### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. The ecological model
- 3. Alternative ecosystem/management outcomes
- 4. Biodiversity value based on simulation results
- 5. Conclusions
- Appendix
- Acknowledgments
- REFERENCES

**Abstract** This paper develops a measure of the contribution of biodiversity in enhancing ecosystem performance that is subject to environmental fluctuation. The analysis draws from an ecological model that relates high phenotypic variance with lower short-term productivity (due to the presence of suboptimal species) and higher long-term productivity (due to better ability to respond to environmental fluctuations). This feature, which is a notable extension to existing economic-ecological models of biodiversity, enables assessment of the interactions between diversity and a range of environmental fluctuations to highlight that biodiversity could be rendered economically disadvantageous when environmental fluctuation is insufficient. The resulting economic-ecological model generates discounted present value of harvests for an ecosystem with diverse set of species. This value is compared with the harvest value of a similar economic-ecological model with no diversity and that of an ecosystem where the dynamics of phenotypes in response to environmental fluctuations is disregarded. The results show that diversity positively contributes to the performance of ecosystems subject to sufficiently large environmental fluctuation. In addition, neglecting an ecosystem's increasing ability to adapt to match environmental conditions is also shown to be more costly than having no diversity in an otherwise identical ecosystem.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. The ecological model
- 3. Alternative ecosystem/management outcomes
- 4. Biodiversity value based on simulation results
- 5. Conclusions
- Appendix
- Acknowledgments
- REFERENCES

The desire to conserve biodiversity^{1} is essentially motivated by two major concerns. One is that when a species becomes extinct, the social value associated with its possible future use is lost. In addition, as Blockstein [1998] argues, the loss of species could lead to cascading changes because natural ecosystems are complex and highly interdependent, and small perturbations can lead to far-reaching changes with unexpected repercussions (Heal [2004]). Thus, individual species possess not only social benefits of their own; they also carry a joint value shared with other species, which is associated with an uncertainty in the functioning of an ecosystem composed of different species (Fromm [2000]).

Ecological studies generally suggest that biodiversity may positively contribute to ecosystem functioning by increasing the mean absolute level at which certain ecosystem services are provided and by enhancing greater stability of ecosystem properties.^{2} However, the quantitative significance of these contributions of diversity might depend on the actual features of the ecosystem. For instance, the strength of species complementarity and interspecific facilitation tends to vary with both the functional characteristics of the species involved and on the exact pattern of environmental change (Symstad and Tilman [2001], Lavorel and Garnier [2002]). Similarly, the contribution of diversity to ecosystem stability may saturate at lower levels of species richness, with certain cases leading to negative contributions of diversity to stability (Petchey et al. [1999], Emmerson et al. [2001], Pfisterer et al. [2004]).

Hence, understanding the contribution of diversity to ecosystem functioning lies in the proper quantification of interspecific and species–environmental factor interactions (Tilman [2000]). A pioneer among economic studies that incorporate significant ecological information into economic analysis of the value of biodiversity is Weitzman [1992]. In his model, the concept of a diversity function is defined in terms of pair-wise distances among species, with the distance measuring dissimilarity among species. The resulting function attaches a diversity value to every conceivable set of biological species and can be used to rank conservation alternatives.^{3}

Mainwaring [2001] questions the role of genetic dissimilarity between a pair of species and diversity value and argues that a theoretically convincing formulation of diversity value should be based on the nature of species interdependences. Similarly, Polasky and Solow [1995] derive the option value of a collection of species, by allowing for imperfect substitution and dependence between species. In addition, Brock and Xepapadeas [2003] show that a more diverse system could attain a higher value, although the genetic distance of the species in the more diverse system could be almost zero.

With uncertainty, biodiversity may confer an insurance value for various ecosystems (Perrings [1995], Perrings et al. [1995], Simpson et al. [1996], Kassar and Lasserre [2004]). A study by Baumgartner [2007] uses observations from ecological studies to show that under uncertainty, biodiversity increases the mean and the standard deviation of the level of the ecosystem service. However, the cost of provision of diversity is assumed to be exogenous. Similarly, Tilman et al. [2005] demonstrate that biodiversity is beneficial both in a homogeneous environment (through the presence of a best performing species) and in a heterogeneous habitat (through increased use of limiting resource and temporal stability).

In any of these studies, however, species adaptiveness and functional diversity are not allowed to fully evolve in response to environmental change. The approach in this paper draws on an ecological model that allows for adaptive interspecies (and species–environmental factor) interactions and, specifically, has the following desirable features.

