Contribution values of biodiversity to ecosystem performances: A viability perspective

E CO L O G I CA L E CO N O MI CS 68 ( 20 0 8 ) 1 4–2 3

a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m

w w w. e l s e v i e r. c o m / l o c a t e / e c o l e c o n

METHODS

Contribution values of biodiversity to ecosystem performances: A viability perspective C. Béné a , L. Doyen b,⁎ a b

WorldFish Center, Consultative Group on International Agricultural Research (CGIAR), Cairo Egypt CNRS, CERSP, Dpt Ecologie et Gestion de la Biodiversité, MNHN, Paris, France

AR TIC LE D ATA

ABSTR ACT

Article history:

This paper deals with the contribution value of biodiversity to ecosystem performances

Received 6 November 2007

under a viability approach. Two contrasted cases are considered. First, a no-exploitation

Received in revised form 20 May 2008

situation where the ecosystem performances are measured through the capacity of species

Accepted 15 August 2008

richness to maintain the ecosystem above a minimum ecological viability threshold

Available online 3 October 2008

measured through a Shannon index. Second, an exploitation situation where the performances of the ecosystem are measured through its economic sustainability, that is,

Keywords:

its capacity to generate direct-use values greater than a minimum guaranteed utility level.

Biodiversity

The analysis, based on numerical simulations, shows that biodiversity has a positive effect

Contribution value

on both ecological and economic performances and that in both cases the marginal

Stochastic viability

contribution of biodiversity is positive. Furthermore, this marginal contribution exhibits

Resource-based model

maximum values. These maximum values seem however to decrease with the level of species richness. These results show interesting links with two of the main current debates on biodiversity: the Noah's Ark problem and the assumption of decreasing marginal value supported by some recent works on bio-prospecting. © 2008 Published by Elsevier B.V.

1.

Introduction

Measuring and valuing biodiversity is a challenging issue. The problem is first complicated by the wide spectrum of biotic scales at which biodiversity operates: genetic diversity within species, species diversity between species and ecological diversity at community or ecosystem level. Even for a given level of diversity, say species, no consensus seems to emerge about the way(s) to define, and subsequently, to measure biodiversity. Consequently a lot of different measurement indicators have been proposed, but those do not always provide consistent or comparable results, making any general interpretations somewhat difficult.

The complexity and uncertainty underlying the functioning of biodiversity further contribute to the difficulty of the assessment. On the ecological side, the scientific knowledge on relationships and interdependencies among species is still relatively incomplete and the effects of biodiversity on the productivity, stability and sustainability of ecosystems are still actively investigated (Tilman et al., 1997; Hooper and Vitousek, 1997; Borrvall et al., 2000; Tilman et al., 2005). Among some insights, the so-called insurance hypothesis (Kinzig et al., 2002; Loreau et al., 2002; Yachi and Loreau, 1999) emphasises that biodiversity may decrease the variability of ecological productivity in a significant way and that it may therefore be extremely desirable to maintain small but viable samples of

⁎ Corresponding author. E-mail addresses: [email protected] (C. Béné), [email protected] (L. Doyen). 0921-8009/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.ecolecon.2008.08.015

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genetically diversified alternatives crops as insurance against catastrophic extinction shocks. The ecological difficulty to evaluate biodiversity is further complicated by our present economic inability to list the detailed roles and economic contributions of living organisms and in particular to predict their potentialities to produce specific goods and services of interest for society in the future (Pagiola et al., 2004). The concept of total economic value as developed in the literature and adopted by the united nations (Moran and Bann, 2000; MEA. 2005) was a first attempt to capture the whole set of various services supplied by ecosystems through the distinction of direct and indirect-use values, non-use values and options values. At the present time however, this whole exercise still remains highly extrapolative as ecosystems are recognized to provide a high variety of different services (biomass production, water run-off prevention, control of pests and diseases, nitrogen and carbon fixation, soil regeneration, etc). In these conditions it is not surprising that many different economic frameworks have been proposed to assess and measure biodiversity. Focusing on non-use values, Weitzman (1992) or Nehring and Puppe (2004) for instance propose to quantify biodiversity through a “diversity function” defined as a measure of genetic differences (distances) between species. Using this diversity function, Weitzman (1992) discusses the so-called “Noah's Ark problem” (Weitzman, 1998; Metrick and Weitzman, 1998) -a parable where the central issues is to develop a cost-effectiveness formula or criterion that can be used to identify optimal biodiversity-preserving strategies under limited budget. Coming from a slightly different angle, Brock and Xepapadeas (2002) argue that the basic principal of valuing biodiversity should be the association of diversity with some directuse values of ecosystem. These authors therefore propose to establish a relationship between a given biodiversity metric (e.g. species richness) and the value of intrinsic characteristics such as productivity (for the more tangible or measurable services), to aesthetic satisfaction, existence value or bequest motives (for the less tangible). Under this approach biodiversity is valued through the change in the ecosystem's characteristics or services that it provides with respect to the biodiversity metric considered. In a somewhat related vein, the positive effects of diversity on the variability of ecosystem services in agriculture or medicine (Swanson and Goeschl, 2003) has led economists to suggest that biodiversity may be seen as a form of insurance against ecological risks and uncertainties (Perrings, 1995; Borrvall et al., 2000; Weitzman, 2000). Like investors in financial markets diversify their asset portfolio to reduce risks, farmers traditionally grow a variety of crops in order to decrease the adverse impact of uncertain environment and market fluctuations (Di Falco and Perrings, 2005; Tichit et al., 2004). Such a concept has deep connections with the insurance hypothesis mentioned above. Option values of biodiversity constitute another important part of the literature. In this approach, biodiversity pricing relies on the probability of making a future commercial discovery. Such a framework aims at linking diversity with measures of economic value through the concept of ’biodiversity prospecting’, that is, the search among naturally occurring organisms for new products of agricultural, indus-

