Title:
Ecological Interaction as a Source of Economic Irreversibility
Authors:
James R. Kahn (corresponding author)
Department of Economics
534 Stokely Management Center
University of Tennessee
Knoxville, TN 37996-0550
jkahn@utk.edu
Robert
V. O’Neill
Environmental
Sciences Division
Oak
Ridge National Laboratory
Proposed
Running Head: Ecological Interaction & Irreversibility
JEL
codes:Q2,
Q3
Acknowledgment Footnote: This research
was funded by the U. S. Department of Energy, Office of Energy Research. Oak
Ridge National Laboratory is managed by Lockheed Martin Energy Research, Inc.,
under contract DE-AC05-84OR21400 for the Department of Energy. Appreciation is expressed to Dina
Franceschi, Jonathan Hamilton, and anonymous reviewers for helpful comments.
Abstract:
Irreversibility can be either physical or
economic in origin. For example, the extinction of a species is physically
irreversible. On the other hand, contamination of lake-bottom sediments by
mercury is not physically irreversible (the mercury and/or sediments can be
physically removed), but the cost is so high that it can be said to be
economically irreversible. This paper argues that economic irreversibility
associated with environmental change is much more common than typically
discussed in the economics literature. The source of the problem is the
inherent complexity of ecological relationships. The paper discusses the origin and policy
importance of these indirect irreversibilities.
1.
Introduction
In a series of
articles during the 1970s, Arrow, Fisher and Krutilla (Arrow and Fisher 1974;
Krutilla and Fisher 1975) discuss the
importance of environmental irreversibility. Some economic actions, such as
damming a river, producing nuclear waste, emitting CO2 into the
atmosphere, or releasing heavy metals, cause damage that simply cannot be
repaired by the ecosystem. In addition,
ecological interactions may amplify the damage and transform seemingly
reversible economic action into irreversible alterations. For example, competitive interactions may
prevent a valuable wildlife population from recovering, even after the damaging
economic activity ceases.
These effects are generated because the
direct impacts of economic activity can cause indirect nonlinear changes in
ecological, social, and physical systems.
Nordhaus (1994), for example, suggests that irreversible thresholds will
determine the socio-economic impacts of global climate change. In fact, it may be that “...low probability
catastrophic events...should be our main concern” (IPCC 1996). Thus,
commonplace non-linear interactions have important implications for economic
activity and economic policy. This paper argues that economic irreversibility
is much more common than typically discussed in the economics literature. The
source of the problem is the inherent complexity of ecological relationships.
Of course, uncertainty regarding irreversibility complicates the decision
making process and generates the need for greater prudence in policy. (Arrow
and Fisher 1974; Krutilla and Fisher 1975; Weisbrod 1964; Dixit and Pindyk
1994)
2. Economic Analysis of Environmental
Change
The economics
literature has shown a relatively narrow focus when measuring the changes in
social welfare associated with improvements or declines in environmental quality. Most of the
literature has focused on either direct use values or existence values.1 Other important values associated with
environmental change include the value of changes in ecological services. Ecological
services include attributes and outputs of ecosystems, including nutrient
cycling, hydrological cycling, maintenance of atmospheric chemistry and global
climate, biodiversity, soil formation and primary productivity. While these
ecological services generally do not directly enter utility functions in a
fashion which is perceivable by the individual, they do contribute greatly to
utility either directly through their life support functions, or indirectly
through their impact on production and consumption activities.
The early
contingent valuation studies (such as Knetsch and Davis 1966) and the early
travel cost studies (such as Clawson 1959) focused on measuring the
recreational benefits associated with environmental resources. The focus of the
valuation literature expanded to include aesthetic benefits of environmental
change (such as the visibility benefits or air pollution reductions (such as
Randall, Ives and Eastman et al 1974) and the morbidity and mortality benefits
of environmental change (see Berger et al. 1987). In fact, both the Clean Air
Act and the Clean Water Act specifically discuss human health benefits as the
primary benefit of environmental improvement. In addition, valuation studies
have tended to focus on existence values, measuring the willingness to pay to
protect individual endangered species, environmental quality in natural parks,
and unique natural environments such as the Grand Canyon or Chesapeake Bay. It
is interesting to note that the studies that come the closest to measuring the
value of ecological services have really only focused on existence values or
direct use values. For example, the study by Rubin, Helfand, and Loomis (1991)
looks at the willingness to pay to preserve the presence of an indicator species,
the spotted owl, rather than the value of the ecological services of the
spotted owl’s’s habitat, the ancient growth forests of the Pacific Northwest.
