Copyright © 1997 by The Resilience Alliance*
Ludwig, D., B. Walker, and C. S. Holling. 1997. Sustainability,
stability, and resilience. Conservation Ecology [online]1(1):
7. Available from the Internet. URL: http://www.consecol.org/vol1/iss1/art7/
A version of this article in which text, figures, tables, and appendices are separate files may be found by following this
link.
Insight
Sustainability, Stability, and Resilience
Donald Ludwig1,
Brian Walker2, and
Crawford S. Holling3
1Department of Mathematics, University of British Columbia;
2Division of Wildlife and Ecology, CSIRO;
3Department of Zoology, University of Florida
ABSTRACT
The purpose of this essay is to define and refine the concepts of stability and resilience and to demonstrate their value in understanding the behavior of exploited systems. Some ecological
systems display several possible stable states. They may also show a hysteresis effect in which, even after a long time, the state of the system may be partly determined by its history. The concept of resilience depends upon our objectives, the types of disturbances that we anticipate, control measures that are available, and the time scale of interest.
KEY WORDS:
bifurcation;
multiple stable states;
resilience;
stability.
1. INTRODUCTION
Humans are dependent upon natural systems for the necessities of life such as air and water, as well as resources that are essential for modern societies (Odum 1993). As humans have imposed greater and greater demands upon natural systems, Arrow et al. (1995) and many others have raised concerns about the sustainability of the resource flows from these systems. The purpose of this exposition is to review some theoretical concepts and present specific examples to illustrate the variety of possible behaviors that natural systems may display under exploitation. The concepts stem from our informal understanding of the ideas of stability, sustainability, and resilience, but clarity requires a more detailed classification of behaviors. The examples that we present do not exhaust the possible behaviors, but each is ``generic,'' in the mathematical sense that small changes in parameter values do not change the qualitative behavior of the system. This implies that the qualitative behavior of each example is typical of a whole class of systems.
Equilibrium
A mechanical system is at equilibrium if the forces acting on it are in balance. For example, when a body floats, the force of gravity is balanced by the buoyant force due to displacement of the liquid. The "balance of nature'' (Pimm 1991) is an extension of this idea to the natural world. The concept usually refers to steady flows of energy and materials, rather than to a system whose components do not change.
Resilience and stability
We are interested in characterizing natural systems that are resilient, i.e., that tend to maintain their integrity when subject to disturbance (Holling 1973). This is related to the idea of stability. The informal concept of stability refers to the tendency of a system to return to a position of equilibrium when disturbed. If a weight is added suddenly to a raft floating on water, the usual response is for the weighted raft to oscillate, but the oscillations gradually decrease in amplitude as the energy of the oscillations is dissipated in waves and, eventually, in heat. The weighted raft will come to rest in a different position than the unweighted raft, but we think of the
new configuration as essentially the same as the old one. The system is stable.
If we gradually increase the weight on the raft, eventually the configuration will change. If the weight is hung below the raft, the raft will sink deeper and deeper into the water as more and more displacement is required to balance the higher gravitational force. Eventually, the buoyant force cannot balance the gravitational force and the whole configuration sinks: the system is no longer stable. On the other hand, if the weight is placed on top of the raft, the raft may flip over suddenly and lose the weight and its other contents long before the point at which the system, as a whole, would sink. This sudden loss of stability may be more dangerous than the gradual sinking, because there may be little warning or opportunity to prepare for it. We may think of the raft system as losing its resilience as more weight is placed on it.
Suppose that we accept the ``balance of nature'' and the steady flows of resources that it implies. As we demand more and more of the products of natural systems, and we load them with more and more of our waste products, are we likely to experience a gradual loss of stability or a sudden one? In order to clarify such questions, we must refine our terminology. To decide whether a system is stable or not, we must first specify what we mean by a change in configuration or loss of integrity. If we don't care whether the raft flips over when weighted, then there is no problem of sudden loss of stability for the floating raft. We must also specify the types and quantities of disturbances that may affect the system. Suppose that a fixed weight is placed on top of an occupied raft. If the occupants of the raft move about, the raft may float at a slightly different angle, but if they move too far or all at once, the raft may tip. The range of possible movements of the occupants that do not lead to tipping is called the domain of stability, or domain of attraction, of the upright
state. If the amount of the fixed weight is gradually increased, the balance becomes more precarious and, hence, the domain of attraction will shrink. Eventually, the weight becomes large enough so that there is no domain of attraction at all, and the raft will flip over no matter what its occupants do.
The preceding example makes a distinction between the weight loading the raft and the positions of the occupants. If the amount of the weight changes very slowly or not at all, we may think of the
"system" as consisting of the raft and weight. The occupants change position relatively quickly, and these changes may be thought of as disturbances of the system. On the other hand, we may adopt a more comprehensive point of view, seeing the raft, the weight, and the occupants as a single system. If the occupants organize themselves to anticipate and correct for external disturbances, then the system may be able to maintain its integrity long enough for them to achieve their objectives. Another possible response to disturbance might be to restructure the raft itself. If it were constructed of several loosely coupled subunits, then excessive weighting or a strong disturbance might flip one part of the system, but leave the rest intact. Such a structure might not require as much vigilance to maintain as the single raft, and it might be able to withstand a greater variety of external disturbances. On the other hand, if the bindings that link the subunits become stiff, then the structure may become brittle and, hence, more prone to failure. This simple example illustrates how the notion of resilience of a system depends upon our objectives, the time scale of interest, the character and magnitude of disturbances, the underlying structure of the system, and the sort of control measures that are feasible.