First, it depicts the nonequilibrium behavior of the ecosystem and depicts possible tradeoffs between short-term productivity (with low variance and optimal phenotypes) and long-term productivity in changing environments (with higher phenotypic variance causing better adaptive capacity).

Second, it summarizes the dynamics of a diverse ecosystem, in terms of the macroscopic characteristics of the system, into just three equations using moment approximation methods. Because the inherent complexity of natural ecosystems, diverse ecosystem models with complex interactions and multidimensions may have the disadvantage in that they need many equations (Crepin [2002], Brock and Xepapadeas [2003], Baumgartner [2007]). The ecological model used here overcomes the computational disadvantage of dealing with several equations each representing individual species dynamics without having to rely on *ad hoc*, simplifying assumptions.

Our approach is to examine the role of diversity in ecosystem functioning using management regimes that take into account the full dynamics of a diverse ecosystem, that assume no diversity, and that disregard the ecosystem dynamics associated with environmental responsiveness, respectively. To provide context for better understanding of the functioning of ecosystems under a changing environment, we take a perennial agroecosystem as an example.

Section 2 presents the ecological model, which is the basis of our analysis. In Section 3, we set up the optimization problems and obtain the corresponding solutions for the value of diversity. Simulation results and discussions are given in Section 4, and Section 5 concludes the paper.

### 2. The ecological model

- Top of page
- Abstract
- 1. Introduction
- 2. The ecological model
- 3. Alternative ecosystem/management outcomes
- 4. Biodiversity value based on simulation results
- 5. Conclusions
- Appendix
- Acknowledgments
- REFERENCES

As mentioned in Introduction, we base our analysis on an ecological model developed by Norberg et al. [2001], which uses moment approximation methods to capture the dynamics of the aggregate characteristics of a group of species. The resulting model represents a diverse ecosystem in terms of total biomass *Q*(*t*)), average phenotype (*X*(*t*)), and phenotypic variance (*V*(*t*)),^{4} as functions of time, *t*. The time dependence will be suppressed from here on for ease of disposition.

The mathematical formulation of the model is based on the single-species growth function *f*(*x*, *E*, *Q*), where *x* is the phenotype of the species and *E* is the environmental factor. In this paper, we use a simple Gaussian growth function

- (1)

The parameters *c* and *d* stand for maximum fertility and death rates, respectively, which are set to *c* = 1.0 and *d* = 0.1 throughout this paper.

Equation (8) describes the dynamics of the environmental factor. In the ecological model, *E* is a time-varying factor that could be characterized by a constant or variable rate of change over time. The variable rate *E* leads to more complicated dynamics (Norberg et al. [2001]: p. 11377) but is also more interesting because it can accommodate unpredictable changes in the environment. Based on this, we take *E*(*t*) to be a stochastic variable, the pattern of which represents a random walk model with standard deviation , where *dz* is the increment to a standard Gauss–Wiener process.

- (8)

Equation (3) gives the rate of change of biomass, *dQ*/*dt*, as a Gaussian growth equation extended to incorporate the role of the environmental factor, phenotypic diversity, and responses to environmental change. It can be noted that because of the factor (1 − *Q*)*Q* in the right-hand side of equation (3), *Q* will always stay between 0 and 1. From the growth function (1), it is clear that the optimal growth is achieved when the average phenotype *X*(*t*) is equal to the environmental factor *E*(*t*). Finally, because is negative when, we see from equation (3) that a large phenotypic variance *V*(*t*) in the aggregate model gives a slower growth under optimal conditions compared to the single-species model (2). This could be seen as the price paid for a high degree of diversity or the presence of a higher proportion of suboptimal species.

Equation (4) shows that a large variance will make the average phenotype change at a higher rate, and thus the system has greater ability to adapt to a new environment. Because of the factor (*E* − *X*) in , the value of the average phenotype *X*(*t*) will always tend to the value of *E*(*t*), and the rate of change is governed by *V*(*t*). Equation (5) represents the dynamics of phenotypic diversity. The second term contains a factor (*E* − *X*)^{4}, which ensures that *dV*/*dt* is large when | *E* − *X* | is large; that is, the farther away the ecosystem is from optimal performance, the higher the rate of change of diversity. On the other hand, the first term ensures that near optimal conditions, the diversity will decrease, which gives an increase in the growth of biomass, as discussed earlier.

