Group Title: Working paper - International Agricultural Trade and Policy Center. University of Florida ; WPTC 05-10
Title: Environmental restoration of invaded ecosystems : how much versus how often?
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Title: Environmental restoration of invaded ecosystems : how much versus how often?
Series Title: Working paper - International Agricultural Trade and Policy Center. University of Florida ; WPTC 05-10
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Language: English
Creator: Ranjan, Ram
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Publication Date: July 2005
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WPTC 05-10


I '-ional Agricultural Trade and Policy Center



ENVIRONMENTAL RESTORATION OF INVADED
ECOSYSTEMS: HOW MUCH VERSUS HOW OFTEN?
By
Ram Ranjan

WPTC 05-10 July 2005


WORKING PAPER SERIES


'V


i~fr


UNIVERSITY OF
FLORIDA


Institute of Food and Agricultural Sciences









INTERNATIONAL AGRICULTURAL TRADE AND POLICY CENTER


THE INTERNATIONAL AGRICULTURAL TRADE AND POLICY CENTER
(IATPC)

The International Agricultural Trade and Policy Center (IATPC) was established in 1990
in the Institute of Food and Agriculture Sciences (IFAS) at the University of Florida
(UF). The mission of the Center is to conduct a multi-disciplinary research, education and
outreach program with a major focus on issues that influence competitiveness of specialty
crop agriculture in support of consumers, industry, resource owners and policy makers.
The Center facilitates collaborative research, education and outreach programs across
colleges of the university, with other universities and with state, national and
international organizations. The Center's objectives are to:

* Serve as the University-wide focal point for research on international trade,
domestic and foreign legal and policy issues influencing specialty crop agriculture.
* Support initiatives that enable a better understanding of state, U.S. and international
policy issues impacting the competitiveness of specialty crops locally, nationally,
and internationally.
* Serve as a nation-wide resource for research on public policy issues concerning
specialty crops.
* Disseminate research results to, and interact with, policymakers; research, business,
industry, and resource groups; and state, federal, and international agencies to
facilitate the policy debate on specialty crop issues.









Environmental Restoration of Invaded Ecosystems: How Much
Versus How Often?

Ram Ranjan
Postdoctoral Associate
International Agricultural and Trade Policy Center
Department of Food and Resource Economics, University of Florida
Email: rranian(@,ifas.ufl.edu, Ph: (352) 392-1881 326; Fax: (352) 392-9898


July 2005

Abstract

This paper derives the optimal level of restorative efforts required to restore
environments degraded by invasive species invasion. Specific attention is focused upon a
case when the restoration efforts face the risk of failure through relapse of the restored
environment. The level of restored environment may also play a role in its future
improvement or susceptibility to failure. The tradeoff between the optimal level of
environmental quality and number of restorative efforts required to attain that given
environmental quality is analyzed.


Selected Paper Prepared for Presentation at the AAEA Meetings July 24-27, 2005,
Providence, Rhode Island






Keywords: Environmental restoration, Resiliency, Restoration failure, Invasive Species







Copyright 2005 by Ram Ranjan. All rights reserved. Readers may make verbatim copies
of this document for non-commercial purposes by any means, provided that this
copyright notice appears on all such copies.









Introduction

Invasive species are a noticeable source of biodiversity degradation (Glowka et al. 94).

Lately, invasive species have become a subject of widespread concern due to the

enormous economic and environmental damages they inflict upon society (Pimentel et al.

1999, 2000). For instance, certain invasive species such as cheat grass cause destruction

of grasslands, forests and the biodiversity within by inducing frequent fires. While a

number of options exist to prevent the advent of invasive species, none of them are

foolproof. Once the species have invaded a given eco-system, steps could be taken to

either control them in part or eradicate them. However, it is rarely economically or

physically viable to eradicate them. Yet, in most cases the invaded environment could be

restored to a certain extent in order that society can continue to derive economic and

environmental services from it.