Section 2 presents the main ideas of stability and resilience for simple, one-dimensional prototype systems. Calculations can be done explicitly for these prototype systems, but their qualitative behavior holds for much more complicated examples. Section 3 illustrates the
ideas of bistable equilibria, hard loss of stability, hysteresis, and resilience with a model for the spruce budworm. There are qualitative similarities between the behavior of the budworm model and a variety of ecological systems, particularly lake ecosystems, the Baltic Sea, and boreal forests, although no attempt is made here to provide a formal model for these systems. Such a model is given in Section 4 for a competitive grazing system. This system has qualitative behavior
analogous to a one-dimensional model, and it also exhibits hysteresis and hard loss of stability. An analogous system that involves fire as a regulating process is presented in Section 5. The latter system also exhibits regular oscillations. The Appendix presents a detailed account of the relationship between return times for a disturbed system and its resilience. There are two conflicting definitions of resilience, which may cause confusion. The definition of Pimm (1991) applies only to behavior of a linear system, or behavior of a nonlinear system in the immediate vicinity of a stable equilibrium where a linear approximation is valid. For Pimm, loss of resilience is due to slow dynamics near a stable equilibrium. The definition of Holling (1973)
that we use here refers to behavior of a nonlinear system near the boundary of a domain of attraction. Loss of resilience, in our sense, is associated with slow dynamics in a region that separates domains of attraction.
2. A SIMPLE PROTOTYPE FOR STABILITY AND RESILIENCE
In order to understand complicated systems, it is often convenient to consider a simpler system that exhibits the type of behavior of interest. A full theory of the floating raft would require a combination of the theories of hydrodynamics and of rigid body dynamics, but the essential features can be captured in a one-dimensional model. We are mainly concerned with the notion of stability and the fact that the domain of attraction of a stable equilibrium may depend upon slowly varying parameters. These features are present in a one-dimensional system.
Global stability
The concept of the balance of nature might be taken to imply that the system will maintain its integrity under any sort of perturbation. Such an assumption may be made (often unconsciously) when we make large modifications to natural systems. Our expectation is that things will proceed more or less as before, and that the response of the system will be approximately proportional to the perturbation. Such behavior is shown by the simplest linear models. Some might argue that
a principle of parsimony dictates that such models be used in the absence of strong evidence to the contrary. The following linear model illustrates the property of global stability, which implies that the system will always return to a certain equilibrium, regardless of how far it is displaced from that equilibrium.
Suppose that the dynamics are given by a relation of the form
where is a
smoothly varying function of an external variable and is the quantity of interest. Then
if ; the system has a
single equilibrium there. This equilibrium is stable, since if , and if . These relations imply that the
system approaches the equilibrium, no matter what the starting point.
A system such as (Eq. 1) cannot fail us or surprise us. It returns to
an equilibrium, no matter how far it is displaced, and the position of
the equilibrium changes smoothly with the exogenous variable . Such a system is
not suitable for a discussion of possible collapses of natural
systems, since such collapses are excluded by assumptions such as
(Eq. 1). Mathematical theory provides numerous examples of different
behavior, and our goal is to investigate their
plausibility. Unfortunately much theory (including most economic
theory) has been based upon assumptions analogous to (Eq. 1). In
particular, ``resilience'' has been defined by Pimm (1991) in terms of
the system (Eq. 1), and this very special assumption may mislead the
unwary. There is a mathematical theory that shows that systems such as
(1) are good approximations to general systems with a stable
equilibrium, but that theory implies that the approximation holds only
in the immediate vicinity of the equilibrium, i.e., that the
approximation is valid only locally. Details are provided in the
Appendix.
Bifurcation
In order to explore the differences between local and global
stability, we must examine nonlinear models, i.e., models in which the
state variable appears in more complicated functions than linear
ones. The following example has three equilibria instead of a single
one. Such a complication requires a cubic or more compliocated
dependence upon the variable, for example,
Here, is a
parameter or a slowly varying quantity whose dynamics are not of
immediate concern. The equilibria of the system are the states where
. These are the
states where either of the two factors in (Eq. 2) vanishes. Hence,
they are points where
If , then there are three equilibria
but if , then there is only the single equilibrium at . Such a change in the configuration or stability of equilibria is a called a ``bifurcation'' (Guckenheimer and Holmes 1983). It implies a change in the qualitative behavior of the system. To explore this feature, we must discuss some additional concepts.
Local stability and domain of attraction
In order to determine the stability of equilibria, it suffices to
examine the sign of the velocity of . For example, if , the second factor in
(Eq. 2) is always positive and, hence, if and if . In this case, the system always
moves away from the state where . We conclude that the equilibrium
at is unstable if
. On the other
hand, if , then
changes sign at
three places:
The equilibrium where is unstable, because the system
always moves away from that point if nearby (according to Eqs. 5 and
6). Similarly, the equilibrium where is unstable (according to Eqs. 7
and 8). On the other hand, the equilibrium where is stable (according to Eqs. 6 and
7), because the motion from nearby points is toward that
point. However, if the system starts outside the interval , it moves away from
the equilibrium at . Therefore, the equilibrium at
is locally
stable, but not globally stable. The system returns to if small perturbations
are made, but larger perturbations take the system into an unstable
domain. The interval is called the ``domain of
attraction'' of the point , because trajectories that start
within that interval eventually return to , but not those that start outside.
It is clear that the domain of attraction of the stable equilibrium at
shrinks as decreases toward
zero. The three equilibria collapse into one where , and only a single unstable
equilibrium remains when . This information is summarized
in Fig. 1
. The diagram looks like a branch and, for this reason, it is
called a ``bifurcation diagram." The domain of attraction of the point
is contained
within the two curved branches, and there is no other domain of
attraction.
FIG. 1. The parameter is plotted on the horizontal axis and the corresponding equilibria in for Eq. (2)are plotted on the vertical axis. The stable equilibrium is plotted with a solid curve, whereas unstable ones are plotted with dotted curves.