In fact, the public debate on the question of harvesting wood in ancient growth
forests tends to focus on the prevention of the extinction of the spotted owl,
rather than the total value of the ecological services which are provided by
these forests. Similarly, studies of the value of biodiversity, such as the
study by Simpson, Sedjo, and Reid (1996), focus on the value of potential
medicinal uses of the species, rather than the total ecological and social
benefit provided by the biodiversity.
If one has a
relatively narrow perspective on what constitutes the benefits associated with
environmental quality, then one will have a correspondingly limited perspective
on what constitutes the optimal level of pollution, which constitutes the basis
for environmental policy goals. The purpose of this paper is to attempt to
broaden economists’ viewpoints on the benefits of environmental improvement, to
focus on the value of ecological services and to show the potential
interruption (and diminution) of these ecological services through irreversible
environmental change. This paper also contributes to our understanding of
environmental change by extending the concept of irreversibility to examine
indirect irreversibilities, which are introduced and defined in the following
section.
3. Nonlinearities and irreversibilities
Irreversibility
can be either physical or economic. For example, the extinction of a species is
physically irreversible. On the other hand, contamination of lake-bottom
sediments by mercury is not physically irreversible (the mercury and/or
sediments can be physically removed), but the cost is so high that it can be
said to be economically irreversible.
Figure 1
presents a schematic of a damage function, which constitutes a functional
relationship between the anthropogenic activity that modifies both the ecological
and socio-economic systems, and resulting change in social welfare.
Irreversibilities can occur at any stage of the process. For example, slash and
burn clearing of tropical forests may result in irreversible (depending on soil
type) loss of forests. Emissions of heavy metals into the environment are
irreversible, since no natural processes exist to decompose the heavy metals.
Carbon dioxide, once emitted into the atmosphere has a residence time of
approximately 500 years.
These types of
irreversibilities can be characterized as direct irreversibilities, as the
original environmental modification (the direct result of the anthropogenic
activity) can not be
reversed. This is the type of
irreversibility that has generally been examined by the environmental economics
literature. For example, Krutilla and Fisher (1975) focus on land use
decisions, and the irreversible decision to allow development in wilderness
areas.
Although direct
irreversibility is important, this paper focuses on what can be termed indirect
irreversibility. Indirect irreversibility does not occur through a direct
impact, but through a behavioral response to a direct impact. In other words,
direct irreversibility refers to the irreversibility of the processes modeled
in the small boxes of Figure 1. Indirect irreversibilities occur as a result of
the impact of these processes on behavioral relationships within the ecologic
or economic system. These ecological and social responses are generated
and exacerbated by the complexity and
nonlinearity of behavioral relationships.
Our objective
in this paper is to demonstrate that this type of indirect irreversibility is
pervasive in both ecological and social systems. This implies that
irreversibility is inherently more common than implied by the direct
irreversibility discussed in Krutilla and Fisher (1975), Porter (1982), and
Arrow and Fisher (1974), which would imply that environmental policy should be
correspondingly more cautious.
Ecological
systems are fundamentally nonlinear and display complex behavior in response to
disturbance. Ecosystems are ordinarily
stable, i.e., when they are disturbed, they recover back toward a stable
equilibrium. Naively considered, an
environmental irreversibility only occurs when a catastrophic disturbance
drives the ecological system beyond its ability to survive or recover. But
viewed at a larger scale, the equilibrium itself is not constant and shifts in
response to changes in background environmental conditions. It is critically
important to realize that irreversible
ecological thresholds can be crossed even with small, gradual changes in the
environment.
As the environment slowly changes, the local
system can enter a region of "metastability". In response to even a minor disturbance, the
local system no longer recovers to the old equilibrium, but moves rapidly to a
new state. The local system now
responds to further disturbances by recovering to the new equilibrium. For example, throughout the world, deserts
are encroaching on grasslands, because of the overgrazing of livestock. This
overgrazing gives a competitive advantage to desert scrub. However, if the
animals are removed from the landscape, it does not return the competitive
advantage to the grasses. The established desert scrub will continue to
out-compete the original vegetation, preventing it from becoming
re-established.