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trial, and particularly pharmaceutical value (Principe, 1989; Pearce and Purushothaman, 1992; Mendelsohn and Balick, 1995; Polasky and Solow, 1995; Simpson et al., 1996). Under this approach, a species value depends not only on the probability that it eventually yields a successful commercial product, but also on the likelihood that other species can, or cannot, lead to the development of this product. This approach points out the specific issue of species substitutability. If species are perfect substitutes, all species but the species in use are valueless because they are redundant. Investigating this hypothesis through various economic models, Simpson confirms that in a deterministic framework the value of the marginal species decreases with the number of available species (Simpson et al., 1996; Craft and Simpson, 2001). This apparent negative effect of substitutability on the marginal contribution of species to biodiversity is critical since, if it were confirmed, it would substantially weaken the incentive for biodiversity conservation. Kassar and Lasserre (2004) challenged this hypothesis. Using a real-option framework and assuming irreversible biodiversity loss, they show that while under certainty their model does confirm Simpson's result, under uncertainty in contrary the flexibility associated with substitutability can be a source of value. In particular, assuming uncertainty about the species which is supposed to produce the good(s) and service(s) of interest in the future, they show that there is a justification for preserving an otherwise useless species because the latter may become the species of choice in the future. In such circumstances, a species becomes valuable precisely because it is substitutable. In sum, while many of these different studies have offered insightful analyses and contributed greatly to advance our understanding about biodiversity measure, it seems that no consensus has yet been reached on the way biodiversity measure should be effectively operationalized. In effect, one of the main issue so far has been the failure to develop frameworks that can accommodate simultaneously ecological and economic objectives — for instance, maximizing species conservation using biodiversity metrics (e.g. species richness or Shannon index) while, at the same time, ensuring societal welfare measured through, e.g., utility function metrics. The present article is in direct line with these different issues. In particular, it is intended to revisit some of the challenging questions related to the measurement of biodiversity and the relationships that is thought to exist between ecosystem performance and biodiversity. For this, we develop a theoretical model of ecosystem represented by a resourcebased model (Tilman, 1982) where n species interact with an evolving resource. In this system, we aim at quantifying the biodiversity value of these n species in two distinct situations. First, in a non-exploited ecosystem situation, the value of biodiversity is quantified by the probability of the species richness to maintain the ecosystem's number of species above a given ecological threshold — measured by its Shannon index. The marginal contribution of species richness is then defined as the marginal change that occurs in this probability when an additional species is added to the system. Second, we consider the case where the ecosystem is exploited and generates direct-use products. In this second situation, the value of biodiversity is measured by the

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probability that the utility derived from the direct-use products remains above a minimum guaranteed utility threshold, and the marginal contribution of species is now measured as the changes in this probability brought by the introduction of an additional species in the system. To keep the mathematical content of the paper as minimal as possible the analyses are made through numerical simulations instead of mathematical proofs and the results are presented in the form of conjectures.

2. A resource-based dynamics for the ecosystem The model used in this analysis to represent n species interacting in an ecosystem En is the Tilman resource-based model (Tilman, 1982). Resource-based models have been recognized recently to be an interesting alternative to the classical Lokta–Volterra model to represent species interaction in an ecosystem. In this model, the species growth is limited by the availability of a resource.1 The resource-based model thus represents a community of species in dynamical interaction in their habitat. Let i = 1,…, n denote these n species and let us assume that all species affect each other through their individual effects (consumption) on the availability of the limiting resource. At each time t, Ni(t) denotes the abundance of species i and R(t) denotes the resource level. The dynamics of these species in competition for the resource can be represented by a system of discrete dynamic equations defined for t ∈ {0,1,…,T} such that 8 > ! i ¼ 1; N ; n < Ni ðt þ 1Þ ¼ Ni ðtÞ þ Δt ðgi RðtÞ  di  ei Þ; n X wi gi RðtÞNi ðtÞ > : Rðt þ 1Þ ¼ RðtÞ þ Δt K  aRðtÞ  i¼1