Increasing evidence has accumulated for the existence of multistable
states in nature: coral reefs (Done 1992, Hughes 1994, McClanahan et
al. 1996), African rangelands (Dublin et al. 1990), shallow lakes
(Schindler 1990, Carpenter and Leavitt 1991, Scheffer et al. 1993),
kelp forests (Estes and Duggins 1995), and grasslands (D'Antonio and
Vitousek 1992, Zimov et al. 1995).
Disturbances and slow parameter changes
We have seen that if , then this system approaches the
stable equilibrium at if it is started within the domain
of attraction. If we envisage disturbances that displace the system a
distance from the
stable equilibrium, they will not affect the integrity of the system
(its tendency to return to the 0 state) as long as . Now, if we allow the parameter
to decrease
slowly toward ,
the system will take longer and longer to return to the state when is displaced to , because motion is very slow near
, and a
disturbance of magnitude may take the system into the region
of slow dynamics. We may think of the decrease in as causing a loss of resilience,
because the integrity of the system is threatened more and more by
disturbances of a given magnitude. A symptom of loss of resilience may
be longer and longer times to return to the vicinity of after disturbance. The
connection between return times and resilience is not completely
straightforward; we address it in some detail in the Appendix.
Two domains of attraction
The preceding system is not a believable model for natural systems,
because it predicts that the state variable may approach infinity
under some circumstances. A more plausible scenario is one in which
the system may change from having a single stable equilibrium to one
with two stable equilibria. We next consider a number of such
``bistable'' systems.
We obtain a simple prototype for such systems by changing the sign of
in (Eq. 2). If
the direction of time is reversed, the stable and unstable equilibria
are interchanged. If , then the single equilibrium at
is globally
stable: the system always returns to that equilibrium no matter where
it starts. Instead of two unstable equilibria when (as in the former case), there will
be two stable equilibria. The new system is
If , for this system, we have
If initially,
then heads toward
the equilibrium at , but if initially, then heads toward the equilibrium at
. Thus, the
domain of attraction of the point is the positive axis, and the domain of attraction
of is the
negative
axis. Each of the stable equilibria is locally stable, but not
globally stable. This system can be flipped from one stable state to
another by crossing the unstable line where . Because this line separates the
two domains of attraction, it is called a ``separatrix." The
equilibria in are
plotted in Fig. 2
.
FIG. 2. Equilibria in for Eq. (9) are plotted vs. , as in
Fig. 1. The direction of increase of time is reversed as compared with Fig. 1; hence,the stable and unstable equilibria are interchanged.
Disturbances and slow parameter change
The bifurcation diagram in Fig. 2
implies a great deal about the
response of the system to disturbance. If , then the system will return to
the stable equilibrium at , no matter how large the
disturbance: there is nowhere else for it to go. However, if and the system starts
near the lower branch, it will tend to return there if displaced by a
small amount. As
decreases toward zero, the distance between the stable equilibria and
the unstable one decreases. Hence, disturbances of a given magnitude
take the system closer and closer to the unstable
equilibrium. Dynamics are slow near the unstable equilibrium and,
hence, the time to return to the vicinity of the lower branch
increases sharply for trajectories that approach the unstable
equilibrium. This point about return times can be made precise by a
calculation analogous to that in the Appendix.
For a higher level of disturbance and , the system may be moved across the
separatrix at ,
and may approach the upper branch of stable equilibria. Under a random
pattern of disturbances, we may expect to see the system spend long
periods of time in the vicinity of one or the other of the stable
equilibria. Every now and then, the random disturbances may combine
and send the system to the other stable equilibrium. The ``Allee
effect'' studied by ecologists provides an example of a bistable
system. A population may suffer reduced survival or reproductive
success at low numbers. For example, schooling fishes tend to suffer
low per capita mortality if their numbers are high enough relative to
the capacity of their predators. If such fish are reduced in numbers
through fishing pressure or environmental degradation, the population
may decline and eventually become extinct locally. On the other hand,
a large population may sustain itself over long periods. Dynamics of
this sort might explain the occasional flips between dominance of
sardine and anchovy, as revealed by deposits of their scales off the
coast of California. A similar pattern appears in such geophysical
features as the polarity of the earth's magnetic field, the ocean
circulation involving the Gulf Stream, and climate fluctuations, but
some of these fluctuations may be too regular to be completely
random. For larger values of , one would expect flips from one
equilibrium to the other to be extremely rare. Thus, we may associate
an increase in
with an increase in resilience.
3. HARD LOSS OF STABILITY AND HYSTERESIS
The two preceding examples illustrate a so-called ``soft loss of
stability." As the exogenous variable changes, the location of the
stable equilibria changes smoothly. The state variable may move from
one domain of attraction to another, but such changes are slow because
dynamics are slow near an unstable equilibrium or a separatrix. The
possibility of such behavior would not ordinarily be cause for alarm,
because slow dynamics may allow for adjustments to new behavior. There
are natural systems, such as outbreaking insect populations, that
sometimes show more abrupt changes.
The following model was used by Ludwig, Jones, and Holling (1978) to understand the dynamics of the spruce budworm. The quantity represents budworm density, measured in larvae per acre. This density is assumed to vary in time according to
where
is an intrinsic growth rate at low densities, is a carrying capacity for the budworm in the absence of predation, and the second term in (Eq. 14) is a predation rate. The predators are assumed to have a Holling type-III functional response, with a maximum predation rate of and a half-saturation budworm density of . This functional form implies that predators have their greatest influence upon dynamics at intermediate ranges of budworm densities. At low densities, the predators search for alternate prey, because returns from foraging for budworm are relatively low. At high densities, budworms swamp their predators; thus, the predators have a small per capita effect, just as predators have a small per capita effect on large schools of fish. The parameter is proportional to a measure of foliage density, because the predators search foliage for the budworms and their response is mediated by the number of budworms per unit of foliage. Hence, is actually a state variable that generally changes on a
slower time scale than the budworm. For the moment, we regard as a constant.
