International Review of Economics and Finance 9 (2000) 69–77
Aspects of ecosystem persistence and the optimal
conservation of species
Amitrajeet A. Batabyal*
Department of Economics, Utah State University, 3530 Old Main Hill, Logan, UT 84322-3530, USA
Received 15 September 1998; accepted 11 April 1999
Abstract
Although ecologists have long recognized the salience of persistence in determining the static and the dynamic behavior of ecological systems, it is only very recently that economists have begun to study this concept in relation to the use of services that are provided by jointly determined ecological-economic systems (ecosystems). As such, there are very few studies of ecosystems that explicitly analyze the ecological and the economic aspects of this use issue. Given this state of affairs, this article has two objectives. First, a new method is used to formally describe and bound the notion of ecosystem persistence. This method explicitly incorporates the stochastic aspects of ecosystems. Second, the bound on persistence is used to study the problem of optimal species conservation. 2000 Elsevier Science Inc. All rights reserved.
JEL classification:Q30; D80
Keywords:Conservation; Ecosystem; Persistence
1. Introduction
The problems posed by desertification, habitat loss, and species extinction are clearly global in scope. However, the solutions to these problems that have been proposed by researchers working within the confines of ecology and economics have generally been narrow in scope. Moreover, as Ludwig et al. (1993) and Holling (1995) have noted, the recent history of ecosystem management tells us that these uni-disciplinary solutions have not worked very well. Consequently, it is now necessary for scholarly research in ecological economics to explicitly recognize the facts that (1) ecological and economic systems are jointly determined, and (2) that if we are to
*Corresponding author. Tel.: 435-797-2314; fax: 435-797-2701.
E-mail address: [email protected] (A.A. Batabyal)
comprehend the functioning of such systems, then we must also understand the many and varied interdependencies between such systems.1
Once it is recognized that ecological-economic systems are jointly determined, it follows that these systems should be studied as one system.2 However, because this
recognition has been recent, a number of issues pertaining to the functioning of jointly determined ecosystems remain inadequately understood. As such, this article has two objectives. First, we use reliability theory to characterize and to bound the ecosystem stability property known as persistence. Next, we use this bound on persistence to study a species conservation problem in which society derives benefits from the persistence of the underlying ecosystem and from the pursuit of economic activities on this ecosystem. The health or the well being of an ecosystem can be described by a number of different concepts. In this article, we shall focus on the notion of persistence. Be-cause persistence refers to “how long a variable lasts before it is changed to another value . . .” (Pimm, 1991, p. 14), it is the appropriate concept to focus on whenever the longevity of the species in an ecosystem is salient. For instance, the work of Costanza et al. (1995) has informed us that in coastal and estuarine ecosystems, the health of the ecosystem depends on how long the composition of a small number of generalist species—that collectively determine the health of the ecosystem—lasts. As such, it should be clear to the reader that for such ecosystems, the notion of persistence provides us with an apposite measure of ecosystem health.
Despite the significance of the concept of persistence, there are very few studies of persistence in the economics literature. In particular, we are aware of only one article that has studied persistence, and linked persistence to the number of species in an ecosystem. Batabyal (1999c) has provided an explicit characterization of ecosystem persistence. However, in his modeling framework, Batabyal makes two key assumptions. First, he supposes that persistence depends only on the keystone3species of an
ecosys-tem. Second, he assumes that there are no interaction effects between the keystone species of an ecosystem. These assumptions detract from the generality of his analysis. Although Batabyal’s definition does provide a link between persistence and the number of species in an ecosystem, this definition does not account for the fact that there will generally exist interaction effects between the different species in an ecosystem.4 Given this state of affairs, in this article we use reliability theory5 to
provide a bound on the persistence of a stylized ecosystem. This bound explicitly accounts for interaction effects between the species of our stylized ecosystem. The rest of this article is organized as follows. Section 2 describes the theoretical framework and then computes an upper bound on ecosystem persistence. Section 3 uses this bound to study a species conservation problem in which society derives benefits from the persistence of the underlying ecosystem and from the pursuit of economic activities on this ecosystem. Finally, section 4 concludes and offers suggestions for future research.