ð1Þ where (N0,R0) are the initial conditions (t = 0) and Δt is the time step of the dynamics. The first equation in system (1) describes the net growth rate of individual species' biomass. Here, giR represents the per capita resource-related growth of species i, while di is the natural death rate and ei is the exploitation rate (e.g. harvesting intensity) on species i. The second equation corresponds to the dynamics of the resource where K-aR stands for the natural evolution of the resource with no consumption by the n species.2 The last term on the right-hand side of the equation reflects the consumption of the resource by the n species, with wi being a constant reflecting the species i's specific rate of consumption. Tilman (1988) has shown that in the long-term, resourcebased models are characterized by an ’exclusion principle’. This principle (the Tilman's rule) states that in the context of a multi-species competition for a limiting factor, under environmental homogeneity, the species with the lowest resource

1 This resource can be thought of, for instance, as an index of sol quality, or subtract availability. 2 With no species the resource’s equilibrium level determined by R(t + 1) = R(t), is R⁎ = K/a.

requirement in equilibrium will competitively displace all the other species.3 Under environmental heterogeneity, however, – i.e. for temporal and/or spatial variation of characteristics of the environment (the resource) or the species, e.g. varying death rate or different colonization (dispersal) rates – the natural exclusion principle is not incompatible with biodiversity preservation (Tilman et al., 2005). In our case, we use nonasymptotic reasoning and short-term dynamics to circumvent the effect of the exclusion principle, thus maintaining biodiversity in the system despite the use of the resource-based framework (see the discussion section for further elaboration on this).

3.

Case 1: Biodiversity without exploitation

In this first case, we assume no exploitation, i.e. ei = 0 for all species and we propose to investigate the contribution of biodiversity to ecosystem viability. The term viability is used here in reference to a series of recent works looking at issues of sustainability and natural resources management where the focus is on identifying the desirable ”space” within which dynamic systems remain viable4 at any time (Cury et al., 2005; Eisenack et al., 2006; DeLara et al., 2007; Doyen et al., 2007; Tichit et al., 2007). From an ecological viewpoint, the so-called population viability analysis (PVA) (Morris, and Doak, 2003) is remarkably close to this viability approach as it focuses on extinction probability in an uncertain (stochastic) environment. The Tolerable Windows Approach (TWA) proposes a similar framework on climatic change issues (Bruckner et al., 1999). In the environmental context, viability analysis has been shown to offer a promising framework to reconciliate economic and conservation constraints as in Béné et al. (2001). In the rest of this section and the subsequent one we present the details of the formal analysis through a series of definitions and conjectures. Readers interested only in the implications of those results can skip this part and resume their readings in Section 5.

3.1.

Ecosystem viability and biodiversity value

In principle, the basic unit of biodiversity can be the molecule, cell, organ, individual, species, habitat, or ecosystem. For the

3

In such a case, the system tends toward the equilibrium:

8 d þ ei di4 þ ei4 > > ¼ R4 ¼ mini¼1; N ;n i > > gi gi 4 < 8 4 K  R a < 4 > 4 if i ¼ i > > N ¼ R4 wi gi > : i : 0 if ipi4 : 4

A system is said to be viable if its trajectories satisfy the set of viability conditions that are associated to its dynamics. For instance in the case of a system representing species abundances over time N(t), one direct viability constraint is that these abundances remain positive at any time, i.e., N(t) ≥ 0,∀t. More generally, this approach turns out to be very close to the maximin or Rawlsian approach (Heal, 1998) and presents some direct links with the concept of intergenerational equity (Martinet and Doyen, 2007).

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purpose of this paper, we choose the species to be the underlying unit of analysis and we use the Shannon index S (t) as a measure of species biodiversity, namely SðNðtÞÞ ¼ 

X

Fi ðtÞlogFi ðtÞ

i

where Fi ðtÞ ¼ PNi ðtÞ is the frequency of population i. i

Ni ðtÞ

We are first interested in measuring the contribution of species richness in maintaining ecosystem biodiversity. For this, we define the ecosystem's ecological viability as the conditions under which a given ecosystem's biodiversity level remains above a guaranteed viability level5 Slim. In other words, the ecosystem En is said to be ecologically viable if its Shannon index S(N(t)) remains greater or equal to Slim, namely: En is ecologically viable if SðNðtÞÞzSlim ;

t ¼ 0; N ; T:

We then use this definition to quantify formally the value of biodiversity through the calculation of viability indexes PS(n,Slim) in a probabilistic framework. These viability indexes represent the probability that En remains ecologically viable under uncertainty. They are computed for each level of biodiversity n through simulations. More formally the biodiversity values is thus defined as follows: Definition 1. Consider a probability P on a set of scenario Ω. Assume that the parameters of ecosystem En = (g,d,w,K,a) defined in system (1) is a random vector on Ω. Assume that the initial conditions N0∈Rn and R0∈R are also random variables on Ω. The biodiversity value of ecosystem En is the probability of ecological viability of En with respect to the minimal guaranteed biodiversity Slim and species richness n, namely  PS ðn; Slim Þ ¼ PðEn ;N0 ;R0 Þ

 min SðNðtÞÞzSlim

t¼0; N ;T

where (N(t),R(t)) evolves according to the dynamics (1) from initial state (N0,R0) at time t = 0. Note that, following Definition 1, the contribution of biodiversity to ecosystem viability is defined intrinsically. This is close to the ideas of direct diversity measure developed for instance in Weitzman (1992), Nehring and Puppe (2002, 2004) and differs from the approach developed in the ecological economics literature where biodiversity is usually defined exogenously through direct-use value.

3.2.

Fig. 1 – Ecological viability: probability PS(n,S) as a function of minimum guaranteed level S and species richness n.

Marginal contribution of species to biodiversity

The concept of biodiversity value as proposed in Definition 1 allows us to define the marginal contribution of species richness as being the marginal change in viability index PS induced by a change in species richness n. More formally, Definition 2. Consider any guaranteed biodiversity level Slim. The marginal contribution CS of species richness n (n ≥ 2) to 5 This guaranteed level Slim can be seen as the minimum biodiversity level necessary to maintain the ecosystem’s services or functionalities. The notation “lim” refers to the ICES precautionary approach for fisheries where a critical biomass Blim is usually assumed to exist, under which the viability of the stock is not maintained.

ecosystem viability with respect to Shanon value Slim is defined by CS ðn; Slim Þ ¼ PS ðn; Slim Þ  PS ðn  1; Slim Þ: As will be discussed in greater detail below, this concept of marginal contribution appears to be very useful in relation to the Noah's ark problem. In particular we will show that it can be used as a potential indicator to rank species according to their contribution to biodiversity conservation.

3.3.

A biodiversity analysis

Once the species marginal contribution CS is defined, we can analyze the relationship between this marginal contribution and ecosystem viability through numerical simulations. The viability indexes PS(n, Slim) and marginal contributions CS(n, Slim) are computerized for different numbers of species n ≤ 150 and minimum guaranteed viability levels Slim, and 10,000 simulations are run over a time horizon T = 2 and for Δt = 0.01 small enough. Using this short time-frame T = 2 greatly reduces dimensionality problems in the computation process. It also allows us to focus on the system's transitional dynamics instead of its asymptotic behavior, thus reducing the potential effect of the exclusion principle embedded into the Tilman resource-based model.6 For each simulation, the initial conditions N0 and R0 and the system's parameters (g,d, w,K,a) are chosen randomly within [0,1] using two types of distribution function: the beta and the uniform distributions. These two distinct functions were initially tested in order to investigate whether different levels/forms of uncertainty

6 Additionally, setting Δt small enough eliminates the risk of violent changes within the dynamics, thus avoiding positivity problems for species’ abundances Ni(t) and resource level R(t). The choice of a small Δt also reinforces our decision to work with T = 2, as it reduces the potential effect of the exclusion principle. Preliminary analysis had showed that the model is relatively sensitive to Δt. In particular, for values of Δt N 0.75 the exclusion principle dominates the dynamics of the system (even with short time-frame T = 2), thus reducing the normative purpose of the simulation exercise. To avoid this effect we set Δt = 0.01 in the rest of this paper.

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generate different responses from the system. Only results obtained with the uniform distribution are presented in the rest of this analysis, as no tangible differences were observed between the two distributions. The results of the simulations are presented in Figs. 1 and 2. Fig. 1 displays the viability indexes PS(n,Slim) as a function of the minimal guaranteed biodiversity level Slim for levels of species richness varying from 1 to 150. Fig. 2 displays the corresponding marginal contributions CS(n,Slim) for the same parameter range. The two figures highlight three important results.

ginal contribution strictly decreases with the level of species richness n, in the sense that: C4S ðn þ 1Þ b1: C4S ðnÞ

3.3.3.

Increasing maximum guaranteed biodiversity

Consider now the optimal biodiversity level S⁎(n) associated to the maximal marginal contribution CS⁎(n). Given any species richness n, this optimal Shannon measure is defined by: S4 ðnÞa Argmax CS ðn; Slim Þ: Slim z0

3.3.1.