Some algebra supplied in Ludwig et al. (1978) shows that there are
either two or four equilibria for the budworm, depending upon the
sizes of the dimensionless parameters and , given by
These equilibria satisfy
where . The
equilibrium is
always unstable, because if is small and positive. The highest
equilibrium is always stable, because if is very large and positive. Thus,
if there are only two equilibria, budworm density always moves toward
the upper equilibrium. When there are four equilibria, they alternate
in stability. A typical case is shown in Fig. 3
. If is between and , may approach either the high
equilibrium or the low equilibrium, depending upon whether the
starting position of is above or below the unstable
equilibrium, which is the separatrix.
FIG. 3. The equilibria in for Eq. (16) are plotted against
for . The points and delimit the interval on the axis where there are three nonzero equilibria for the system.
Hard loss of stability
Imagine that the parameter begins at a low value and gradually
increases as the forest grows. It turns out that does not change with forest growth;
hence, Fig. 3
applies. Because is proportional to , will increase. At first (when ), budworm numbers
will remain low, as the only stable equilibrium is the low one. Even
when increases
beyond , the
budworm numbers will remain low, because they lie below the unstable
equilibrium, which determines the domain of attraction of the low
equilibrium. The stability of the low equilibrium becomes precarious
as approaches
, because the
domain of attraction shrinks. Finally, at , the lower two equilibria
disappear and budworm density jumps to the high value: an outbreak
occurs. This abrupt change in the attracting state is called a "hard
loss of stability." It should be contrasted with the soft loss of
stability displayed by the system in (Eq. 9). In the case of the
budworm, once density has reached the high equilibrium, there is no
easy way to reduce it to the lower equilibrium. If the variable is reduced below , the budworm remains
at the high equilibrium. As is further reduced, there is a
second hard loss of stability as declines below . In this case, there is a jump
down to the low equilibrium, which is not reversed as increases again.
Hysteresis and cycles
If we now connect the dynamics of the trees and the dynamics of the
budworm, a new phenomenon appears. If the system starts with low
foliage density and low budworm numbers, the foliage density slowly
increases until it surpasses . At this point, an outbreak
occurs, as shown previously. High budworm numbers eventually cause
death of trees, so begins to decrease when the budworm
has an outbreak. Budworm numbers remain high even though declines, because
budworm density lies above the separatrix. As continues to decline to , budworm density
declines slowly and then jumps to a low value when decreases below . The different paths followed by
the total system for increasing vs. decreasing constitute the "hysteresis effect."
The combination of budworm and forest dynamics produces stable cycles
with long periods. Such stable cycles that are maintained through
alternations of rapid transions and slow changes are called
``relaxation oscillations." They are common in many physical,
chemical, and physiological systems (Edelstein-Keshet 1988).
Disturbances and resilience
If the objective of management is to keep budworm numbers and foliage
damage low, the loss of stability as increases beyond may be regarded as a loss of
resilience. This model suggests that small disturbances near the lower
stable equilibrium may exhibit long return times if they approach the
unstable branch. Such long return times may be a useful diagnostic
indicator. However, because increases as trees grow, a loss of
stability accompanied by a budworm outbreak seems inevitable.
We may adopt a different perspective and regad periodic budworm
outbreaks as part of a stable system that renews the forest from time
to time. Indeed, systems analogous to the budworm-forest system
frequently appear as stable oscillators. The advantage of such
oscillators is that they continue to oscillate more or less with the
same frequency and amplitude under a wide variety of
disturbances. Hence, physiological oscillators are important in
maintaining integrity of the organism, which is another kind of
resilience. According to this perspective, an attempt to halt the
oscillations may lead to a disastrous breakdown in the long term. Will
human interventions to increase productivity in natural systems suffer
a similar fate?
Lake dynamics
Carpenter et al. (1996) have discussed the applicability of these
ideas to lake ecosystems. They characterize lake dynamics as either
"normal" or "pathological." Normal lakes have high numbers of game
fish, effective grazing upon phytoplankton, and low incidence of algal
blooms. The normal system maintains its integrity when subjected to
perturbations such as phosphorus pulses, because phosphorus moves
rapidly into the higher trophic levels (Carpenter and Kitchell 1993)
and humics constrain algal growth (Jones 1992). However, heavy
phosphate loading, removal of macrophytes, overfishing, and removal of
wetlands and riparian vegetation may lead to the pathological state in
which there are few game fish, less grazing, no macrophytes, and
extensive and frequent algal blooms (Harper 1992). This may be a rapid
transition and it is not easily reversed (National Research Council
1992).
This situation appears to fit the definition of a hard loss of
stability, because the change is rapid and large, and is sometimes not
reversed even if phosphate loads are decreased (National Research
Council 1992). One may say that the normal lake is resilient because
it maintains its integrity under perturbation, but resilience is lost
as phosphate loading and other stresses are increased. If critical
levels of phosphate and other environmental variables could be
identified, we might attempt to measure resilience in terms of the
difference from the critical levels (Vollenweider 1976). Perhaps the
question of whether or not the lake ecosystem fits our definitions may
be answered by statistical analysis of long-term data.
The Baltic Sea
Jansson and Velner (1995) describe the Baltic system in terms that
show many similarities to the lake system. The Baltic Sea is partially
enclosed and, consequently, has a residence time of water on the order
of 20 years. Algae form the base of a diverse food web, with higher
trophic levels occupied by commercially important species such as
herring, flounder, pike, and perch. There may be long periods when
there is weak vertical mixing of the water column, due to lack of
inflow from the North Sea. During such periods, oxygen levels at
greater depths may be very low and sulphur bacteria may
predominate. The latter put large quantities of phosphorus into
solution, which then upwell and cause plankton blooms.