The impact of the total biomass on both average phenotype (in equation (4)) and diversity (equation (5)) is negative and is explained by the environmental change and relationship with the total biomass. For any value of the environmental variable, the species characterized by the optimal phenotype is the one with the highest growth rate at that time. In slowly changing phases of the environmental variable, the best competitor dominates the community at the expense of all other species, leading to increase in the total biomass. This increase is followed by reduction in average phenotype and subsequently to reduction in phenotypic diversity, due to density dependent effects of an increase in biomass. During a subsequent acceleration phase of the environmental variability, however, the species distribution fails to track the accelerating optimum species type set by the level of the environmental variable at every point in time, leading to lower total biomass and increased average phenotype, and subsequently increased phenotypic diversity (Norberg et al. [2001]).

It should also be noted that ecosystems are almost always subject to more than one environmental factor. However, our analysis focuses only on one environmental factor because adding more than one environmental factor would complicate the model without adding much insight into the working of diverse ecosystems. Hence, the implicit assumption is that there is one environmental factor that is of major importance to the functioning of the ecosystem.

To enable better understanding of the functioning of diverse ecosystems under a changing environment, we give some examples of different components of such ecosystems below.

The environmental factor could be represented by temperature, resource concentration, or predator abundance (Norberg et al. [2001]). Kooi and Troost [2006] consider a system consisting of the nutrient, the resident population, and the mutant population, with nutrient representing the environmental factor. Brock and Xepapadeas [2003] represent the environmental factor interacting with the ecosystem as a predator population.

Examples of species phenotypes include species structures that facilitate the uptake of nutrients or antipredatory defenses (Norberg et al. [2001]). Andalo et al. [2005] use height measures of phenotypes to assess adaptation to precipitation and temperature of populations in local locations and new planting sites. Liu et al. [2007] identify carbon isotopic composition, which determines the water use efficiency of plants. Madritch and Hunter [2002] use nutrient dynamics, that is, the ability to mobilize or immobilize carbon and nitrogen as phenotypic features of plants. Labokas and Bagadonte [2005] use the number of flowers per inflorescence, flower diameter, length of inflorescence and length of petiole, leaf length, and leaf width as phenotypic characteristics plants. Similarly, phenotypic diversity within the plant community could be measured by differences in the nutrient uptake and mortality rates of the two species (Loreau and Behara [1999]).

As we mentioned in Introduction, the focus of our analysis is an agroecosystem with a variety of perennial plants. Agroecosystems, particularly those in the developing world, are highly rainfall dependent and are subject to stresses such as heat and drought that affect their productivity. Having plants that respond differently to weather and temperature randomness contributes to resilience and ensures that there will be plants of given functional types that thrive under differing environmental conditions (Heal [2004]). These highlight the importance of diversity–environment interactions in the productivity of agroecosystems.

Given that we had to replicate the ecological relationships using the specific Gaussian growth model and the random walk model of the environmental factor, and because the whole set of the parameters were difficult to find for a specific agroecosystem, we used the parameters that were used in Norberg et al. [2001].

In the next section, we will present a discussion of the structure of the bioeconomic model under alternative management regimes in general terms, to maintain continuity with Section 2 and to highlight the applicability of the model to a wide range of complex adaptive ecosystems. The application of the model with respect to perennial agroecosystems will be introduced in Section 4 where specific parameters are employed to study the behavior of the model.

### 3. Alternative ecosystem/management outcomes

- Top of page
- Abstract
- 1. Introduction
- 2. The ecological model
- 3. Alternative ecosystem/management outcomes
- 4. Biodiversity value based on simulation results
- 5. Conclusions
- Appendix
- Acknowledgments
- REFERENCES

This section develops a framework that enables derivation of the value of biodiversity by comparing alternative management regimes. As we argued in Section 2, a diverse ecosystem is represented by a set of three interdependent dynamic variables (biomass, average phenotype, and diversity) and one exogenous variable (the environmental factor). We use this ecological model as a representation of our first state of the ecosystem. The second state of the ecosystem is also based on this same ecological model, modified to take into account its no-diversity (single-species) feature.

To analyze a situation where the adaptive responsiveness of a diverse ecosystem might be disregarded by the manager, we analyze what we call a myopic management of a diverse ecosystem. Our premise is that because biomass is a source of harvest, its dynamics is of direct economic interest. However, because average phenotype is not a direct factor in the system's immediate productivity and harvest does not (directly) depend on it, it is not of direct economic importance. If optimization only considers biomass dynamics, then it leaves out an important indirect effect. Thus, by disregarding the dynamics of the average phenotype, the myopic management fails to account for the indirect effect, which captures the ecosystem's adaptive response to environmental stress.