The
phenomenon can be explained relatively simply in mathematical theory
(Tikhonov 1950). A model is solved for
equilibrium by setting the differential equations to zero. Under conditions of metastability, the
system has moved into a region of parameter space where setting the equations
equal to zero yields a higher order equation. That is, the equations are
quadratic or higher in the state variables and there is more than one possible
equilibrium state. Empirically, one
often observes a stable system moving rapidly to a new stable state, without
the intervention of a major external disturbance.
An Example
It is important
to emphasize that nonlinear phenomena can occur even in very simple
environmental systems, without the complexities associate with order state
equations. Consider the dynamics of a
population, N1, valued as a food resource, and N2, an
inedible competitor. Their dynamics are
expressed as:
dN1/dt
= r1N1 - bN1N2
(1)
dN2/dt
= r2N2 - bN1N2
where ri is the potential rate
of increase in the population and b represents a competition coefficient. For simplicity, we will consider the rates
of increase to be equal, that is, r1 = r2 = 1. The two populations are competing for a
common resource that supports a total combined equilibrium of K = N1
+ N2. At this equilibrium, Nj
= K - Ni , further growth is
impossible and
dNi/dt
= Ni - bNi(K - Ni) = 0, so that
b = 1 / (K - Ni)
(2)
Substituting the equilibrium expression
for b (Eqn. 2) back into Equations 1 yields,
dN1/dt
= N1 (1- N2 / (K - N1))
(3)
dN2/dt
= N2 (1- N1 / (K - N2))
The reason for
deriving Equations 3 is to illustrate a counterintuitive response one often
gets from an ecological system. The
system is asymptotically stable to the equilibrium point, K. That is, if we harvest a few of the food
organisms, the system replaces the harvest by growing back to N1 + N2
= K. However, the system is neutrally
stable in the sense that the new equilibrium will have a different proportion
of N1 and N2.
Consider the
simple case in which initially N1 = N2 = 50, so that K =
100. Now we will harvest a given
number, H, of the food organism at a single point in time and allow the system
to asymptotically approach a new equilibrium of K = N1 (¥) + N2
(¥) = 100. To simplify the presentation, we will stop
the simulation of Eqs 3 when N1 + N2 > 99.5. The resulting equilibria for different
levels of harvest are shown in Table 1.
Simply
explained, what happens is that the harvest of the food organism reduces the
system below its carrying capacity and both edible and inedible populations
grow back in response to this opportunity.
But the result is that the system does not have any single equilibrium
value for the edible population. The
populations grow at the same rate and the final ratio N1(¥) / N2(¥) is sensitive
to the initial conditions and, therefore, to the harvest. Because of the nonlinear response, each
harvest leads to a new equilibrium value for N1.
At first
thought, it may seem that the dilemma of harvesting the food organism is simply
resolved by applying some pest or weed control to the inedible population. However, the underlying nonlinear dynamics
make this strategy difficult. Consider
the situation in which K = 100 and N1 (0) = N2 (0) =
50. Assume that one would like to harvest H individuals of the
food organism and allow 10 years for the system to recover before another H
individuals are harvested. What
percentage of the inedible population would one then have to kill each year in
order to ensure that N1 = 50 after 10 years? The results are given in Table 2. The results show that there is a maximum
harvest of about 30 food organisms every 10 years, even if the competitor is almost
completely killed back each year.
Thus, even
though Equations 3 are a trivial representation of an ecological system, the
nonlinear dynamics leads to important differences in the response of the system
to human impact. We have developed this
example for competing fish, but readers will readily see the analogy with more
complex situations such as the clear-cutting of forests, modification of the pH
of an aquatic system, destruction of habitat, global climate change or
introduction of non-native species. We have chosen a simple model of species
interaction to illustrate that nonlinearities and indirect irreversibilities
can arise from even a simple nonlinear dynamic model. As the system becomes
more realistic and more complex, as the number of species and the number of
interactions increase, the potential for the generation of indirect irreversibilities also increases.
Nonlinear Irreversibilities
So much for theory. Is there any reason to
believe that such nonlinear phenomena occur in biological and environmental
systems (O'Neill, Gardner, and Weiler 1982; O’Neill, Johnson and King
1989)? At the smallest scale, it is
generally accepted that this type of response causes periodic bursting in nerve
cells (Plant and Kim 1975) and rapid transients in microbial colonies (Rozich
and Gaudy 1985; Worden and Donaldson 1987).