In historical times, the Baltic has experienced several extended
anoxic periods, but the system has not been permanently altered. Since
the industrial revolution, the Baltic has been loaded with increasing
amounts of phosphorus, and there are indications of a change of
configuration to a detritus-based system. This would imply more turbid
water and a fish community consisting mainly of sluggish species such
as bream, roach, and ruffe, which are much less valuable than those
previously listed. The possibility exists that the Baltic might reach
a point at which even reducing phosphate inputs might not return the
system to its earlier, more desirable state. Such a turn of events
would correspond to a hard loss of stability, analogous to the
behavior of the lake ecosystem. Unfortunately, there are no replicates
of the Baltic system; hence, we have only analogies to guide action. A
purposeful demonstration that the Baltic is actually capable of a
sudden change corresponding to a hard loss of stability is unthinkable
as an experiment. Nevertheless, it may be occurring as a result of
human negligence. The earlier ability of the Baltic to recover from
anoxic periods may correspond to resilience, but we cannot be sure
whether this resilience is being lost.
The boreal forest
Carpenter et al. (1996) characterize the boreal forest as a system
with relatively few species and complicated interactions and
dynamics. In upland regions, forests dominated by aspen and birch
alternate with forests dominated by spruce and fir. Browsing by moose
over a period of 20-40 years can convert an aspen stand into one
dominated by conifers. As stands of conifers mature, they become
increasingly favorable for reproduction of the spruce
budworm. Eventually, outbreaks occur and portions of the system are
converted into early successional aspen. The budworm outbreak
corresponds to a hard loss of stability, and the combined upland
system undergoes stable, long-period oscillations analogous to those
described previously.
As the upland regions undergo these oscillations, the valley bottoms
alternate between flooded plains and moist meadows. The flooded state
is maintained by beavers, which cut aspen bordering streams for food
and dam the streams to create ponds. When the supply of aspen is
insufficient, the beavers abandon their dams, the dams break, and the
ponds are soon replaced by meadows. This relatively rapid change, a
consequence of decreasing supply of aspen, may be thought of as a hard
loss of stability. The upland and lowland cycles tend to entrain each
other because of the interaction between beavers and aspen. Fires also
play a role in synchronizing cycles over large spatial areas, because
conifers killed by the spruce budworm provide an abundance of fuel.
Although this system undergoes large alterations, sometimes very
quickly, it may be thought of as resilient, maintaining its character
over many centuries. Conditions at any given site may change abruptly,
but the system is usually a mosaic of patches at differing stages of
the cycle. When considered as a whole, it maintains considerable
diversity.
4. A COMPETITIVE GRAZING SYSTEM
In this section, we describe a natural system that may be
bistable. Competition between grasses and woody vegetation in a
semiarid environment is described in Walker et al. (1981). Suppose
that either the grass or the woody vegetation has an advantage when at
high densities relative to the other. In such a case, the system has
stable equilibria that correspond to high levels of grass and woody
vegetation, respectively. The competition is also influenced by the
stocking rate of cattle, which consume grass but not woody
vegetation. We shall regard the two plant forms as the dynamic state
variables, and the stocking rate as a slowly varying parameter.
Imagine starting with high levels of grass and low levels of woody
vegetation. At low levels of stocking, there is only a small
difference from the ungrazed system: if the system starts out with
grass dominant, grass will continue to dominate. As stocking
increases, the competition may favor woody vegetation. Eventually,
there may be a collapse of the grass, and woody vegetation will
dominate. Thus, the effect of grazing is to move the system from a
state in which grass dominates to one in which woody vegetation
dominates. Even when grazing pressure is relaxed, there may be little
change in composition, because of the advantage enjoyed by woody
vegetation over grass when the former is dominant. The effect of
grazing is to move the system into the domain of attraction of woody
vegetation for the ungrazed system.
If one plots grass density vs. the stocking level, the behavior may
appear to be inexplicable: the grass level declines as grazing
increases, but does not return to former levels when grazing returns
to its former level. The apparent paradox is resolved if we realize
that the density of grass depends not only on the stocking level, but
also on competition with woody vegetation. These phenomena may be
illustrated by a modification of the Lotka-Volterra competition model.
Mathematical model
Let represent the
density of grass, and let represent the density of woody
vegetation. The rate of change of grass density is assumed to be
where is a growth
rate, and and
are competition
coefficients. The parameter is determined by the stocking rate
of cattle. The rate of increase of the woody vegetation is assumed to
be
where is a growth
rate, and are competition
coefficients, and
is a source term. In the illustrations that follow, the parameters
were chosen as:
The case of light grazing corresponds to = 1/10.
In order to understand the behavior of this system, it is helpful to
plot some curves in the plane, as in Fig. 4
. In the phase
plane, the direction and speed of change of the system are given by
the vector .
This vector is vertical on the curve where (the null isocline), and it is horizontal on
the curve where
(the null
isocline). As can be seen, , where either , or
This locus is a straight line, and it shifts to the left as increases. The null
-isocline is a
hyperbola according to (Eq. 18). One of the asymptotes is the -axis, and the locus
passes through the point , labelled "W." This locus is
independent of the stocking parameter . Figure 4
shows in detail how a
system may approach more than one steady state, depending upon the
starting conditions. Although this system is much more complicated
than (Eq. 9), its qualitative behavior is the same if . This illustrates how
the very simple one-dimensional models may, nevertheless, be a
valuable heuristic guide.
FIG. 4. The phase plane for grass and trees, according to
Eq. (17), Eq. (18), and Eq. (19). The null and isoclines are plotted with dashed
lines. These are the curves where or , respectively. The equilibria are
points where the isoclines cross. The stable equilibria are labelled
as "G'' and "W," respectively. The unstable equilibrium is labelled as
"S." The trajectories that enter S form a separatrix: they are
plotted with dotted lines. Trajectories to the right of the separatrix
eventually reach G, and those to the left eventually reach W.