Given this, we formulate the economic problem as follows. The benefit from harvesting is a function of price, *p*, and biomass harvest. Harvest is the product of harvesting effort, *y*, and total biomass, *Q*. Hence, harvest is defined as *q* = *yQ*. The total benefit from harvesting is, thus, *pyQ* = *pq*. The cost of harvesting is given as *sy*^{2}, where *s* is a constant. The net benefit from harvesting at a specific point in time (where the time dependence is omitted) is the difference between the total benefit and the cost of harvesting, *B*(*y*) = *pyQ* − *sy*^{2}. The quadratic form of this expression implies the cost of harvesting increases with the amount of harvest. The choice of quadratic cost function is also in line with several similar studies looking into optimal effort and harvest rules in the management of natural resources. Baumgartner [2007] assumes a strictly convex cost function, and a quadratic cost function to derive general and specific harvest rules, respectively. Although it is common to use a quadratic cost function, Brock and Xepapadeas [2003] assume zero harvesting cost for simplicity. The sole manager would seek to maximize the sum of the discounted stream of net benefit from harvesting the biomass with a risk-free, positive discount rate given by *r.*

To value the harvest, we have assumed a single price corresponding to the total biomass. In our analysis, species derive their distinct features from their individual contribution to the total biomass^{5} and their individual response to environmental stress. Thus, different species contribute different amounts of biomass and have different levels of environmental responsiveness, at every point in time. An additional difference could be that the qualities of biomass contributed by different species may be different leading to different market prices of the biomass corresponding to the different species (Tilman et al. [2005]).^{6} Our analytical framework, which is realistic in many respects and hence complex, did not allow us to incorporate the possible price differences of the species with respect to biomass. Hence, we assumed identical prices per unit of harvest for all the different species.

#### 3.1. Management of a diverse ecosystem

In this setting, the manager maximizes the present value of net benefits from harvest, subject to the dynamics of the full features of the ecological model presented in Section 2. The mathematical formulation of the model is

- (9)

s.t.

- (10)

- (11)

- (12)

- (13)

where *Q*(0) = *Q*_{0}, *E*(0) = *E*_{0}, *X*(0) = *X*_{0}, *V*(0) = *V*_{0}, and *T* represents the end time and *W* is the value of the opportunity to exploit the diverse ecosystem. Following Malliaris and Brock [1982], the Hamilton–Jacobi–Bellman's (HJB) equation for the above problem is given in equation (14)

- (14)

where *W* = *W*(*t*, *Q*, *X*, *V*, *E*) and indices on *W* denote partial derivatives.

From equation (12), we solve for *y*, which is the optimal level of effort corresponding to the optimal harvesting rule, obtaining *y** as

- (15)

Substituting the optimal effort into the HJB equation transforms the expression into

- (16)

It should be noted that the functional form of the value function, *W*(*t*, *Q*, *X*, *V*, *E*), is not known. Because our problem is not in the class of stochastic optimization problems that are quadratic in the objective function and linear in the constraints, the functional form also cannot be approximated by the methods of Dockner et al. [2000]. A solver was therefore implemented in C++ using finite differences and implicit time-stepping to obtain the numerical solution on a regular grid. From the solution *W*, the derivative was computed and inserted into equation (15).

#### 3.2. The no-diversity (single species) ecosystem

Setting the level of diversity to zero in the ecological model implies that the equations representing the dynamics of the average phenotype and diversity and the environmental factor disappear. Moreover, this modification reduces the biomass equation to a logistic growth function. Given this, the optimization problem becomes

- (17)

s.t.

- (18)

- (19)

where *Q*(0) = *Q*_{0}, *E*(0) = *E*_{0}, *x* is the phenotype of the species, *T* represents the end time, and *W* is the value of the opportunity to exploit the no-diversity ecosystem.