Recent studies on the economics of invasive species management include those

by Shogren (2000), Knowler and Barbier (2000), Olson and Roy (2002), Eiswerth and

Van Kooten (2002), Perrings (2003), etc. While these studies focus mostly on the optimal

combination of prevention and control options, one possible option is also to take

restoration measures to bring the invaded eco-system close to it's pre-invaded state. This

paper looks at the important issue of the extent of optimal restoration of an invaded

environment that provides economic amenities to the society. The extent of restorative

efforts is analyzed for an environment that exhibits 'hysteresis' in environmental quality

and is faced with continuous risk of future invasions. Further, the risk of re-invasion is

considered that might lead to failure of the restoration project, causing the restored

environment to relapse back to its initial invaded state. In light of these limitations of the









invaded environment, the optimal extent of restoration is analyzed and policy

implications are derived.

Current work on restoring invaded ecosystems has been mostly confined to the

field of restoration ecology. Yet, there are significant issues of economic importance that

come into play while deciding the extent of restoration. Total restoration may neither be

feasible nor desirable in most cases of invaded ecosystems due to high costs involved in

achieving and maintaining them. Further, restored eco-systems face the risk of falling

back into their degraded states from repeated invasions. Therefore, restoration efforts

that do not incorporate this possibility of failure are bound to lead to inefficient

outcomes. Most restoration efforts after the initial investment require substantial

subsequent efforts to constantly monitor and fight the invasives for sustained periods of

time. This is an essential feature of restorative efforts that are specifically targeted

against invasions. The restored environment may face continuous threats from invasion

even as restoration efforts are undergoing1.

There are numerous cases of biodiversity restoration where restoration efforts

need to be sustained for long periods of time and despite that chances exist of reversal of

the restored ecosystem back into degraded states. One case is that of grasslands of the

Great Basin region in the US, which have been invaded by an alien species of grass,

Bromus Tectorum (cheat grass). This grass is 500 times more likely to catch fire and lead

to destruction of grassland as compared to the native grass of the region (BLM, 2000).

As a result grassland fires have been occurring every 3 to 5 years instead of their natural

wild land fire/annual grass cycle of 60 to 100 years (Kaczmarski 2003). Another


1 For example, invasive plant species may survive through the next season through their seeds, which may
be hard to eliminate.









example is that of invasion of wetlands from the Pacific coast to Saskatchewan to

Arkansas by invasive weeds (aquatic macrophytes) Typha Spp. (cattails) that cause

significant loss of biodiversity (Milklovic 2003). Restoration efforts include flooding,

mowing, drainage, burning, chemical and biological control. However, due to their fast

reproduction rate and colonizing skills, these species re-establish themselves in restored

ecosystems time and again.

A crucial economic issue is then over the extent of restorative efforts to be

undertaken per period when risks of failure of restoration projects are real. For instance,

invasive species that lead to frequent fires may be countered by planting other species

that compete with them and are fire resistant. However, in case of a fire break-out

species of both kinds would get eliminated, therefore, negating all the previous efforts of

restoration. Another related issue is over the level of restorative efforts when risks are

stock-dependent. In the above case, the more the species of fire-resistant kind are

planted; the lower would be the risks of failure of restorative efforts. Further, higher

stock of fire-resistant species may exhibit stock-dependent resilience, i.e. once a

threshold level of fire-resistant species has been reached, there may be a sharp decline in

the level of other restorative efforts required to preserve the level of restored

environment. Experimental work on restoration ecology has revealed that degraded eco-

systems may be resilient to restoration efforts owing to changes in landscape connectivity

and changes in native species pools from invasion by exotics (Suding et al. 2004).

Restoration and resiliency improving measures under risk have been found to be

at the center of issues that deal with invaded ecosystems in the ecology literature.