2. Theoretical framework
2.1. Preliminaries
and fires result in shocks of varying magnitudes to this ecosystem. While these shocks are in general detrimental to the various species of the ecosystem, they need not have the same impact on all the species. For instance, a shock resulting from excessive hunting will generally affect an ecosystem quite differently than will a shock that results from a fire. The presence of these shocks means that the lifetimes of the different species in our ecosystem are random. Let us denote the lifetime of theith species, 1 # i # n by Li. Tilman (1996) has noted that as a result of interspecific
competition and other factors, one can generally expect there to be interaction effects between the various species of an ecosystem. To account for these interaction effects, we note that the shocks make the species lifetimes random variables; moreover, we suppose that these random variables, that is, the Li, are not necessarily statistically
independent.
There will generally be some substitutability between species in the performance of ecological functions. Hence, we will need to make an assumption about the degree of this substitutability. The cases of zero substitutability and partial substitutability have been studied in Batabyal (1998, 1999a). Consequently, in this article, we shall study the case of perfect substitutability. In other words, we shall say that our ecosystem is functional at timetif and only if at least one of thenspecies is alive. The reader should note the precise sense in which we are using the word “functional.” By functional, we are referring to the very minimal condition under which our ecosystem is able to provide a flow of economic services to society over time. We are not saying that our ecosystem’s persistence depends on the survival of a single species. In fact as we shall soon see, persistence depends on all the species of the ecosystem.6Let us now formally
describe and compute an upper bound on the persistence of our ecosystem.
2.2. Persistence
By virtue of the previous paragraph’s substitutability assumption, we can now provide an expression for the lifetime of our ecosystem. That expression is
ecosystem lifetime 5max(L1, . . . ,Ln). (1)
The persistence of our ecosystem is given by the expected lifetime of our ecosystem (i.e., by the expectation of the left hand side of Eq. (1)). To the best of our knowledge, for the case in which the Li are not necessarily statistically independent, an exact
expression forE[ecosystem lifetime] cannot be computed. Consequently, we now follow Ross (1997, pp. 509–510) and compute an upper bound for the persistence of our ecosystem. First note that
ecosystem lifetime #k 1
o
i5n
i51
(Li2k)1, (2)
wherekPR1is a constant andL15max(L,0). Our goal now is to computeE[
ecosys-tem lifetime]. Taking expectations of both sides of Eq. (2), we get
Persistence; E[ecosystem lifetime] #k 1
o
i5n
i51
E[(Li2k)1]. (3)
hand side of Eq. (3) can be simplified. This simplification yields
E[(Li2k)1] 5
#
∞
k Prob{Li.s}ds. (4)
Using Eq. (4), the upper bound for ecosystem persistence becomes
Persistence#k1
o
i5n
i51
#
∞
k Prob{Li. s}ds. (5)
Note that Eq. (5) holds for every value of the constant k. Consequently, to make the above bound as tight as possible, it will be necessary to choose k optimally. In other words, our goal now, shown in Eq. (6), is to solve
minkPR1[k1
o
i5ni51
#
∞k Prob{Lt. s}ds]. (6)
The first order necessary condition to this problem is7
o
i5n
i51
Prob{Li.k*}5 1. (7)
We can now rewrite Eq. (5). The best upper bound for the persistence of our ecosystem is given by
Eq. (8) is the upper bound for the persistence of our stylized ecosystem. Now let the indicator variableVi5 1 whenLi .k*, and let Vi5 0 when Li #k*. Then we
see thatE[Ri5n
i51Vi] 5Rii55n1Prob{Li.k*}51. This tells us that ifkis chosen optimally,
then the expected number of species lifetimes that will exceed k* equals one. This result is noteworthy because if in fact exactly one of theLiexceedsk*, then we can
replace the inequality in Eq. (8) with an equality. In other words, in some circumstances, the expression in Eq. (8) will give us an exact characterization of ecosystem persistence. In this article we have not focused on the computation of a lower bound for ecosystem persistence. However, it should be clear to the reader that we can always choose zero to be the lower bound on the persistence of our stylized ecosystem.