The HJB equation corresponding to equations (17)–(19) is given by

- (20)

with *W* = *W*(*t*, *Q*, *E*). Solving for the maximum in equation (20) gives the harvest schedule (15) as in the full foresight case, which, inserted into equation (20) gives

- (21)

Figure 1 presents the behavior of the diverse ecosystem compared to the no-diversity system. The first panel in the figure shows the dynamics of the total biomass over time, up to the end time 20 units. The second panel depicts the environmental variable and the average phenotype, with the random environmental change depicted by the thick solid line. The third panel shows the dynamics of the phenotypic variance. The average phenotype roughly follows a similar direction as the environmental variable, albeit with a smoother pattern. The total biomass decreases when | *E*(*t*) − *X*(*t*) | is large and recovers again when | *E*(*t*) − *X*(*t*) | is small. The phenotypic variance increases when | *E*(*t*) − *X*(*t*) | is large, making the average phenotype change more quickly approach the optimal value. The single-species system does not adapt to the environment and thus suffers loss in growth rate when the environment changes.

#### 3.3. The myopic management

The myopic manager disregards the impact of a changing environment on the performance of each of the different species. Although the manager perceives the environmental factor as a variable that has an impact on the dynamics of the biomass, they treat the average phenotype and the phenotypic diversity of the ecosystem as constants. Nevertheless, the actual average phenotype and diversity of the ecosystem evolve over time. Therefore, the effect of perceiving these variables as constants primarily distorts harvest decisions and the evolution of other components of the ecosystem.

Mathematically, the problem is stated as

- (22)

s.t. equations (10)–(13) hold, and where, as before, *Q*(0) = *Q*_{0}, *E*(0) = *E*_{0}, *X*(0) = *X*_{0}, *V*(0) = *V*_{0}, *T* represents the end time, and *W* is the value of the opportunity to exploit the diverse ecosystem. Below, *x* and *v* are the assumed values for the phenotypic diversity and phenotypic variance, respectively.

After solving for *y* and inserting from equation (11), the HJB equation for the myopic management scenario becomes

- (23)

The solution *W*(*t*, *Q*, *E*) was obtained and the dynamical system (10)–(13) was solved as described for the full foresight scenario but using the solution to equation (23) to compute and insert this into equation (10).

### 4. Biodiversity value based on simulation results

- Top of page
- Abstract
- 1. Introduction
- 2. The ecological model
- 3. Alternative ecosystem/management outcomes
- 4. Biodiversity value based on simulation results
- 5. Conclusions
- Appendix
- Acknowledgments
- REFERENCES

In this section, we present the contribution of biodiversity to ecosystem functioning by comparing diverse, single-species, and myopic management systems and using agroecosystems as an example. Accordingly, the parameter values used, *p* = 0.25, s = 0.05, , *r* = 0.0, and the initial values *Q*_{0}= 0.5, *X*_{0}= *E*_{0}= 0, and *V*_{0}= 0.2 represent prices of the perennial harvest, the cost of harvesting, the variability of rainfall, the individual farmer's discount rate, the initial total biomass, initial phenotype, and the diversity of the perennial plants.

Because the total biomass, and thus also the harvest, may vary significantly between simulations, depending on the realization of the Brownian motion for *E*(*t*), it is not suitable to compare and compute means of absolute values between runs. However, identical realized values of the environmental factor are used for corresponding diverse and single-species (or myopic) simulations, making the corresponding single simulations comparable. Hence, for a single realization of the environmental factor, we solve the dynamical system and compute the value of harvest for a single species, *W*_{ss}, from equation (21) and the value of harvest for a diverse ecosystem, *W*_{ff}, from equation (15). For each set of parameter values, 100 simulations were run with different realizations of the environmental variable, *E*(*t*), and the value function ratio *W*_{ff}/*W*_{ss} was computed for each simulation. The mean of these ratios is then computed and is then used to determine the value of biodiversity for different parameter settings. A ratio above 1 indicates added value when biodiversity is present, whereas a ratio below 1 indicates that a single species ecosystem outperforms its diverse ecosystem counterpart.

We start assessing the impact of the different parameters by looking at the standard deviation of the environmental factor. The value of the environmental factor is critical in determining the distance between the value of the phenotype and the optimum environment, which is a measure of how far below a current environmentally determined optimum the performance of the ecosystem is. For a diverse ecosystem, a higher environmental fluctuation leads to faster adaption of the average phenotype. This higher adaptive capacity leads to increased growth of the biomass and in turn to increased harvest. For a single species, this effect will not be observed because the average phenotype is constant throughout. This will lead to slower growth of biomass and smaller harvest overall.

A stable level of the environmental factor, for a diverse ecosystem, means that the average phenotype will remain constant and the total biomass will grow at a stable rate.^{7} However, because some of the species will not be optimal for the environment, the growth of biomass will not be as high as for a single species in its optimal environment. Thus, in this case, harvest will be larger in a single-species ecosystem.