However, these issues also make the economic analysis fairly complicated, as the non-









linear attributes of the ecological processes must be included in a traditional cost-benefit

approach. Currently there are no known applications of restoration risks in the

economics literature on invasives species and restoration, however, there has been some

work related to threshold effects, such as hysteresis, in the recent past (Maler et al. 2000)

that may be similar to the approach adopted in this paper.

In this paper, a model of environmental restoration is designed that incorporates

the risk and resiliency effects associated with environmental restoration. The issue of

how much restoration effort to undertake is then looked at in an inter-temporal cost-

benefit analysis setting. When risks of failure may be stock dependent, the question of

how much restoration versus how often becomes relevant, as the costs of continual but

lower restoration must be weighed against the costs of less frequent by larger restorative

efforts leading to a higher environmental quality. This also determines under what

circumstances a more resilient state is desirable given the higher costs associated with its

attainment. Numerical simulations reinforce the analysis.

The paper first starts with a deterministic model, where restoration efforts are not

faced with the threat of failure, in order to understand the role of resiliency associated

with environmental restoration. The analysis delves over the existence of multiple

equilibriums with respect to environmental restoration. Next, risk of failure is introduced

into the model. Finally, the trade off between the level of restoration and the frequency

of failure of restoration is taken up in the above setting.









Basic Model

Consider a degraded environment that could provide recreational and environmental

benefits upon restoration. There may be multiple options available for its restoration;

however, in order to simplify things, here we assume that it is possible to combine these

options together into a single restoration variable (/). The environmental quality

q improves due to restoration efforts net of any natural rate of decay given by 6 The

amount of environmental quality lost to decay increases as the level of environmental

quality improves. This assumption is made in order to make unlimited improvements in

environmental quality difficult. Perhaps, a more realistic assumption would be where the

environmental quality stabilizes beyond a certain level; however, incorporating such

dynamics may add unnecessary complexity to the model.


(1) q=cal + 9-q
q +b

The second term in equation (1) leads to a sharp upward jump in the environmental

quality once a threshold level has been crossed. This term captures the resiliency aspect

of degraded ecosystems. Conventionally, resilience has been defined in two ways in the

ecology literature. First one, termed as the 'engineering resilience' defines it as the speed

of bouncing back of any perturbed system (Pimm 1984). The other one, termed the

'ecological resilience', is about the amount of stress that the system can tolerate before

flipping from its original state to another stable but degraded state (Holling 1995,

Carpenter and Cottingham 1997). In this paper we follow the 'ecological resilience'

definition to model the impact of restoration. Parameters q a and b define the rate and

magnitude of this effect. This functional form is associated with the process of









hysteresis in environmental literature and is characterized by a sharp jump (but not

irreversible) in the states of the ecosystem that make it costlier to revert back to. For

instance Maler et al. (2000) use this formulation to study the process of eutrophication of

lakes where a lake turns from a clean state into a turbid state with an increase in the

Phosphorous content. However, in this paper restoration induced jump in environmental

quality is defined in a positive sense, as beyond a certain threshold of environmental

restoration the environment shifts into a better state and is more responsive to restoration

efforts. Alternatively, this formulation mandates that a willful restorative perturbation in

the environmental quality would not lead a system out of its degraded state unless some

threshold is crossed2.

Note that the restorative efforts do not necessarily have to add in more of the

environmental stock from outside. In most cases restorative efforts are simply about

removing the cause of trouble. In most cases, even the degraded environments may have

a capacity to grow back to their full potential, but are overshadowed by the negative

forces such as pests that cause its degradation through a complex interaction involving

natural forces such as fire, droughts, floods, diseases etc. One particular example is the

case of Buffel grass invasion in Queensland, Australia on the native species such as the

Brigalow and Gridgee. Buffel grass pastures increase the risk of fires amongst these

native species, and the more fire-infested the surrounding gets, the higher is the density of

the Buffel grass over time. Thus, in a positive feedback relationship with the fire and the

native species, Buffel grass has been able to wipe out a large chunk of these species over




2 This way to define resiliency may be taken as a cross between the conventional definitions of resiliency
and hysteresis.









time (Butler and Fairfax, 2003). Other examples of models involving resiliency in

grasslands can be found in Perrings and Walker (1997, 2004).