Species richness favors ecosystem viability

First Fig. 1 shows that for any guaranteed level Slim the viability index PS(n,Slim) increases with species richness n, indicating that the probability of the ecosystem En to remain viable increases with the number of species n present in En. This result suggests that species richness promotes ecosystem viability. A direct corollary of this is that the species marginal contributions CS(n,Slim) are positive or nil for all n and Slim values. This is confirmed in Fig. 2. Based on these first results, we propose a first conjecture: Conjecture 1. For Δ t small enough, ecosystem viability increases with species richness, and the marginal contribution of biodiversity is positive or nil for all n, in the sense that CS ðn; Slim Þz0;

3.3.2.

8nz2;

8Slim N0:

Decreasing marginal species contribution

Fig. 2 also reveals that the marginal contribution CS(n,Slim) displays a maximum value for every richness n: C4S ðnÞ ¼ max CS ðn; Slim Þ: Slim z0

Note however that this maximum value CS⁎(n) tends to decrease strictly with n, suggesting a decreasing marginal contribution of species richness to ecological viability. Based on these various findings we propose a second conjecture: Conjecture 2. For Δt small enough, the marginal contribution of biodiversity to ecological viability displays a maximum value for any given species richness n. This maximum mar-

This value S⁎(n) represents the guaranteed biodiversity at which the marginal contribution is maximum for a given level of species richness n. This value S⁎(n) is shown in Fig. 3 for levels of species richness ranging from 2 to 150. The figure shows that S⁎(n) increases with n following a concave pattern. This leads to the third conjecture of this analysis: Conjecture 3. For Δt small enough, the maximal guaranteed biodiversity level S⁎ increases with the level of species richness n, in the sense that: ST ðn þ 1Þ N1: ST ðnÞ This conjecture means that the maximum guaranteed level of biodiversity that can be reached in an ecosystem with n species in interaction increases with the number of species n. Furthermore, for a given biodiversity requirement Slim, let us consider the optimal species richness n⁎(Slim) defined by: nT ðSlim ÞaArg max CS ðn; Slim Þ: nz2

n⁎(Slim) represents the level of species richness at which the marginal contribution of biodiversity is maximal7 for a given Slim. This optimal richness level n⁎(Slim) is shown in Fig. 4. The figure indicates that n⁎(Slim) increases with the guaranteed viability levels Slim and that this increasing trend displays a convex shape. The implications of these different results will be discussed in detail in the last section of the paper.

4.

Case 2: Biodiversity under direct-use values

In this second section, we now propose to extend the analysis by looking at the case of an exploited ecosystem. This means that the assumption ei = 0 is now released in system (1), and that a certain level of outputs (or catches) H(t) is now generated. Following the conventional approach, we assume a proportional relation between these outputs H(t) and the extraction rate e, in the sense that Hi ðtÞ ¼ ei Ni ðtÞ: We then adopt a direct-use valuation framework so that the total utility derived from the exploitation of the ecosystem Fig. 2 – Ecological viability: marginal contribution CS (n,S) as a function of minimum guaranteed level S and species richness n.

7

More precisely, the maximal species richness n⁎(S) corresponds to the inverse of maximal biodiversity S⁎(n) in the sense that S⁎(n⁎(S)) = S.

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(H(t)) derived from En's direct-uses remains above the guaranteed utility Ulim: Definition 3. Consider a probability P on a set of scenario Ω. Assume that the parameters of the ecosystem En = (g,d,w,K,a) defined in system (1) is a random vector on Ω. Assume further that initial conditions N0 and R0 and the economic parameters α ∈ Rn are also defined randomly. The economic sustainability of ecosystem En is measured through the probability that the ecosystem En remains sustainable with respect to the minimal guaranteed utility Ulim, namely  PU ðn; Ulim Þ ¼ PðEn ;N0 ;R0 ;aÞ

4.2. Fig. 3 – The maximal biodiversity S(n) as a function of species richness n.

En is defined by the constant elasticity substitution (CES) function: UðH1 ðtÞ; H2 ðtÞ; N ; Hn ðtÞÞ ¼

n X

!1 ai Hbi

b

with bN0:

i¼1

This CES function is a generalised form of utility function. It includes in particular the linear case (β = 1) and the Cobb Douglass case (β → + ∞).

 min UðHðtÞÞzUlim :

t¼0; N ;T

Marginal contribution of species to utility

Drawing upon Definition 3 above, we define the marginal contribution of utility of the biodiversity level n as follows: Definition 4. Consider any guaranteed utility Ulim. The marginal contribution CU (n,Ulim) of biodiversity level n (n ≥ 2) with respect to Ulim is defined by CU ðn; Ulim Þ ¼ PU ðn; Ulim Þ  PU ðn  1; Ulim Þ:

4.3.