At the ecosystem scale, Jones (1975) argues that outbreaks of pests
follow these dynamics. At the global
scale, Crowley and North (1988) show that in very simple models of ice cap
dynamics; that the system can jump from one stable state to another. They argue
that this accounts for rapid climate changes in glacial‑interglacial
transitions.
The most
convincing evidence for nonlinear behavior in the global system is provided by
mass faunal extinctions (Donovan 1989).
The fossil record documents nine major extinction events with climate
change and/or sea level change implicated in eight. The ninth event was probably precipitated by an asteroid which,
in turn, created climate change.
However, the other eight extinctions were not simply caused by a major
abiotic event. Internal dynamics, i.e.,
biotic interactions, were also involved.
For example, the evolution of biomineralized (bony) jaws changed the efficiency
of predators. Land bridges caused
faunal exchanges and extinctions of less efficient fauna. These internal dynamics moved the ecological
system to a new stable state, exactly as predicted by nonlinear theory. While there appears to be little risk of
mass extinctions due to projected CO2 increases, many candidates for
nonlinear responses appear at the landscape, ecosystem, and population
scales. The criteria include: (1)
systems near the limits of their geographic range, (2) systems already under
severe stress, (3) systems that have impaired recovery ability due to other impacts.
The most
conspicuous threshold at the landscape scale occurs at the ecotone, the tension
zone where one vegetation type changes suddenly into another, e.g., grassland
into forest (Hansen, di Castri and Naiman 1994). These sharp transitions have long attracted the attention of
ecologists (e.g., Clements 1897, 1905; Livingston 1903; Griggs 1914). Changes
occur as disturbances destroy the existing vegetation and open the opportunity
for new vegetation to take over the site.
As with other
threshold phenomena, some ecotones are simply explained by sharp
discontinuities in the abiotic environment.
The simplest example is seen on mountains in the northern hemisphere
where the northern (colder) slope differs in vegetation from the adjacent south-facing
(warmer) slope, yielding a sharp ecotone along the ridgeline. But as with other nonlinear threshold
phenomena, some ecotones occur as sharp transitions even along gentle gradients
in abiotic factors (Hobbs 1986). These
ecotones are sharp because of competitive interactions within the system
(Daubenmire 1968) and a small change in environment causes the system to move
to a new stable state. As environmental
changes occur through time, the ecotone responds by moving in space (Boaler and
Hodge 1962; Spugel 1976; Delcourt and Delcourt 1987). Pollen records indicate that past climate changes have caused a
slow (5 - 200 km per century) migration of ecotones. The IPCC (1996) reports a consensus that biomes may migrate
150-500 km north due to global warming.
The concern for
nonlinear responses is also motivated by the broad spectrum of stresses being
imposed by society (Goodland 1991).
Average temperature is increasing faster than it has in the last 10,000
years (Arrhenius and Waltz 1990). The
human economy uses 40% of net primary production (Vitousek et al. 1986). Natural vegetation has been fragmented,
making it more difficult to recover from natural disturbances (Gardner ,
O’Neill and Turner 1993). The ozone
shield has been damaged. Soil erosion
is nearly universal, with soil losses exceeding soil formation rates by at
least 10-fold (Pimentel et al. 1987).
More than 50% of the tropical forests have been cut with the current
rate of deforestation exceeding 168,000 square kilometers per year (Goodland
1991). Some impacts, such as species
loss, are irreversible (Krutilla and Fisher 1975) and technology is not
available to repair large-scale damage (Norton 1991).
Phillips (1995)
discusses the importance of the time interval between successive disturbances
and its implications for the time required to recover from a disturbance. The multiple stresses currently being
imposed on the global system are both increasing the frequency and intensity of
disturbances and decreasing the ability of ecosystems to recover. This twofold impact increases the risk of
threshold phenomena and irreversibility.
4. Policy Implications of Ecological
Irreversibilities
Nonlinearities
and irreversibilities have long been recognized as having important implications
for policy. In the past, ecological and
economic models have incorporated thresholds, but as discrete and isolated
phenomena. The present study points out that these nonlinearities and indirect
irreversibilities may result from common place, but complex interaction among
ecologic and economic variables. The potential existence of these
nonlinearities and indirect irreversibilities has important implications for
policy.
One set of
policy implications arises from nonlinear thresholds associated with renewable
resources within the ecological system.