Benefits m(q)are derived per period from environmental quality3. The cost of

restoration c(l)is convex in restorative efforts, thus making unlimited restoration

prohibitive. Let p be the shadow price of the environmental quality and r the social

discount rate. Society maximizes benefits from improved environmental quality net of

restorative costs:


(2) Max {m(q)
0


c(l)}e 'dt


subject to the constraints posed on environmental restoration by equation (1). The

current value Hamiltonian is written as:

a
(3) cvh = m(q) c(l) + d(al + qr q)
qa +b

First order condition with respect to restorative efforts implies that the per unit cost of

restoration must be equated to the shadow value of that marginal unit of restoration.


(4) c'(1)
a

Co-state variable pu evolves as:


(5) t = -m'(q)+ (r + )/p


(rabq" 1
S(q +b)2


From (4) and (5), the time path of restorative efforts could be derived as:




3 These benefits are ecological benefits that do not deplete from public consumption. Ecosystems such as
grasslands, forests and fisheries are also subjected to direct harvests that lead to a reduction in the
environmental stock. This has not been modeled here as the primary goal of restoration may not be
immediate consumption in most cases.










(6) m'(q)ac c'(1) abqa 1
(6) + (r+ 3 2-)
c"(1) c"(1) (qa +b)2

In a steady state, restorative efforts and the environmental quality are held constant.

From (1) and (6) we get:

rqa
(7) al + = q
q +b

(8) '(q) c'(1) rabq '1
c"(1) c"(1) (q + b)2

Equations (7) and (8) define a relationship between environmental quality and restorative

efforts, which could be solved to derive their steady state values. The isoclines for which

the levels of restorative effort and the environmental quality are constant are represented

in figure 1 below4.

Note that there exist three possible equilibriumsL ,U, and R, the low, middle

and the high environmental qualities respectively. Of the three, the low and the high

equilibriums are the stable ones with the middle one being unstable. The resiliency

effect is depicted by a jump in the environmental quality once the environmental quality

crosses the threshold given by the crest in the 4 = 0 curve. The state below this threshold

is the degraded state. Also notice that the R is the resilient equilibrium as environmental

quality can be reduced significantly without letting the system flip to the low quality

steady state. The threshold below, which the environmental quality falls into the

'degraded' state, is given by the trough in theq = 0 curve. The state above this threshold

is the high-quality state or the resilient state. Also notice that the 'high equilibrium',

which is the resilient state, may not be possible to reach from a degraded state in some

4 The shapes of the cost and benefit curves are assumed to be non-linear and the relevant parameters are
shown in the Appendix.









cases. If the benefits from environmental restoration are lower than the costs incurred, or

if the discount rate is high, or if the resiliency effect is not very significant, the high

equilibrium may not be desirable. Figure 2 depicts a case where the benefits of

restoration exceed their costs, thus leading to the high equilibrium as the only possibility.

The effect of varying levels of discount rate is depicted in figure 3 below. As the

discount rate increases, only equilibrium that is possible is the low quality one, on the

other hand, with low discounting high resiliency equilibrium is the only possible

equilibrium. Consequently, time preferences play an important role in deciding the level

of environmental restoration.