We now turn to the effect of increased stocking. The effect of an
increase in is to
shift the null -isocline down and to the
left. Hence, the points S and G will approach each other along the
null
isocline. Because the separatrix passes through the point S, the
domain of attraction of G must shrink, whereas the domain of
attraction of W will expand. Qualitatively, the representation of
Fig. 4
still holds. For a still higher value, , the points S and G coincide. The
values of
corresponding to the roots S and G are shown in Fig. 5
. A similar
diagram could be drawn to show the corresponding values of . This is a
bifurcation diagram analogous to Fig. 3
. If the stocking rate changes
slowly, we may expect the grass density to be given by the upper curve
in Fig. 5
. However, if , the grass density must crash,
since there is no stable equilibrium with a nonzero grass
density. This is a hard loss of stability, and indeed there is a
striking similarity between Figs. 5
and 3.
FIG. 5. A bifurcation diagram showing grass equilibria as a
function of the grazing parameter for the system (17)-(19).If , there are three
roots. The lowest branch corresponds to the point W where , which is always a
stable equilibrium. The middle branch corresponds to the locus of
point S as
varies. Those points are unstable equilibria, so they are plotted with
a dotted line. The highest branch is the locus of point G as varies. This point is
always stable.
For values of ,
the qualitative form of Fig. 6
applies. The only equilibrium is point
W, and all trajectories approach W. The domain of attraction of W is
the whole quadrant where and .
FIG. 6. The phase plane for grass and trees for the system
(17)-(19), with . In this case, there is only a
single stable equilibrium for the system at . All trajectories approach that
point.
We may imagine the system beginning with the stocking rate . According to Fig. 5
,
the unstable equilibrium S and the stable equilibrium W are very close
together. Hence, the separatrix in a figure analogous to Fig. 4
is in
the extreme upper left corner: virtually any initial combination of
and will lead to a high
density of , given
by the upper branch in Fig. 5
, with a correspondingly small density of
. Now, if is increased slowly,
the density of
will move downward along the upper branch until the bifurcation point
is
reached. Beyond that point, the grass density must crash, and woody
plants will dominate. Even if grazing pressure is withdrawn ( ), the grass cannot
recover because it will be to the left of the separatrix. This is the
hysteresis effect.
5. FIRE IN A SAVANNA SYSTEM
For two reasons, the preceding model does not describe most savanna
systems. (1) Generally, neither grass nor woody vegetation can
completely exclude the other. Instead, there is a single stable
equilibrium for the system (which changes over time, depending upon
rainfall, grazing, and fire) where grass and trees coexist. (2) Woody
vegetation cannot exclude grass indefinitely: after a long period of
low grazing, the grass may return. This is due to a combination of two
effects. First, the older woody vegetation may die and leave gaps that
may be colonized by grass, and then fire gets into the system. Second,
woody vegetation dies back very quickly in dry years, but recovers
only slowly in wet years, too slowly to use all the water. Grass, on
the other hand, can increase 10-fold in a season, quickly enough to
fully use all of the available water. Because of this, the combination
of wet and dry years keeps woody vegetation at lower levels than the
average rain would sustain, and permits grass to remain in the system
in significant amounts.
When viewed at a long time scale, a brief period of grazing may cause
a rapid collapse of the grass, followed by a slow recovery. What
appears to be an equilibrium with high woody vegetation, when viewed
at a short time scale, corresponds to a region in which the system
spends a long time when viewed at a long time scale. This sort of
qualitative behavior is analogous to "excitable systems." Such systems
are best known as models of the nerve impulse, according to the theory
of Hodgkin and Huxley, as modified by Fitzhugh. The system has a
single stable equilibrium, but when perturbed in an appropriate
direction, it may undergo a very large excursion (firing of the
neuron), followed by a long recovery (refractory) period. Details are
given in Edelstein-Keshet (1988).
The effect of fire on dynamics
To model aging properly is complicated, but for present purposes, it
suffices to find a simple system that has the required qualitative
behavior. We do not contend that the following model is an accurate
representation of the true dynamics. We must keep track of surplus
grass that may serve as fuel for fires. Hence, we let gross grass
production be given by , where
where is a source
term, and is the
rate of grass production available for grazing. The grass that is not
consumed by grazing cattle is potential fuel for fires, and is denoted
by , where
The parameter
determines the proportion of grass consumed by cattle. Now (Eq.17) is
replaced by
The dynamics of
will be influenced by fires, and the age of trees influences their
susceptibility to fire. Let a new variable denote the product of the woody
plant density and the average age of the woody plants. A first
approximation yields , but that relationship neglects
the influence of fire on the average age. The dynamics of and are given by
where the fire risk is defined as follows: let the fire
potential be
proportional to the available fuel:
We assume that the fire risk is given by
where the parameter is an age at which the fire risk is
half of its maximum, and the parameter determines the sharpness of the
increase of fire risk with age. We have used to give a sharp increase, and . The remaining
parameters were chosen as
The stocking rate
may be chosen as a parameter or control variable. In the following, we
chose .
Because there are three state variables in this system, one cannot
make a meaningful phase plot. However, if the age variable is fixed, we may gain
an impression of the dynamics. Figure 7
shows how a low value of leads the system to an
equilibrium with high . On the other hand, a higher
value of leads to
an equilibrium with much lower , as shown in Fig. 8
. Now, if increases with time,
the system will first have high , then lower as it ages. Fires decrease the
average age of trees as well as their density. Hence, the system gets
reset to a state analogous to that shown in Fig. 7
after a fire. The
cycling of with
time in the full system, with changing, is shown in Fig. 9
.
FIG. 7. The phase plane for grass and trees for the system
(21)-(28), neglecting the dynamics of the age variable , with a low value of
. There is only a
single equilibrium with a high value of and a lower value for , and it is stable. All
trajectories approach this equilibrium.