Nonlinear thresholds, for example, appear to be a reasonable explanation
for the global collapse of fisheries which has been occurring during the last
decade. This is important to emphasize,
because the existence of indirect irreversibilities implies that increasing the
demand will eventually cause the collapse of the fishery. While increasing
fishery effort will initially lead to a new equilibria associated with a
smaller stock and smaller populations, indirect irreversibilities can prevent
the return to the old equilibrium. These indirect irreversibilities may also
trigger the crossing of a threshold, implying the collapse of the fishery.
There are further implications for fishery and renewable resource management.
For example, “pulse fishing” is often suggested as an appropriate management
strategy for some fisheries (see Clark 1985 for a discussion of pulse fishing.)
Following this policy, a species is fished extremely hard for several periods
and then allowed to recover. During
this recovery period, attention moves to a different species which is fished
extremely hard and then allowed to recover, and son on. However, in the
population growth models which underlie the potential efficiency of pulse
fishing, each species is assumed to be independent with no nonlinear
competitive interactions and conventional management. More complex models have been developed which focus on
inter-species interaction, but fishery management policy continues to operate
as if species were independent and as if it were feasible to restore collapsed
species simply with a cessation of fishing effort.
Analogous examples can be constructed for
forests and other renewable resources.
Shortsighted fire-prevention policies, for example, simply stop small
fires. These small fires are a normal
part of the evolutionary history of the forest and not only do not kill mature
trees but aid in the dispersion and germination of the seeds of the climax
trees. The small fires also act to
decrease the fuel level of dead wood and form natural fire-breaks. Without the small fires, a threshold is
reached where fuel accumulation permits hotter fires that can spread over large
continuous regions and create vast stretches where the forest is completely
decimated. Thus, inevitable fires
associated with a fire prevention policy are larger and far more costly.
Policy must
also consider irreversibilities when setting standards for pollutants that
affect a renewable resource. The direct
impact may be acceptable in itself, but may move the renewable resource across
a threshold. This occurs, for example,
when a pollutant impacts the competitive interactions between species. If a fish species, valued for its
recreational potential, is more
sensitive than its competitors, then acid precipitation may have a large
indirect impact. As sulfur dioxide
emissions increase the acidity of the system, other fish are given a
competitive advantage and displace the valued species. Our analysis of even the very simple
interactions of Equations 3 suggests
that even if the original pH of the system were restored, the valuable species
may not recover because of the nonlinearities generated by competitive
interaction
A similar
scenario appears to be occurring in the coniferous forests of the southern
Appalachian mountains. Although acid
precipitation has not directly killed the ridge-top spruce forests, the
sublethal impact has weakened the trees and made them susceptible to attacks of
a pest, the balsam wooly adelgid. The
mature trees are dying and other species, unable to compete under normal
conditions, are growing into the gaps
created by the dead trees. The
sublethal impact, combined with an irreversibility due to competition, may eliminate
this forest type from the region. It is
important to recognize that many other types of environmental impacts, such as
nutrient enrichment, conversion of wetlands, introduction of non-native species
and global warming also have the potential to cause such disruptions and
associated indirect irreversibilities.
The implication
of these phenomena is that even pollutants with short environmental residence
times may create damage over very long periods. This creates a very different
optimization problem for these short-lived pollutants than traditionally
utilized. For example, the typical optimization problem for short-lived
pollutants such as sulfur dioxide is to minimize
|
where the total social cost (TSC(t)) of
sulfur dioxide emissions (E(t)) in period t is
the sum of the total cost of abatement and the damages from the
pollution that remains. In this example, TSC is a function of emissions, and
TSD is directly a function of emissions and indirectly a function of emissions
through its affect on the fish stock (N1). The first order
conditions indicate that the optimal level of emissions in each period is
independent of the level of emissions in other periods.
Nonlinearities
change the nature of the optimization, because emissions in period t also
affect a stock through the state equations which govern the competitive
interactions. Hence the problem shifts from a simple static optimization
problem to a dynamic problem. The
optimal control problem becomes
|
Thus, the policy decision no longer
involves simply choosing today’s optimal level of emissions. The policy must consider the optimal time
path of emissions, taking into account the interdependencies among time periods.
The necessity to determine an optimal time path of emissions has long been
recognized for persistent pollutants which accumulate in the environment (such
as heavy metals, DDT, or chlorflourmethanes) but has not been discussed for
short-lived pollutants such as sulfur dioxide.2
5. Policy Implications of Economic
Irreversibilities
The complex
competitive interactions of ecological systems also occur in economic systems.