Restoration with Relapse

One issue that restoration projects are faced with is the relapse of restored

ecosystems into their original degraded states. This could be caused by a number of

factors such as renewed infestations which could be seasonal, climate-induced or man-

made. Further, once the system flips back into the degraded state, one has to start all over

again as the environmental quality built up in the past is gone. Therefore, the manager is

faced with the challenge of incorporating such possibilities into her optimization

framework. The manager's task is to maximize her long term value:


(9) V(q) = Max (m(q) c() + pV(q))e -z('dt
0

subject to (1), where p is the constant hazard rate of invasion characterized by a Poisson

process. The equation of motion of the hazard rate given by:

(10) A(t)= p









In equation (9), the third term represents the expected value from the system flipping

back into the original state and the manager having to start all over again. V(q)

represents the value function from starting all over again from the initial level of

environmental quality q0. Equation (9) in its extended form can be re-written as:


(11) V(qo) = Max (m(q) c(l))e- (t) rdt + (pV(qo))e -(t) dt
0 0

which can be further re-written after integrating the second integral on right hand side as:


(12) V(q) = Maxf (m(q) c(l))e -(t -dt + pV(qo)-
o r+p

given that A(t) = pt, the above relation can be further simplified as:


(13) V(qo) = r+PMax (m(q) -c())e- -rtdt
r 0

Setting up the current value Hamiltonian for the above problem, we get:


(14) (m(q) c(1))e-( r + + (al + q q)
r qa +b

where is the shadow price of quality

The first order condition with respect to restorative effort yields:


(15) = lc'()er + p
a r

Let {e = p, be the adjusted shadow price of quality. The rate of evolution of the

shadow price is determined by the no-arbitrage condition as:

r + p abq"'1
(16) = -m'(q)e + q(-r + + r)
r (q +b)2

Therefore, the rate of change of the adjusted shadow price / is given by:









r+p abq-1
(17) P = -m'(q) + P(-r abqa + r + p)
r (qa + b)2

In steady state, = 0, implying:

m'(q) r + p
(18) P -- abq
abqc- r
(-r + r + p)
(q + b)2

Substituting for/p from (15) above, we get the steady state relationship between

restoration efforts and environmental quality as:

m'(q)a
(19) c'(l) =m'(q)
abq
(-r- + +r + p)
(q + b)2

Notice that in the no-risk case derived before, the steady state evaluation of equation (6)

would yield:


(20) c'(1) = m'(q)
(- abq r)
(-r 2 r)
(q + b)2

Equation (20) is similar to equation (19) except for the extra term p in the denominator

of equation (19). When the restoration efforts are faced with an ever present constant

exogenous risk of invasion, the risk acts as an additional discounting term. Consequently

steady state restorative efforts are lower in the case when there is a risk of relapse as

compared to no-risk case.

Notice that in the above equations (19 & 20), the increment in the resiliency from

a change in stock serves as an adjustment to the discount rate which is also augmented by

the natural rate of decay of the environmental quality. From the way this resiliency effect

has been specified in the model some interesting implications can be deduced for the









optimal restoration path. The environmental quality shows a sharp jump upwards once a

certain threshold level has been reached. Due to this reason, as long as the environmental

quality is lower than this threshold, the resiliency effect will not be that significant.

Therefore, the discounting effect brought by a change in resiliency due to environmental

stock, kicks in only beyond that threshold level of stock. As a consequence, the change

in the optimal steady state level of restoration effort and the environmental quality from

some external disturbance in parameters would be significantly higher if the steady state

is closer to this threshold. In lay terms, the incentives for restoration efforts are higher;

the closer is the system to the threshold.



Stock Dependent Risk

In the stock-independent risk case, the relationship between the hazard rate and

environmental quality is given by:


(21) = p(q)

The value function can be specified as before as:


(22) V(q) = Max(m(q) c(1) + p(q)V(q))e
0

which can be further expanded for a starting level of environmental quality as:


(23) V(qo)= Max (m(q) -c(l))e --rtdt+ Max (p(q)V(qo))e --rtdt
0 0

Rewriting above we get:










j(m(q) c(l))e "drt
(24) V(qo) = Max
(1-(p(q))e -"r)
0

subject to the equations of motion for the hazard function as given by (21) and the

environmental stock as given by (1)

It is not very straightforward to analytically perform dynamic optimization on the above

problem using the Pontryagin's maximum principle; therefore, we take recourse to

numerical simulations5.