Figure 2 presents various values of the standard deviation, , and the resulting value function ratio of the diverse ecosystem and single species outcomes. In our agroecosystem example, rainfall variability represents this standard deviation. The results are grouped according to the maximum magnitude of the environmental factor throughout the run. It is easily seen from the figure that a large gives a higher probability of large environmental changes.

Looking at the left-hand side of Figure 2 (small ), we see perhaps the most important result of the study. For small changes in the environmental factor, the average ratio of the value functions is less than one, implying that a single-species ecosystem outperforms the diverse one. However, for larger changes in *E*, caused by a larger , the diverse ecosystem outperforms.

The result that a diverse ecosystem outperforms a single species ecosystem is in line with many similar studies, which find that biodiversity has an insurance value under uncertainty. For instance, Baumgartner [2007] shows that for a given level of environmental variation, biodiversity has a positive value. In addition, a fluctuating environment is shown to favor diverse ecosystem characterized by different biomass, energy storage, and nutrient intake structures (Tilman et al. [2005], Kooi and Troost [2006]).

However, what distinguishes our finding from these findings is the result that insufficient levels of environmental fluctuation may render biodiversity invaluable or even disadvantageous to the performance of the ecosystem. This finding is made possible with a feature of our model that allows for varied level of environmental fluctuation, which drives the variation in phenotypes and diversity itself. This feature allows for different levels of environment–diversity interaction. Particular to agroecosystems, this result indicates that crop biodiversity positively contributes to agricultural productivity as the patterns of rainfall and drought are sufficiently erratic.

There is one more interesting fact to note in Figure 2, namely that when the environmental factor becomes too large, in particular for large , the value function ratio decreases again towards 1, and the diverse system has little advantage over the single-species system. This indicates that in these cases the environment is changing so quickly that none of the species in the diverse ecosystem can proliferate, and thus both single-species and diverse ecosystems produce a very low harvest, as would be the case under various kinds of natural disasters.

The impact of initial value of diversity is shown in Figure 3. The upward sloping pattern indicates that the value function ratios between the diverse and single-species ecosystems increase with increased initial level of diversity. Because the single species ecosystem exhibits zero diversity throughout, the difference between the two value functions is fully dictated by the pattern of the harvest in a diverse ecosystem. For a diverse ecosystem, a higher initial level of diversity leads to accelerated adaptation of the average phenotype, which in turn enhances the growth of total biomass in a changing environment and eventually increases harvest. As diversity decreases (i.e., moving left along the curve), the value function quotient drops. This is because a less diverse ecosystem will have a slower adaptation of the average phenotype to changing environments. This result is similar to Brock and Xepapadeas [2003], who find that there is a gain in biomass from an ecosystem that has a positive level of diversity.

The results in Figures 2 and 3 highlight an important result in the contribution of diversity to ecosystem functioning—namely that diversity is beneficial to the functioning of the ecosystem. However, from Figure 2 we can observe that it is not diversity *per se*, it is the interaction of diversity with the environmental fluctuations that is critical in determining the role of diversity in ecosystem functioning. Hence the gains from diversity will be positive as long as the environmental fluctuations are large enough, and this gain will increase with an increase in the level of diversity. However, irrespective of the level of diversity, a stable pattern of the environmental factor might mean diversity will not be beneficial to overall ecosystem productivity.

Figure 4 shows how the average value function ratio varies with initial value of the phenotype. The constant single-species phenotype, *x*, was always chosen equal to the initial average phenotype, *X*_{0}, of the diverse ecosystem, whereas the initial value of the environmental factor was always set to 0. Thus, when the initial average phenotype is far from 0, there is a negative impact on the growth of biomass from the start. When the difference is small, the diverse system may still have the possibility to adapt to the environment and achieve a higher biomass growth, whereas the single-species system is unable to adapt, giving very large average values of the value function ratio. When the initial difference is large, however, the diverse system is too far from optimum to be able to fully adapt to the environment, which is the reason that the ratio decreases toward the left and right ends of Figure 4. This illustrates the importance of having a single-species phenotype adapted to the environment because otherwise the diverse system significantly outperforms the single-species system, even in a moderately varying environment.