Eq. (8) tells us that the upper bound on persistence depends on the number of species in the ecosystem (n) and on the probabilities that the lifetimes of the various species in this ecosystem will exceed an optimally chosen constantk*. From this we conclude that if the number of ecosystem species can be counted and if the above probabilities can be determined, then we will be able to compute a numerical measure of persistence. Precise measures of ecosystem persistence will generally be difficult to estimate empirically. In such instances, numerical measures of persistence that are based on the best scientific evidence can be useful in the design of ecosystem manage-ment policies.
be very important to the functioning of an ecosystem. Consequently, before we end this section, let us briefly discuss the way in which our previous analysis can be modified to account for the keystone species of an ecosystem. Suppose that of then
species in an ecosystem,m—wherem PNand m,n—are keystone species. Then, one possible way to study ecosystem persistence would be to focus on themkeystone species and abstract away from the remaining (n 2 m) species. In our theoretical framework, this would involve replacingnwith min all the previous equations.
We now use the upper bound in Eq. (8) as a proxy for ecosystem persistence and we study an aspect of the optimal conservation of species. The reader should note that in our framework, a social planner/ecosystem manager takes the ecological and the economic aspects of the conservation question into account.
3. Ecosystem persistence and species conservation
In the past decade, the question of what to conserve has received a great deal of attention from ecologists and economists.8 Although these studies have certainly
furthered our understanding of the many and varied complexities of the conservation question, these studies have not considered the nexus between ecosystem persistence and species conservation. Consequently, we now study a simple model of optimal species conservation that incorporates the ecological and the economic dimensions of the question into the analysis.
Let us suppose that the ecosystem of section 2 provides ecological and economic benefits to society. The economic benefits include the flow of services provided by activities such as fishing, grazing, and hunting. Clearly, the continuance of these benefits depends, in part, on the persistence of the ecosystem. As well, society derives benefits from the existence of this ecosystem. To this end, letB[x→,P] denote society’s
benefit function. The vectorx→
5(x1, . . . ,xr) denotes therpossible economic activities
that society may engage in and P, the upper bound from Eq. (8), is our proxy for ecosystem persistence. We suppose that the r possible economic activities can be varied continuously, and that the benefit functionB[·,·] is concave and increasing in both its arguments. This means that increasing the level of economic activities and/ or ecosystem persistence raises social benefits, but at a decreasing rate.
Economic activities are costly to undertake and these activities have varied effects on thenspecies in our ecosystem. Consequently, there is a cost involved in conserving these species. LetC[x→,n] denote society’s cost function. In this formulation,C[x→,·] is
the cost of engaging in economic activities andC[·,n] is the cost of species conservation. We assume thatC[·,·] is convex and increasing in its arguments. In other words, an increase in either the level of an economic activity or the number of species will lead to higher costs, at an increasing rate.