Figure 4 below shows the time paths of restorative efforts for two starting levels

of environmental quality when there is no risk of project failure (q0 =2.8 & 7.8). Notice

that the higher quality steady state is reachable only when the starting value of

environmental quality is high. This is because the hysteresis effect in environmental

quality is not very significant, thus requiring higher restorative efforts in order to

maintain the high steady state level of environmental quality. This, however, may not be


In order to reduce the problem into a standard framework one may define two more state variables as z,
and z2, whose rate of change is defined as:
(25) iz = (m(q)- c(l))e -rt &

(26) i2 p(q)e rt)e -
Now the above problem in equation (23) reduces to:

(27) V(qo) = Maximize subject to
(1 z2)

' = al + 6Sq, = p(q),z =(m(q) -c(l))e -A-rand z2 =p(q)e- )e-
q +b
The current value Hamiltonian of the above problem that would maximize V(qo) is defined as:
(28)

Z1 + (al + 7 q) + 2p(q) + y, (m(q) c(l))e t y4(m(q) c(l))e--
(1- z) q" +b
The first order conditions along with the equations of motion for the co-state variable would yield a time
path for the restorative efforts and the environmental quality.









feasible when the discount rate is high or the benefits from environment do not exceed

their costs in the long run.

However, with a slight increase in the hysteresis effect of q =.06 (as shown

below in Figure 5) and with a lower discount rate of r =.03, we can see that the higher

steady state equilibrium is attainable even when the starting level of environmental

quality is lower (q, = 4.8).

Next we compare the time paths of restorative efforts under constant and quality-

dependent risk of failure with the no risk case. In the constant risk case, the hazard rate is

assumed to be 0.1, where as in the quality dependent risk case the hazard function is

defined as:

(29) p = p,(1- 9*q(t)), where p,=.01, and 9 =.1 &q(t) <10

In the above equation p, is constant component of the hazard rate that is capable of

falling further with an increase in environmental quality. Figure 6 shows the time paths

of restorative efforts and environmental quality under the three cases for a starting level

of environmental quality of 0.8. Notice that the highest level of environmental quality is

attained when there is no risk of failure. Under a constant risk of failure, the

environmental quality attained is the lowest, whereas the endogenous risk case has a

higher environmental quality. The effect of risk is primarily to discount the future

benefits from environmental quality. However, when the risk is stock dependent,

environmental quality is increased to capitalize on its risk reducing impacts.

How Much Versus How Often

When restoration projects are faced with the risk of collapsing back into a

degraded state, the question of how much effort to put in becomes important. If the









ecosystem keeps collapsing into the degraded state time and again, it may take a long

time before the desirable level of environmental stock is attained. Therefore, it may

happen that systems that require a low level of restorative effort but are faced with high

risks of reversal may take a longer time to reach their steady states as compared to

systems that may require a higher level of restorative effort but are faced with a lower

level of risk. Note that the risk of project failure has been accounted for in the above

models. However, the above models do not say anything about the number of times the

project would fail before a steady state is reached. The time taken to reach the steady

state in the above formulations of the problem is the one when there are no setbacks to

the restoration project. However, the actual time taken to reach a desirable level of

environmental restoration would also depend upon the number of times the relapses

happen during restoration. This concept is explored further in the setting of the model

described above.

Let t* be the time it takes for the ecosystem to reach the steady state level of

environmental quality without collapsing when there is a constant risk of reversal to the

initial degraded level6. In presence of a constant risk of reversal, the expected time E(t)

taken to reach the steady state would be given by:


(30) E(t) = {s + E(t)}pe"ds
0

Notice that in the above formulation, once the system reverts back into the degraded state

it has to start all over again and therefore, would take the same amount of expected time

thereafter. s is the time at which the restoration effort fails, thus sending the system

6 Analytically, in most steady state problems it may take an infinite amount of time for the system to reach
the steady state. However, for practical purposes, t* can be decided to be the time taken to reach a point
very close to the target.