Biodiversity analysis

When accounting for uncertainty, the economic sustainability of ecosystem En is the probability that the total utility U

Sustainability indexes PU(n,Ulim) and marginal contributions CU(n,Ulim) of En are computed through numerical simulations for different levels of species richness n ≤ 150 and minimum guaranteed utility levels Ulim. For each simulation, as in the previous section, the initial conditions N0 and R0 and the system's parameters (g,d,w,K,a) are chosen randomly within [0,1]. Additional economic parameter αi (which would correspond to the prices in the case of a linear CES function) are also chosen randomly within [0,1]. We use the intermediate case β = 2 to run the simulations -shown in Figs. 5 and 6. Fig. 5 shows the sustainability indexes PU(n,Ulim) as a function of the species richness n and the minimal guaranteed utility level Ulim. Fig. 6 displays the associated marginal contributions CU(n,Ulim). Finally, to explore the sensibility of those results to the CES parameter β, Fig. 7 shows the maximal utility U⁎(n) in 3 cases (β = 1, β = 2 and β = 5). The different results

Fig. 4 – The maximal species richness n(S) as a function of biodiversity minimum guaranteed level S.

Fig. 5 – Economic sustainability: probability PU (n,U) as a function of utility of catch U and species richness n for a CES utility function β = 2.

4.1.

Ecosystem sustainability under exploitation

We now look at the economic sustainability of ecosystem En. Following an approach comparable to that adopted in the previous section, we propose to define the economic sustainability of the ecosystem En by the existence of a minimum guaranteed threshold measured in terms of utility Ulim under which the ecosystem En is said to be not economically viable, namely: En is economically sustainable if UðHðtÞÞzUlim ; t ¼ 0; N ; T

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U⁎(n) represents the guaranteed utility at which the marginal contribution is maximum for a given level of species richness n. When computed, this maximum guaranteed utility U⁎ appears to increase with the level of species richness (Fig. 7). Furthermore, over the range of values considered in these simulations (2 ≤ n ≤ 150 and β = 1;2;5) the increment in U⁎(n) seems concave for β N 1. We propose below the last conjecture of this analysis:

Fig. 6 – Economic sustainability: marginal contribution CU (n, U) as a function of utility of catch U and species richness n for a CES utility function β = 2.

of those simulations suggest a series of conjectures which are presented below.

4.3.1.

Species richness strengthens economic sustainability

Fig. 5 shows that for any given minimal guaranteed level Ulim the sustainability index PU(n,Ulim) increases with n, suggesting that species richness promotes economic sustainability. Fig. 6 confirms that the marginal contributions CU(n,Ulim) of the species is positive or nil for any combination (n,Ulim). Conjecture 4. Ecosystem sustainability increases with species richness, and the marginal contribution of biodiversity is positive or nil for all n, in the sense that CU ðn; Ulim Þz0;

4.3.2.

8nz2;

8Ulim N0:

Decreasing marginal contribution

Fig. 6 also shows that the marginal contribution of biodiversity CU(n,Ulim) is characterized by a maximum value with respect to guaranteed utility Ulim for any level of species richness n: CTU ðnÞ ¼ max CU ðn; Ulim Þ: Ulim z0

This maximum value, however, decreases with n, suggesting a decreasing marginal contribution of species richness to economic sustainability. Conjecture 5. The marginal contribution of biodiversity CU(n, Ulim) exhibits a maximum value CU⁎(n) with respect to guaranteed utility Ulim. This maximum contribution decreases with the species richness level n, in the sense that: CTU ðn þ 1Þ b1: CTU ðnÞ

4.3.3.

Increasing minimum guaranteed utility

Drawing upon Conjecture 5, one can compute the maximum guaranteed utility U⁎(n) defined as: T

U ðnÞa Argmax CU ðn; Ulim Þ: Ulim z0

Fig. 7 – The maximal guaranteed utility U(n) as a function of species richness n for three distinct values of parameter β for CES utility function U (a) β = 1 (linear), (b) β = 2 and (c) β = 5.

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Conjecture 6. The maximum guaranteed utility level U⁎ increases with the species richness in the sense that UT ðn þ 1Þ z1: UT ðnÞ Like for n⁎(S) in the ecological case, a maximal richness n⁎(U) could be computed as the inverse function of the maximal utility U⁎(n). As an inverse function, this maximal richness n⁎(U) would display an increasing trend (convex for β N 1).

5.