Figure 5 shows that the value function quotient between single-species and diverse ecosystems falls with increase in the initial value of the total biomass. For a diverse agroecosystem, a lower initial biomass leads to a higher variability in the average phenotype, due to density dependent effects. This will lead to a higher degree of adaptation to the environment, leading to higher growth of the total biomass, and thereby larger harvest. However, with initial biomass increasing, the variability in the average phenotype decreases (again due to density dependent effects), leading to a lower value function quotient of the diverse and single-species ecosystems.

The results from the preceding three analyses show that the benefit from diversity is highly state dependent. This result is similar to the finding by Eichner and Pethig [2006], who show that ecosystem features such as the size of habitat are critical determinants of the contribution of diversity to ecosystem functioning.

To assess the impact of the harvesting cost and harvest price on the value functions of the diverse ecosystem and single species harvests, we varied the coefficients of the net benefit function, *p* and *s*. However, no conclusions could be drawn from the results, as the random fluctuations were larger than the effect of varying the parameters. The results of different price-cost combinations are shown in Appendix.

In Figure 6, the impact of the discount rate, *r*, shows that the value function ratio decreases with increasing *r*. This means that for a lower discount rate the difference between the diverse and single-species value functions is higher. This is because lower discount rates will increase the future value of resources and reduce current harvest. For a diverse ecosystem, the change in the average phenotype (adaptiveness) will contribute more to the productivity of the ecosystem. For a single species ecosystem, however, this benefit will not be realized because the average phenotype is constant. At a higher discount rate, the future value of resources falls, leading to diminished ratios of value functions.

As mentioned earlier, these results are based on the assumption that the unit harvest prices of different species are identical. Introducing species-specific prices would alter the proportion of instantaneous harvest across species with higher-valued species becoming more attractive to harvest. Given that species with higher market prices are also more responsive to environmental fluctuation, for instance, modified harvesting patterns would lead to reduction the adaptive capacity of the ecosystem. Changes in harvest, resulting from introducing differentiated prices, would also affect the relationships between the other parameters in the ecosystem, and its productivity. Hence, the impact of introducing differentiated prices needs to be thoroughly addressed in similar further research.

#### 4.1. Discussion on the impact of myopic management of a diverse ecosystem

We turn now to the case of a diverse ecosystem, where the manager does not have an accurate perception of the responsiveness of the ecosystem to environmental fluctuation. To assess the impact of such an oversight, we consider a management regime that ignores the dynamics of average phenotype and diversity—the myopic management regime. This management outcome is then compared with a management regime in which there is a full insight into the working of the diverse ecosystem—the full foresighted management regime.

It should be noted that the full foresighted management in this section is identical to the management of the diverse ecosystem in the previous section and thus gives similar results. Therefore, differences in the results between this section and the previous one are only due to differences between the myopic management of a diverse ecosystem on one hand and a single-species ecosystem on the other hand.

The impact of varying the standard deviation of the environmental factor on the value function ratio between fully foresighted and myopic outcomes is shown in Figure 7 and shows similar patterns as the corresponding case in Section 4.1 (cf. Figure 2). For a low level of environmental fluctuation, the current value of the environmental factor, and therefore also the average phenotype, stays close to its initial value, meaning that the assumed value in the myopic management model is close to the real value. This implies that the harvest patterns under the two management regimes remain similar. For intermediate standard deviation, the ratio becomes larger but decreases again for large standard deviation. This is because, for larger environmental fluctuation, the myopic manager will harvest more than is optimal due to lack of feedback from the dynamics of the average phenotype. However, when the environmental fluctuation is very high, higher harvest (by the myopic manager) will depress growth of biomass, leading to an even higher responsiveness to environmental change. When the environment changes too rapidly, the ecosystem also suffers a slow (or even negative) growth of biomass, with very little the manager can do to improve matters, resulting in approximately equal performance for the two managers, although the full foresighted manager has a slightly better ability to deal with the situation.

As can be seen in Figure 8a, the value function ratio uniformly increases with an increase in initial diversity, indicating that an increase in diversity will lead to an advantage for the full foresighted manager. This is not surprising, as the diversity is not taken into account in the myopic manager's decision making, and the larger the actual diversity is, the larger the effect of this neglect.

The results from the analysis of the initial average phenotype show that the value function ratio fluctuates heavily as the initial average phenotype is varied (see Figure 8b). It is hard to draw any conclusions about the dependence on the initial average phenotype, but instead the large fluctuations indicate a strong dependence on the stochastic environmental factor. This is because with an initial average phenotype far from the optimal phenotype, the outcome is heavily dependent on whether the environment changes to make the average phenotype more optimal or less optimal, yielding completely different conditions under which the managers operate.