back to its initial level. s ranges from 0 to t" and the probability of failure is

exponentially distributed with hazard rate p. Moreover, the system faces risks of

reversal even after the steady state has been reached, however, by the optimal nature of

the steady state it would mean that restorative efforts and environmental stock are

optimally chosen at that level of risk. Integrating the above term we get:

t8
(31) E(t)= spe- + E(t)(-e )
P Jo

Solving the above equation one derives the expected time as:

ept* 1
(32) E(t) = t
P P

The figure below plots the contours for E(t*) for a range of values for the hazard rate p

and t*. Notice that the expected time it takes is much higher when either t* or p are

higher.

The case of stock-dependent risks is slightly complicated. Note that the time

taken to reach the steady state without any interruptions is a function of the rate of

discount, the marginal benefits and costs of restoration, the rates of decay of

environmental stock and the resiliency parameter. For example, a high rate of discount

would require a lower stock and thus would take less time to reach as compared to a case

when the benefits from environmental stock are high or the costs of restoration are low.

Whereas, a low rate of discount would make the resilient state more desirable thus

requiring more time to traverse. Similarly, a lower level of po (the constant component

of the hazard rate) would make a higher environmental quality feasible. This is shown in

figure 8 below, where maximum possible level of environmental quality falls with an









increase inp0. However it can be numerically verified that the expected time to steady

state t actually is lower for the case when po is 0.1 (420 time units) as compared to the

case when po is 0.5 (37279 time units). Using a time horizon of 250, the time to reach

the steady state without relapses is 211 units for p = .1 and 162 units forp = .5. This

is because the steady state level of environmental quality falls as po rises. However, an

increase in po also increases the number of relapses, thus increasing the total expected

time.

If the hazard rate falls quickly with an increase in the environmental stock, it

would reduce the expected number of relapses over the same period, as the expected

duration before for a single relapse increases. This would have an effect of reducing the

expected time to reach the steady state. However, the negative effect of environmental

quality hazard rate would also make it beneficial to strive for a higher environmental

quality as the hazard rate comprises one of the elements of the adjusted discount rate as

derived in equation (14) above. This is shown in figure 9 below where an increase in 9,

the parameter that influences the impact of stock of quality on hazard rate, leads to an

increase in the maximum possible environmental stock. It can also be numerically

verified that the expected time to steady state falls as 9 increases even as the time taken

to reach the steady state without relapses is higher for higher levels of 9 .

A reduced discount rate would mean that future benefits from environment get a

higher weightage than before and therefore more environmental quality would be strived

for. As a consequence, whether the stock dependant resiliency effect leads to higher

expected time to steady state than the stock independent one would depend upon whether









the effect of reduction in discounting achieved through lower hazard rate (which leads to

an increased time to steady state) dominates the effect of a reduction in the expected

number of relapses through a reduced hazard rate. The net effect could go in either

direction. The dilemma in this case when stock of environmental quality could have a

negative influence on project completion time is obvious. On one hand it offers the

incentive to attain higher stock of environmental quality, as a higher quality yields direct

utility and also reduces the risk of relapse. However, on the other hand a higher quality

also means that a higher restoration effort is required to reach there. If costs are convex

in restoration efforts, restoration efforts may need to be stretched over a longer period of

time. However, the more time that is required for reaching the steady state, the higher

would be the expected number of relapses. Therefore, a trade off between how much

quality to strive for and how many failures in order to reach it is highlighted in the case of

stock dependent risks of restoration.

The issue of expected time to steady state is important to policy makers as one

important goal of restoration projects is to bring the system back to a level at which it

could be exploited for direct economic uses. In the case when consumption of

environmental quality leads to a reduction in its stock, additional restorative efforts will

need to be taken in order to maintain the optimal steady state level.




Conclusion

In this paper, the role of restoration measures in improving environmental quality was

looked at through the application of the concept of resiliency. Optimal restoration efforts

were derived when environmental quality impacts the risks of failure of the restoration









projects. It was shown that the environmental and economic parameters determine the

desirability of the level of resiliency, and a highly resilient environment may not be

always desirable.