This has many implications for economic policy, such as anti-trust and trade
policy, but it also has specific
implications for environmental policy. Environmental change may alter the
competitive relationship among economic activities, such as production of
specific agricultural crops.
Nonlinearities may allow new activities to keep a competitive advantage
despite amelioration of the environmental problem. Assume, for example, that cotton is particularly sensitive to
tropospheric ozone. As tropospheric ozone increases, cotton yields fall which
simultaneously causes cotton farming profits to fall while cotton price
increases. Two types of adaptive reactions take place in the economy. Garment
manufacturers switch to other fabrics and cotton farmers switch to other crops
which are less sensitive. Although a series of adjustments has been made, the
level of social welfare falls because the environmental change distorted
economic activity.
Moreover, a
whole set of indirect reactions take place. Fashion designers develop styles
more suited to the new fabrics. Department store buyers begin to think in terms
of the new fabrics and fashions. Farmers redesign their farms towards the new
crops, changing cropping schemes and capital equipment. Some farmers may even
invest in perennial crops, such as pecan trees, to replace the cotton. All
these reactions create barriers to the economy returning to its original
position, once tropospheric ozone is reduced and cotton grows more
prolifically. Hence, there is a legacy of damages even after the environmental
problem is eliminated.
Many economists
argue that these adaptations are good because they ameliorate the damages
associated with environmental change. At one level of analysis, this is
certainly true and this point is made persuasively in the global climate change
literature (NAS 1991). However, if
tropospheric ozone levels dropped so that cotton again became the most
efficient fabric, the altered state of the economy has restructured the market
and created barriers to reintroducing cotton. In short, the reaction to the
environmental change limits future options, as economic conditions adapt to
environmental conditions. Again, this argues that intertemporal dependencies
must be explicitly considered when developing environmental standards. Comparing present period marginal damages
and abatement costs is insufficient to maximize social welfare.
6. Conclusions
This paper
argues that environmental systems are far more complex systems than
traditionally perceived in economic analysis. This complexity can be measured
both in terms of the diversity of ecological services that the ecological
systems provide, and in terms of the nonlinearity of ecological interactions
and interactions between the economic and environmental systems. This
complexity and nonlinearity can lead to indirect irreversibilities, where
seemingly small and inconsequential actions have very large and irreversible
impacts.
The
implications of this different conception of ecological processes for
environmental policies follow in a relatively straightforward fashion following
Krutilla and Fisher’s (1975) approaches to direct irreversibility. The
potential for indirect irreversibilities and the uncertainty associated with
their trigger points implies an additional need to be cautious with regard to
environmental degradation. Instead of adopting a “wait and see” attitude
towards measuring the costs of environmental degradation, the importance of
ecological services and the existence of indirect irreversibilities implies
that proactive policy is necessary and that environmental quality goals should
be more stringent than traditionally conceived. This is true with respect
to all environmental policy, but particularly true with environmental problems
that have the greatest potential for system-wide change, such as policy towards
non-native species introductions, global climate change, tropical
deforestation, desert encroachment and loss of habitat in general.
Footnotes:
1. Direct use values arise when
environmental quality is directly used in activities such as water quality for
swimming. Existence values occur when the knowledge of the existence of an
environmental resource increases an individual’s utility, such as people
gaining utility from the presence of whales in the ocean.
2. An exception is Farmer, et al. (1998)
who examine the abatement cost side of the pollution problem and show how
intertemporal dependencies related to technological innovation in abatement
technology generate a dynamic optimization problem for short-lived pollutants.
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Figure Captions:
Figure 1: Schematic of Marginal Damage Function
Table 1.
Results of harvesting a food fish, N1, that shares resources
with an inedible competitor, N2.
Shown are the equilibria (Eqs 3) achieved after a single harvest.
Harvest N1 (¥) N2 (¥)
1 49.5 50.5
2 49 51
4 47.8 52.2
8 45.3 54.7
16 39.1 60.9
20 35.4 68.6
30 24.6 75.4
40 12.5 87.5
42 10.0 90.0
44
7.5 92.5
46
5.0 95.0
48
2.8 97.9
Table 2. The percentage of the inedible fish, N2,
that must be killed each year to ensure that the food fish, N1,
recovers fully from a harvest, H, after 10 years.
H, harvest of N1 every 10
years % of N2 that
must be killed each year
1 0.5
5 3
10 6
15
10
20 20
25
30
29 100