Analysis of effect of the level of initial total biomass, presented in Figure 8c, shows that the average ratios of the value functions fluctuate slightly with increase in biomass. The fluctuations are so small that the effect of initial total biomass on the ratios must be regarded as negligible.

The effect of varying the price–cost parameters is shown in Appendix. In the high-cost and low-price region, the value functions are approximately equal, whereas in the high-price and low-cost region, the value function ratio increases, most likely because the value of harvest rises, leading to increased harvest and faster dynamic in the growth of the total biomass and subsequently to increased average phenotype. This is in contrast to the fluctuating pattern seen when varying the economic parameters in the case of single-species management. Similar to the case in Section 4.1, the difference between the value functions of full foresighted and myopic management also decreases as the discount rate increases. As with the single species case, this is mainly because the benefit of the dynamics of average phenotype, which contributes to long-term productivity, will be discounted highly with larger discount rate (because the outcomes are similar to the case in Section 4.1, we have not added a figure demonstrating the impact of varying discount rate). The combined effects of price and cost are shown in the Appendix. Figure A1 shows that, for a diverse ecosystem, the alternative price and cost combinations do not lead to a specific pattern of changes in the value of the total biomass. When the dynamics of diversity is ignored, as in Figure A2, high price–cost combinations tend to be associated with high value of biomass and vice versa.

### 5. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. The ecological model
- 3. Alternative ecosystem/management outcomes
- 4. Biodiversity value based on simulation results
- 5. Conclusions
- Appendix
- Acknowledgments
- REFERENCES

A proper assessment of the contribution of biodiversity to ecosystem functioning relies on the availability of an operational conceptual framework that links economic and ecological relationships. Accordingly this paper builds on a growing interest in incorporating ecological information into assessment of the contribution of diversity. By drawing from an ecological model that quantifies the interspecific tradeoffs plant species face given the constraints of their environment, and by taking agroecosystems as examples, the analysis assesses management regimes that consider the potential costs and benefits of diversity.

The optimal effort/harvest rules for each of the management regimes are generated using techniques of stochastic dynamic optimization and numerical methods suitable for analyzing highly nonlinear problems with unknown forms of the value function.

The results show that, for a sufficiently high environmental fluctuation, higher diversity always leads to higher gains in terms of harvest. However, diverse ecosystems may underperform compared to single-species ecosystems when environmental fluctuation is low. Hence, biodiversity enhances the long-term productivity of the ecosystem by providing insurance against unpredictable changes in the external environment, given that external shocks are significant. The main distinction of this study from similar studies on the relationships between diversity and the environmental factor is that our approach enables assessment of the interactions between a positive level of diversity and a range of environmental fluctuations to highlight that biodiversity could be rendered economically disadvantageous when environmental fluctuation is insufficient.

This result also implies that it is not diversity *per se*, but the interaction of diversity with the environmental fluctuation and the nature of species phenotypes that are critical in determining the role of diversity in ecosystem functioning. The major policy implication of this result is that biodiversity conservation is economically viable in cases where external shocks are large enough to make the adaptive function of biodiversity significant.

The results also show that the value of diversity depends on the initial values of the key ecosystem variables, such as initial biomass, initial average phenotype, and initial diversity, indicating that the ecosystem's long-term performance is history dependent. This finding also points to the need for case-specific diversity management and conservation policies. In line with this, in their study of the role of crop biodiversity in agricultural productivity of drought prone Ethiopian highlands, Di Falco and Chavas [2009] find that diversity may enhance the stability and resilience of the agroecosystems when rainfall fluctuation is severe, but with limited economic gains. This implies that maintaining crop biodiversity tends to provide the agroecosystem with a wider range of productive responses to weather shocks, but with varying degrees of economic significance. Further empirical analyses would also assist policy makers in designing targeted policies.

The major limitation of the study is that the economic analysis assumes a constant price for all species of the ecosystem. This was necessary due to the use of a moments-based aggregation, meaning that it was not possible to introduce individual prices of the biomasses of the different species. Future research that uses diversified prices to examine multispecies relationships would further improve our understanding of the economic value of biodiversity.

In addition, agroecosystems are increasingly subject to erratic climatic conditions, implying that the assumption of the environmental factor being pure random walk needs to be relaxed to capture possible extreme values of the environmental factor. The use of more economic and ecological parameters, which are based on specific experimental or field surveys would also generate more accurate information on the role of species diversity in particular agroecosystems.