Discussion

This paper contributes to the ongoing debate on how biodiversity can be measured and valued in relation to ecosystem performances and services. Drawing upon Brock and Xepapadeas (2003), Nehring and Puppe (2002) and others, we adopt the view that the basic principle of valuing biodiversity should be through the association of diversity with some useful characteristics that ecosystem possesses -such as productivity, utility of direct uses or existence satisfaction. The idea is to establish a relationship between a biodiversity metric (for instance ecosystem's species richness) and the ecosystem's characteristics under consideration. Once this relationship is established, changes in the characteristics (e.g. change in the productivity or in the market values of service) can be associated with changes in biodiversity metrics — thus revealing the value of biodiversity. In our case, we propose to value biodiversity through its capacity to maintain ecosystem performances above a guaranteed level, adopting a stochastic viability approach. To conduct this valuation exercise, we use the Tilman's resourcebased model describing an ecosystem as a community of n species competing for a limiting resource in an ecosystem. Two cases are considered. In the first part of the paper, we consider a non-exploitation situation where the ecosystem performances are assessed through the capacity of species richness in maintaining the ecosystem above an ecological viability threshold Slim. In the second part of the paper, the ecosystem is then assumed to be exploited and its performances are measured through its economic sustainability, that is, its capacity to generate direct-use values greater than a minimum utility level Ulim. In both cases, a probabilistic framework is adopted and performances of the system are examined through numerical simulations. The analysis of the numerical simulations suggests that biodiversity promotes conjointly the ecological and economic performances of the ecosystem as both ecological viability and economic sustainability increase with species richness (Conjectures 1 and 4). These results add pertinent elements to the current literature on biodiversity as they confirm the positive effect of biodiversity on ecosystem performances. While this conclusion in itself is not totally new (see e.g. Kinzig et al., 2002; Loreau et al., 2002; Tilman et al., 2005), the innovative part of our work comes from the fact that these conclusions are observed in parallel for ecological and economic criteria, using one single analytical framework. In both cases, the biodiversity value is measured through the capacity of species in maintaining the ecosystem performance above a minimum

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guaranteed level, taking into account uncertainty. In both situations, the behavior of the model remains qualitatively the same. Increasing species richness n also increases the performances of the system. The analysis is then extended by analyzing the marginal contribution of species richness to these ecosystem performances. For this, we define the marginal contribution of biodiversity as being the change in the index considered (ecosystem's viability PS or sustainability PU) induced by a change in the species richness n. Other authors have proposed to look at this concept of marginal contribution of species through different but comparable approaches. Weitzman (1992) for instance defines the value of marginal diversity as the value of improving survival probabilities for different species. The marginal species' value proposed by Kassar and Lasserre (2004) is also related to this approach except that the focus is on the ’survival’ of the commodity or service provided by the species rather than the survival of the species itself. The concept of marginal contribution proposed here can accommodate both dimensions: survival of the ecosystem (ecological viability) and ’survival’ of the direct-use product derived from this ecosystem (economic sustainability). In both cases, the marginal contribution of species is positive and displays some maximum values (Conjectures 2 and 5). The existence of these maximum values offers interesting links with some ongoing discussions on biodiversity conservation and, in particular, the “Noah's Ark problem” (Weitzman, 1998; Metrick and Weitzman, 1998). In economic terms, one of the main issues for conservation projects is to develop cost-effectiveness criteria that can help identify the “optimal” level of biodiversity to conserve under budget constraint. In the Noah Ark's parable, the question becomes: how big the ark should be and what criterion should Noah use to determine the optimum number of species to be brought on board? We argue that the maximal marginal contribution n⁎(S) as identified in our analysis could be a relevant choice. In the case of non-exploited ecosystem, the n⁎(S) (or similarly n⁎(U) in the case of exploited ecosystem) indicates the level of species richness at which the contribution of an additional species in enhancing the ecosystem's ’survival’ is optimal. For any level of species richness lower than n⁎(S), boarding an additional species into Noah Ark's limited space would increase the chance of the ecosystem to remain above a given Shannon index by an increasing probability; whereas for any level of species richness greater than n⁎(S), boarding an additional species would still increase the probability of survival of the ecosystem (since the marginal contribution is always positive or nil) but by a diminishing quantity. In both cases however, the ’price’ would remain the same: it would be the space occupied by the additional species. The optimal species richness is thus n⁎(S) and so is the optimal size of Noah's Ark. Note however that the exponential shape of the relationship in Fig. 4 means that, in order to satisfy the n⁎(S) optimal criterion, the size of Noah's Ark will eventually have to become ‘exponentially’ large as the minimum guaranteed biodiversity level S increases. The analysis of marginal contribution highlights other relevant results. In particular, the simulations indicate that in both exploited and non-exploited cases, the maximum value of the marginal contribution decreases with the level of