FIG. 8. The phase plane for grass and trees for the system
(21)-(28), neglecting the dynamics of the age variable , with a high value of
. The single
equilibrium has a low value for and a moderately high value for .
FIG. 9. Tree density (W) vs. time (T, in
years) for the system (21)-(28), including dynamics for the age
variable. A similar figure could be drawn for the grass density, which
tends to be high when woody vegetation is low, and vice versa.
If one were to observe this system over a time of 5-20 years, it would
appear that woody plants would eventually dominate grasses because of
the combination of competition and grazing. Over the next 30 years,
however, the effect of fire and aging of trees leads to a collapse of
the trees, making way for the next cycle. This example illustrates how
the time scale over which we observe the system may have a decisive
influence upon our classification of its behavior. Unfortunately, the
scale over which we are able to observe the system is often much
shorter than the scale over which it exhibits its characteristic
behavior.
6. CONCLUDING REMARKS
The examples presented here illustrate the complexity of the task
facing us as we attempt to clarify the concepts of sustainability and
resilience for natural systems. Mathematical theory presents a wealth
of possibilities, but the limitations of our understanding and the
data available make it difficult to distinguish between them.
Here, we have proceeded from simple conceptual models, such as the
raft analogy, to simple but abstract mathematical models, to ecosystem
analogues, and finally to a fairly detailed model of a savanna
system. In no case can these analogies be considered to be complete,
nor is our knowledge of the systems detailed enough to support a
full-blown model with statistical justification. Perhaps one might
conclude that all this is merely speculation, unworthy of serious
attention. On the other hand, prudent decision making requires that we
take account of a variety of plausible hypotheses about the responses
to our actions. The examples presented here do not encourage
complacency about the ability of natural systems to support us and our
habits in the lavish fashion we have enjoyed in the past. If we refuse
to contemplate possibilities that are only dimly perceived, we may
miss the opportunity to learn about the world and adapt our behavior
accordingly. If we insist that the simplest and most convenient
hypotheses have priority when choosing actions, we run the risk of
allowing sloth and pride to bring us to disaster.
RESPONSES TO THIS ARTICLE
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Acknowledgments
This research was the product of a Resilience Network Planning Workshop
supported by the Beijer Institute, a Division of the Royal Swedish Academy of
Sciences.
LITERATURE CITED
- Arrow, K., B. Bolin, R. Costanza, P. Dasgupta, C. Folke, C. S. Holling, B.-O. Janssen, S. Levin, K.-. Mäler, C. Perrings, and D. Pimentel. 1995. Economic growth, carrying capacity, and the environment. Science 268: 520-521.
- Carpenter, S. R., B. O. Jannson, D. Ludwig, J. Pastor, G. Peterson, and B. Walker. 1996. A comparative analysis of total system resilience. Notes from the Beijer Institute Resilience Network Planning Workshop April 10-16.
- Carpenter, S. R., and J. F. Kitchell. 1993. The trophic cascade in lakes.
Cambridge University Press, Cambridge, UK.
- Carpenter, S. R., and P. R. Leavitt. 1991. Temporal variation in the paleolimnological record arising from a trophic cascade. Ecology 72(1): 277-285.
- D'Antonio, C. M., and P. M. Vitousek. 1992. Biological invasions by
exotic grasses, the grass-fire cycle, and global change. Annual Review of Ecology and Systematics 23: 63-87.
- Done, T. J. 1992. Phase shifts in coral reef communities and their ecological significance. Hydrobiologia 247: 121-132.
- Dublin, H. T., A. R. E. Sinclair, and J. McGlade. 1990. Elephants and
fire as causes of multiple stable states in the Serengeti-Mara woodlands. Journal of Animal Ecology 59: 1147-1164.
- Edelstein-Keshet, L. 1988. Mathematical models in biology.
Random House, New York, New York, USA.
- Estes, J. A., and D. Duggins. 1995. Sea otters and kelp forests in Alaska: generality and variation in a community ecological paradigm. Ecological Monographs 65(1): 75-100.
- Guckenheimer, J., and P. Holmes. 1983. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, New York, New York, USA.
- Harper, D. 1992. Eutrophication of freshwaters. Chapman and
Hall, London, UK.
- Holling, C. S. 1973. Resilience and stability of ecological systems. Annual Review of Ecology and Systematics 4: 1-23.
- Hughes, T. P. 1994. Catastrophes, phase shifts, and large-scale degradation of a Caribbean coral reef. Science 265: 1547-1551.
- Jansson, B.-O., and H. Velner. 1995. The Baltic: the sea of surprises. Pages 292-372 in L. H. Gunderson, C. S. Holling, and S. S. Light, editors. Barriers and bridges to the renewal of ecosystems and institutions. Columbia University Press, New York, New York, USA.
- Jones, R. I. 1992. The influence of humic substances on lacustrine planktonic food chains. Hydrobiologia 229: 73-91.
- Ludwig, D., D. Jones, and C. S. Holling. 1978. Qualitative analysis of insect outbreak systems: the spruce budworm and the forest. Journal of Animal Ecology 47: 315-332.
- McClanahan, T. R., A. T. Kamukuru, N. A. Muthiga, M. Gilagabher Yebio, and D. Obura. 1996. Effect of sea urchin reductions on algae, coral, and fish populations. Conservation Biology 10(1): 136-154.
- National Research Council. 1992. Restoration of aquatic ecosystems. National Academy Press, Washington, D.C., USA.
- Odum, E. P. 1993. Ecology and our endangered life support systems. Second edition. Sinauer, Sunderland, Massachusetts, USA.
- Pimm, S. L. 1991. The balance of nature? University of Chicago Press, Chicago, Illinois, USA.