The tradeoff between the extent of restoration and the number of restorations was

derived. It was shown that the expected time to reach the desirable state in the event of

multiple relapses is a function of both the hazard rate (p) and the time taken to reach the

steady state under no relapse (t*). This relationship between p and t* is convex, implying

that the expected time to steady state under the possibility of relapses could be same for

high risks of collapse but lower t* and low risks of collapse but a higher t*. Note that t*

could be low due to several factors such as the discount rate, benefits and costs

restoration, etc. It also turned out that no straightforward derivation of expected time to

steady state is possible when risks are stock dependent.

There exist several other challenges to restoration projects. Some even oppose the

idea of human interference in degraded environments. Holling and Meffe (1996) in an

influential paper argue in favor of natural disturbances that help build the resiliency of a

system rather than human interventions that shield it against them. Conflicting opinions

exist towards the choice of restoration tools, with some even claiming that exotic species

themselves may play beneficial roles in restoration of the environment as human

interference lead to further disturbances (Antonio and Meyerson 2002). However, when

restorative options are available and their advantages are clear, it may be worthwhile to

apply them, especially when the benefits from their restoration span economic and

environmental goods. In case of environments invaded by alien species, the need for

restoration is an urgent one, as invasive species pose serious threats of extinction of









valuable native ecosystems. It must also be kept in mind that restoration projects need to

incorporate longer time horizons and utilize the resiliency effects offered by higher levels

of environmental quality in order to be able to ward off current and future threats of

invasion.









References

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Table 1: Parameters


used for Simulation


Parameters a b Y 6 n r a r

Values 6 10000 2 .01 4 .05 .02 .05



Table 2: Functional forms of Cost and Benefit Functions

m(q) = qY

c(l) = I"







Figure 1: Steady State Levels of Environmental Quality and Restoration



1
2 -


1.- j=0

1 /



02 4 6 8 10 12 14



2 4 6 8 10 12 14








Figure 2: A Case of Resilient Steady State (high stock benefits)

1


i=0


2 4 6 8 10 12 14








Figure 3: Restorative Effort Isoclines for various Discount Rates

1

3.5

3

2.5- 0 =o

2 = 0,r =.0oo

1.5
0, r = .05

1= o,r= .08
0.5,

2 4 6 8 10










Figure 4: Time paths of restorative Efforts from two Starting Levels of


Environmental Quality


1.75

1.5

1.25

1

0.75

0.5

0.25


Note: Restorative efforts fall to zero even before they reach the steady state due a higher discount rate that
reduces the time horizon for optimization.












Figure 5: Time paths of restorative Efforts from two Starting Levels of
Environmental Quality When Hysteresis Effect is Substantial

1
2


1.5


1


0.5


2
Note: rI


4 6
.06, r =.03 and q0


8
0.8& 4.8











Figure 6: Restoration and Quality Levels under no-risk, constant-risk and
quality stock dependent-risk



1.2








0.8




0.6



0.4

- q(constant-risk)
I(constant-risk)
0.2 q(stock-dependent risk)
I(stock-dependent risk)
-m- q (no-risk)
-*-(no-risk) time

,,,, . . . . . . . . . . . . . . . . 0D







Figure 7: Contours of Expected Time to Steady State in Presence of Risk

3-


2. (t)=5

t* \E(t)=2
2-

E(t)=.5
1.5


1


0.5


0 P
0 0.5 1 1.5 2













Figure 8: Environmental Quality Levels under p,






12









q08




06




04




02


21 41 61 81 101 121 141 161 181 201 221 241

















Figure 9: Environmental Quality Levels under 9


q

0.8



0.6



0.4



0.2


time
0
1 21 41 61 81 101 121 141 161 181 201 221 241


--mu=0.2
- mu=0.5
mu=0.1




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