Our social planner/ecosystem manager’s problem can now be stated. This individual solves
maxx→
,nB[x
→
,P] 2C[x→
The reader will note that Eq. (9) is a mixed integer programming problem. This is because→xis a continuous decision variable (by assumption) andnis an integer decision
variable.9 To apply the calculus to this problem, we shall interpret n as a rate of
species conservation and we shall suppose that there exists a continuous approximation ofPinn. Then the first order necessary conditions to Eq. (9) are10
]B[·,·] ]xq
5 ]C[·,·]
]xq
, q51, . . . , r, (10)
and
]B[·,·] ]P
]P
]n 5
]C[·,·]
]n . (11)
Note that the optimal values of ther 1 1 decision variables are given jointly by Eqs. (10) and (11). Eq. (10), the “economic” first order condition, says that each of the r economic activities should be pursued to the point where the marginal social benefit from this activity equals its marginal social cost. Of greater interest is Eq. (11), the “ecological” first order condition. This equation has implications for species conservation. The equation says that the social planner/ecosystem manager should conserve species at a rate such that the marginal social cost of conservation equals the marginal social benefit. Note that the marginal social benefit is the product of two terms. The first term captures the effect of a marginal increase in persistence on social benefit and the second term captures the effect of an incremental increase in the number of species on ecosystem persistence. The reader will note that because persistence is a function of, inter alia, the number of species in the ecosystem, it is possible to study the question of optimal species conservation in a way that considers the ecological and the economic aspects of the problem jointly.
It should be noted that our specification of society’s benefit and cost functions implies that the optimal values of thexq,q5 1, . . . ,r, depend on the optimal value
ofn, and vice versa. This means that the optimal level at which a particular economic activity should be carried out depends on the optimal rate of species conservation. In turn, the optimal rate of species conservation depends on the optimal levels of the
rpossible economic activities. As opposed to this, if the societal benefit and the cost functions were separable in their arguments, then this mutual dependence would disappear. However, because economic activities have varied effects on the species of an ecosystem and because these (generally) non-identical effects have different implications for the conservation of species, we believe that it is necessary to account for this dependence between the optimal levels of the various economic activities and the optimal rate of species conservation.
4. Conclusions
bound is a function of the number of species in the ecosystem, and on the probabilities that the various species lifetimes will exceed an endogenously chosen constant. We then used this upper bound to study the question of optimal species conservation. Our analysis shows that if the number of species in an ecosystem can be counted and if the relevant probabilities can be determined, then one can, in principle, use Eq. (8) to provide a very tight numerical upper bound on the persistence of an ecosystem. Ceteris paribus, very persistent ecosystems are more able to withstand the deleteri-ous effects of shocks stemming from natural events and from the conduct of economic activities. As such, it seems sensible to suppose that for such ecosystems, social plan-ners/ecosystem managers should be less concerned about the effects of expansions in economic activities on ecosystem health. In other words, rigid restrictions on resource development in very persistent ecosystems may impose unnecessary costs on the regional sectors of an economy. As opposed to this, a great deal of care must be exercised when dealing with ecosystems whose persistence is low. This is because in such ecosystems, the shocks from increases in economic activity may prevent ecosystem species from performing salient ecological functions.
The analysis of this article can be generalized in a number of different directions. In what follows, we suggest two possible generalizations. First, the discussion of persistence provided in this article did not allow for the possibility that society may take steps to ensure that endangered species become less endangered over time. Examples of such steps include moratoriums on fishing in aquatic ecosystems, bans on grazing in public lands, and the cessation of logging in forest ecosystems. Second, this article’s discussion of the species conservation problem did not incorporate into the analysis the effects that specific constraints—such as the existence of limited conservation budgets—might have on the optimal levels of the various economic activities and on the optimal rate of species conservation. Consequently, formal studies of persistence and species conservation that incorporate these aspects of the problem into the analysis will provide richer and more realistic characterizations of persistence, and permit more elaborate analyses of the connections between species conservation and ecosystem persistence.
Acknowledgments
We thank Hamid Beladi and two anonymous referees for their helpful comments on a previous version of this article. We acknowledge financial support from the Faculty Research Grant program at Utah State University and from the Utah Agricultural Experiment Station, Utah State University, Logan, UT 84322-4810, by way of grant UTA 024. The usual disclaimer applies.
Notes
2. For more on this line of thinking, see Batabyal (1998, 1999a, 1999b), Perrings et al. (1995), and Levin et al. (1998).
3. The structure and the dynamics of an ecosystem often depend on the existence of certain key species. Hence, these “keystone species” are very important to the functioning of the ecosystem. The term “keystone species” is due to Paine (1966).
