we learn to conduct
experiments and use appropriate analyses that escape the old
thought limitations of subdivision and continuity. As an
example, we might start with resource maps. 'JYpically, such
maps subdivide the world into "homogeneous" units. The nice
thing about these homogeneous units is that our old models and
concepts work well within the units. In fact, it is the boundaries
between the map units that are the most interesting and contain
the most challenges in management, as these areas are most
likely to possess discontinuous responses or ecotones (Casey
and Jameson 1988, Holland et al. 1991). A new resource
mapping concept that emphasizes boundaries rather than
homogeneous land units would at least indicate that we know
were the challenging problems lie.
Displays of land area based on discontinuous responses or
ecotones will lead to examination of the relationship of
bifurcation and multiple stability zones of one "map cell" to
behavior of another map cell. Are stability zones 'contagious? If
the land area represented by one map cell shifts to a lower
production stability zone, will this shift be absOIbed by adjacent
land areas, or will the shift spread to neatby land areas? The
implications of these alternatives will cause us either to shrug
off desertification and global climatic change, or to feel that
desertification and global climatic collapse are inevitable.
Each of these strategies has its own cost to be considered in
choosing the best action The cost of strategy (1) is largely the
cost of benefits foregone by operating at a conselVative level of
control. The cost of strategy (2) is the cost of the recovery action
The cost of strategy (3), initiation of slow control Ut, depends
on waiting for the slow control to be effective, and largely results
from discounting future benefits of the restored system
In other cases, the best we can hope for until suitable
techniques are developed is to depend on the experience and
memory of human managers. Historically, human nature is such
that memory selVes only those that have experienced such
catastrophes in their own life, and .little wisdom is transferred
to others.
FUTURE DIRECTIONS
•
Almost any natural resource system could be perceived and
modeled as a single steady state system, or conversely could be
modeled as multiple steady state. However, it is not always clear
that the choice of models makes any practical difference. A
useful approach is to model the system both ways, then use
experiments on the models to detennine if the multiple steady
state properties of the model lead to management decisions that
are different from those reached using single steady state
assumptions.
An alternative approach is to directly examine the behavior
of the natural system to see if responses are such that a multiple
steady state approach must be used. For some systems, the single
steady state approach may lead only to management
inefficiencies rather than to "catastrophes"; for other systems
the consequences are more severe than mere inefficiency. If
multiple steady state properties can be found experimentally, it
would clearly indicate that the system cannot be managed
incrementally (i.e., with a passive adaptive approach) without
catastrophic results. If catastrophe conditions are not found, then
a simple incremental approach will at worst lead to
inefficiencies. For natural resource management, a lack of
catastrophe behavior means that satisfactory management
corrections could be based on obselVations of ecosystem
responses without leading to ecosystem destruction. If
catastrophe conditions are found, then errors in management
cannot be corrected merely by reversing management direction
A handicap that we must face is that most of our history of
natural resource research and management is based on concepts
that we can subdivide the world and conduct experiments to
determine what we need to know about system behavior. Walters
(1986) has nicely documented some problems that arise when
the system under study cannot be subdivided, but must be
" probed" in order to learn the necessary aspects of system
behavior in an active adaptive management scheme. Studies in
global climatic change certainly will face problems of this kind:
how does one probe the world?
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