- Scheffer, M., S. H. Hosper, M.-L. Meijer, B. Moss, and E. Jeppesen. 1993.
Alternative equilibria in shallow lakes. TREE 8(8): 275-279.
- Schindler, D. W. 1990. Experimental perturbations of whole lakes as tests of hypotheses concerning ecosystem structure and function. Oikos 57: 25-41.
- Vollenwieder, R. A. 1976. Advances in defining critical loading levels for phosphorus in lake eutrophication. Memorie dell'Istituto Italiano di Idrobiologia 33: 53-83.
- Walker, B. H., D. Ludwig, C. S. Holling, and R. M. Peterman. 1981. Stability of semiarid savanna grazing systems. Journal of Ecology 69:473-498.
- Zimov, S. A., V. I. Chuprynin, A. P. Oreshko, F. S. Chapin, III, J. F. Reynolds, and M. C. Chapin.1995. Steppe-tundra transition: a herbivore-driven biome shift at the end of the Pleistocene. American Naturalist 146(5): 765-794.
APPENDIX
Return times and resilience
It is important to distinguish between behavior near a stable
equilibrium and behavior near the boundary of a domain of attraction,
which is an unstable equilibrium or separatrix. As discussed in
Section 2, the long return times associated with a loss of resilience
are caused by slow dynamics near the unstable
equilibrium, not by slow dynamics near the stable
equilibrium point. Unfortunately, there are two conflicting
definitions of resilience and consequent confusion about the
connection between resilience and return times.
Pimm (1991:13) defines resilience as "how fast a variable that has
been displaced from equilibrium returns to it. Resilience could be
estimated by a return time, the amount of time taken for the
displacement to decay to some specified fraction of its initial
value.'' Pimm (1991: 33) describes return to equilibrium by the
equation
where is the
population density at time , is the initial population density,
and is the
equilibrium density. The differential equation for that corresponds to this formula is
A similar model with discrete time could be given instead, but that
would not alter the following argument. If we measure displacement
from by , then satisfies
which is equivalent to Eq. (A.1) if is replaced by . Strictly speaking,
Pimm's definition depends upon this simplicity, because the amount of
time required for
to decay to some specified fraction of its initial value is only
constant if the model (A.1) is used. In
fact, if the initial displacement is and the fraction is , then (A.1) implies that
from which we conclude that the return time is given by
The remarkable feature is that the magnitude does not appear in this
formula. This is a feature of this model only, as we shall see
below. In more general circumstances, such a result can be expected to
hold only in the limit as . Such results are called "local."
As pointed out in Section 2, a common error is to extrapolate local
results to global ones. In the present context, it amounts to
replacing a complicated function by a linear approximation. Such
approximations are certainly easy to work with, but they may miss
essential features of the dynamics. In fact, failure to recognize the
distinction between local stability and global stability can lead to
unwarrented optimism about the likely consequences of interventions in
natural systems. If we think that stability to small perturbations
necessarily implies stability to large perturbations, then precautions
are never required.
In order to distinguish behavior near the equilibrium at from behavior near an
unstable equilibrium, we must use a model with more parameters than (A.2). We set
This particular form leads to an especially simple equation for the
return time: the time to reach a position starting at is given by
and the form for
was chosen so that
as can be verified algebraically. In view of (A.7) and (A.8),
Now, if we replace by , (A.9)
becomes
where
If the last two terms in (A.10) are
omitted, this result is identical to Pimm's assumption (A.1). Our more complicated dynamical
assumption (A.6) is the analogue of Pimm's
assumption if there are three equilibria. Under what conditions does
(A.10) imply large return times? The first
term, which corresponds to Pimm's model, implies a long return time if
the ratio is
small or if is
small. In Pimm's discussion, is a parameter that describes a
probe or observation of the system. Ordinarily, is fixed, and the return time
provides an estimate for .
The second term in (A.10) implies a long
return time if is
small or is
small. Our previous discussion was concerned with a possibly variable
and disturbances
that might take the system near an unstable equilibrium. That
corresponds to
near , or near . In such a case, will be large even if the parameter
is large. That
is, return times may be long, even for systems that show very rapid
return when close to the stable equilibrium. According to this point
of view, long return times may be diagnostic for a small or for disturbances
that are large enough to take the system near an unstable
equilibrium. They may also correspond to weak repulsion from the
unstable equilibrium, i. e., small . If a disturbance takes the
system beyond the unstable equilibrium, there is no return at all.
In summary, according to Pimm (1991) and according to us, long return
times may be diagnostic for a loss of resilience, but the meanings of
the terms are quite different in the two cases. Pimm is concerned with
behavior near a stable equilibrium. In that case, a long return time
for a given displacement from the equilibrium indicates a small
coefficient or,
equivalently, a small derivative of . We are concerned with behavior
of a system with two or three equilibria, one of which is
stable. Resilience describes the tendency of the system to return to
its stable equilibrium. A long return time is due to disturbances that
bring the system near an unstable equilibrium, or possibly to a weak
repulsion from an unstable equilibrium.
Address of Correspondent:
Donald Ludwig
Department of Mathematics
University of British Columbia
Vancouver, British Columbia, Canada V6T 1Z2
phone: 604-541-9409
fax: 604-822-6074
ludwig@math.ubc.ca
Addresses of Coauthors:
Brian Walker
CSIRO, Division of Wildlife and Ecology
G.P.O. Box 84
Lynham, ACT 2602, Australia
Phone: 6-242-1742
Fax: 6-241-1742
brian.walker@dwe.csiro.au
Crawford S. Holling
University of Florida
Department of Zoology, P.O. Box 118525
Gainesville, Florida 32611 USA
Phone: 352-392-6914
Fax: 352-392-3704
holling@zoo.ufl.edu
*The copyright to this article
passed from the Ecological Society of America to the Resilience Alliance on
1 January 2000.
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