4. For more on this, see Tilman (1996).
5. For more on reliability theory, see Gertsbakh (1989), Kovalenko et al. (1997), and Ross (1997).
6. The reader will note that in this article we have not explicitly modeled changes in the populations of the existing species. However, it should be noted that conservation policy, which we study in section 3, is generally concerned with individual species and with the populations of these species.
7. We assume that the second order condition is satisfied.
8. See Peters et al. (1989), Balick and Mendelsohn (1992), Weitzman (1993), and Simpson and Sedjo (1996).
9. For other examples of such problems, see Batabyal (1996). 10. We assume that the second order conditions are satisfied.
References
Balick, M., & Mendelsohn, R. O. (1992). Assessing the economic value of traditional medicines from tropical rain forests.Conservation Biology 6, 128–130.
Batabyal, A. A. (1996). The queuing theoretic approach to groundwater management,Ecological Model-ling 85, 219–227.
Batabyal, A. A. (1998). An ecological economic perspective on resilience and the conservation of species.
Unpublished manuscript, Utah State University.
Batabyal, A. A. (1999a). Species substitutability, resilience, and the optimal management of ecological-economic systems.Mathematical and Computer Modelling 29, 35–43.
Batabyal, A. A. (1999b). Aspects of the optimal management of cyclical ecological-economic systems.
Ecological Economics 30, 285–292.
Batabyal, A. A. (1999c). An analysis of persistence, resilience and the conservation of keystone species.
Unpublished manuscript, Utah State University.
Costanza, R., Kemp, M., & Boynton, W. (1995). Scale and biodiversity in coastal and estuarine ecosystems. In C. Perrings, K. Maler, C. Folke, C. S. Holling, & B. Jansson (Eds.),Biodiversity Loss: Economic and Ecological Issues(pp. 84–125). Cambridge, UK: Cambridge University Press.
Gertsbakh, I. B. (1989).Statistical Reliability Theory.New York: Marcel Dekker.
Holling, C. S. (1995). What barriers? What bridges? In L. H. Gunderson, C. S. Holling, & S. S. Light (Eds.),Barriers and Bridges to the Renewal of Ecosystems and Institutions(pp. 3–34). New York: Columbia University Press.
Kovalenko, I. N., Kuznetsov, N. Y., & Pegg, P.A. (1997).Mathematical Theory of Reliability of Time Dependent Systems with Practical Applications.New York: Wiley.
Levin, S., Barrett, S., Aniyar, S., Baumol, W., Bliss, C., Bolin, B., Dasgupta, P., Ehrlich, P., Folke, C., Gren, I., Holling, C. S., Jansson, A., Jansson, B., Maler, K., Martin, D., Perrings, C., & Sheshinski, E. (1998). Resilience in natural and socioeconomic systems.Environment and Development Economics 3, 222–235.
Paine, R. T. (1966). Food web complexity and species diversity.American Naturalist 100, 65–75. Perrings, C., Maler, K., Folke, C., Holling, C. S., & Jansson, B. (Eds.). (1995).Biodiversity Loss: Economic
and Ecological Issues.Cambridge, UK: Cambridge University Press.
Peters, C., Gentry, A., & Mendelsohn, R. O. (1989). Valuation of an Amazonian rainforest.Nature 339, 655–656.
Pimm, S. L. (1991).The Balance of Nature?Chicago: University of Chicago Press.
Ross, S. M. (1997).Introduction to Probability Models, 6thedition. San Diego: Academic Press.
Simpson, D., & Sedjo, R. (1996). Paying for the conservation of endangered ecosystems: a comparison of direct and indirect approaches.Environment and Development Economics 1, 241–258.
Tilman, D. (1996). Biodiversity: population versus ecosystem stability.Ecology 77, 350–363.
Weitzman, M. (1993). What to preserve? An application of diversity theory to crane conservation.