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E CO L O G I CA L E CO N O MI CS 68 ( 20 0 8 ) 1 4–2 3

species richness (Conjectures 2 and 5). In other words, although the model suggests that the contribution of an additional species to the ecosystem performances is always positive, the maximal marginal contribution decreases when the number of species present in the system increases. This result is worth mentioning as it provides additional elements to the ongoing debate about the marginal species' contribution to biodiversity conservation. In particular our results corroborates Simpson's findings that species marginal value is expected to decrease when the number of species available in the ecosystem increases (Simpson et al., 1996; Craft and Simpson, 2001). The major improvement brought by our study, however, is that our conclusion derived from a model which has an explicit ecological dimension embedded into its functional formalism (through the resource-based equations). This contrasts strongly with the models used by Simpson and his colleagues, which are ecologically unrealistic as they contain no specific formalism to account for species interactions. This limitation reduces greatly the significance of their conclusions, specifically since those conclusions refer to situations where species richness is assumed to vary (and tend toward infinity). Our model rectifies this important flaw. Unfortunately it still confirms Simpson's claim – and even extends it to the case of non-exploited ecosystem – as it shows that beyond a certain level of biodiversity the contribution of the marginal species becomes weak or even negligible. This results alters considerably the argument made by some authors who claim that linking diversity with the measure of economic values through bio-prospecting can play an important role to ensure species conservation through marketbased incentives. Other results presented in this study suggest, however, that the effect of this decreasing contribution of species richness may be overcome by other mechanisms, in particular those related to the existence of S⁎ Let's recall that S⁎ indicates the maximum biodiversity level that can be generated by n species in resource-based competition in an ecosystem. Under limited time-horizon, the simulations (Fig. 3) show that S⁎ increases with n (Conjecture 3). This result has important implications for conservation policy as it suggests that conserving species richness can be one effective way to increase the maximal guaranteed level of ecosystem biodiversity. While laboratory microcosms, field experiments and theoretical models have already shown that increased diversity can improve ecosystem functional performances (Naeem and Li, 1997; Hooper and Vitousek, 1997; Kinzig et al., 2002; Loreau et al., 2002; Tilman et al., 2005), the positive relation between S⁎ and n shown in this analysis means that the larger the number of species initially present in the ecosystems, the higher the level of maximum guaranteed biodiversity. Note however that the concavity of this S⁎(n) relationship also means that, as n tends toward large numbers, the guaranteed biodiversity levels S⁎ increases at a decreasing rate. In other words, under uncertainty, it takes more and more species to ensure a constant increase in the maximum guaranteed biodiversity level as ecosystem species richness increases. Symmetrically – and perhaps more importantly in the context of the increasing number of ecosystems experiencing rapid loss of species richness – this result means that under uncertainty the maximum biodiversity level that can be

guaranteed in an ecosystem that is losing species declines at an increasing pace. The lower the species richness, the faster the decline in guaranteed biodiversity. In the case of direct-use value, the analysis indicates that the maximal utility level U⁎ that can be guaranteed increases with n (Fig. 7). In essence, this last result is not different from the conclusions reached above for the ecosystem's ecological viability – compare for instance Conjectures 6 and 3 – but it highlights important additional potential implications for conservation policy. In particular, it provides a strong (economic) incentive for biodiversity conservation as it predicts that under uncertainty the total utility derived from the direct uses of the harvested products increases with the level of richness n of the ecosystem. The larger the ecosystem (in species number), the higher the maximum guaranteed utility derived from this ecosystem. Furthermore this conclusion appears to be robust for a large range of different utility functions.

6.

Conclusion

It is probably necessary to start this conclusion by highlighting a couple of caveats. First, as in many cases where one attempts to investigate complex issues with simple, theoretical, models, the results presented in this study depend to a large extent on the nature of the model and the subsequent assumptions used in the analysis. Likewise the fact that a very short time-frame (T = 2) is used for the simulations allowed us to investigate more thoroughly some of the questions but it also reduced our capacity to predict the behavior of the model on longer time horizons.8 The different results proposed in this paper should therefore be considered with these caveats in mind and our main conclusion is probably that more research is needed in several of the domains investigated in this initial research. In that respect, a pertinent extension of this work should be to investigate whether the nature of the relationship between ecosystem performances and biodiversity depends on the type of biodiversity metrics considered. In particular, an obvious option would be to revisit some of these results using other biodiversity metrics.9 Despite these limitations, this study has demonstrated the potential relevance of the viability framework in unfolding some of the complex issues related to the challenging task of valuing biodiversity and its ecological and economic contribution to ecosystem performances under uncertainty. Our analysis allowed in particular to identify potential criteria for optimizing conservation interventions under the framework of the Noah's Ark parable, and to revisit -and confirm under a more robust ecological formalism-the hypothesis of

8

The implications of releasing the constraint on the exclusion principle (i.e. T greater than 2 and Δt larger than 0.01) are being investigated in an on-going research. 9 For instance the marginal contribution of species richness (see Definitions 2 and 4) could be defined as an unit increase in the ecosystem’s Shannon index instead of an unit increase in its species number.

EC O LO G I CA L E C O N O M I CS 6 8 ( 2 00 8 ) 1 4–2 3

decreasing contribution of marginal species suggested by Simpson's work.

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