Additivity properties in metapopulation models:implications for the assessment of marine reserves

Transcription

1 Journal of Environmental Economics and Management 49 (2005) Additivity properties in metapopulation models:implications for the assessment of marine reserves James N. Sanchirico Resources for the Future, 1616 P Street, NW, Washington, DC 20036,USA Received 28 January 2003; received in revised form 20 October 2003 Available online 21 August 2004 Abstract Marine reserves increase both aggregate (system-wide) catches and population levels when the dispersal benefits from the reserve are greater than the opportunity cost of closing the area to fishing. Although the general nature of this condition is clear, how the underlying bioeconomic drivers interact to determine the outcome is not. In this paper, we develop a class of spatially explicit models that enable us to explore how different assumptions regarding connectivity and dispersal can lead to different assessments of marine reserves. The analysis also illustrates how the incorporation of space into models of resource exploitation raises fundamental questions about the relationship of aggregate biological production to the production levels in each patch. We show that models with the supraadditivity property system-wide production is greater than the sum of patch production with no dispersal are more likely to predict that the benefits of marine reserves outweigh the costs. r 2004 Elsevier Inc. All rights reserved. JEL Classification: Q0; Q2; Q22 Keywords: Additivity properties; Fisheries; Marine reserves; Metapopulation Corresponding author. Fax: address: sanchirico@rff.org (J.N. Sanchirico) /$ - see front matter r 2004 Elsevier Inc. All rights reserved. doi: /j.jeem

2 2 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) Introduction In 2001, more than 150 marine scientists signed a Scientific Consensus Statement [35] summarizing the potentially significant conservation and fishery-enhancement benefits of marine reserves areas of the ocean closed to all extractive uses. These benefits can include enhanced biodiversity, additional sources of larvae and biomass, greater quantities of biomass throughout the fishery, increased catches, and a hedge against management failures [26,35]. All of these benefits, moreover, are thought to enhance the long-run sustainability of fisheries. With proposals to close up to 90% of the fishable habitat [24], current users would no doubt be directly and immediately affected, and they d likely fear losing the option of using particular fishing grounds in the future. Because these concerns have legitimacy in fishery-policy arenas, and because compromises over the location, size, and number of marine reserves are inevitable, it is prudent to search for cases that yield the greatest gain in the biological health of the system at lowest cost to the fishing industry. In fact, most of the literature on marine reserves has assessed their effects on measures, such as increased fish populations and aggregate (system-wide) catches that appeal to both conservationists and fishermen. Reserves are typically found, both theoretically and empirically, to increase the level of biomass contained within their own boundaries [16], but whether aggregate catches increase depends on whether the dispersal benefits from the reserves ( spillovers ) are greater than the opportunity cost of closing the area to fishing. 1 In general, dispersal benefits are a function of the connectivity of the system, population levels in the reserve, and dispersal mechanisms and rates all of which depend on the species that the reserve is set up to protect. 2 The opportunity costs are a function of the bioeconomic conditions in the area prior to closure and of fishermen s responses:whether they contract or expand their efforts, and where they focus their efforts after the area is set aside. With momentum building behind the establishment of marine reserves, policymakers will no doubt be required to predict the benefits and costs, both biological and economic, of setting aside areas before detailed empirical analysis can be completed e.g., [22,37]. This task will not be easy, for two reasons. First, although the general nature of the cost benefit relationship is clear, how the underlying bioeconomic drivers interact to determine the outcome is not the specifics depend on the particular fishery under consideration. Second, current biological and economic research assessing marine reserve creation does little to describe how certain assumptions affect results or to compare results across models. For example, it is not clear how results derived from models based only on adult dispersal compare with results derived from models that consider larvaldispersal processes. In this paper, we develop a bioeconomic framework that nests multiple models found in the literature and enables us to explore how different assumptions regarding connectivity and dispersal can lead to different conclusions regarding the potential for marine reserves to benefit all 1 For a review of the literature on marine reserves, see, for example, [4,6,15,26,29]. 2 The dispersal of larvae and adults (and juveniles) is complex and poorly understood, but there is optimism among marine scientists that improvements in genetic analysis, mark-and-recapture methods, otolith geochemistry, and oceanographic models of currents, gyres, and coastal upwelling processes will lead to a better understanding of these processes. (Otoliths are fish ear bones that provide, via analysis of chemical signatures left behind, a natural way to determine the environment in which fish live. Mark-and-recapture studies use artificial tags to determine dispersal pathways.)

3 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) stakeholders. In particular, our biological model includes implicit larval dispersal and adult and juvenile dispersal and is coupled to a model of spatial exploitation. We also illustrate analytically how the incorporation of space into models of resource exploitation raises fundamental questions about the relationship between aggregate biological production and the production in each patch. While a patch within a spatially explicit model of a population is space-limited, 3 it is possible that connectivity increases the level and rate of biological growth in the system. For example, areas with high populations could be contributing larvae to areas with low populations. With less competition for habitat in the receiving area, overall larval settlement rates in the system could be higher than if the patches were not connected, thereby increasing the system-wide biological production. To measure the effects of spatial linkages on aggregate biological production, we define supraadditivity, additivity, and subadditivity properties. In particular, supraadditivity characterizes a system in which there are increasing returns to combining patch populations, and subadditivity occurs when there are diminishing returns to combining patch populations. We then investigate how certain assumptions regarding dispersal processes can lead to the different properties, and find that the density dependence or independence of larval production along with the natural mortality rate during the dispersal process are critical determinants. We also show that the nature of such additive properties determine whether and with what likelihood certain models will predict that marine reserves can benefit both the commercial fishery and the population of fish, thereby yielding a win win outcome. In general, marine reserves are corner solutions that lead to fishing efforts in some areas and biological production in other areas [8,12]. Helfand and Rubin [21] illustrate that such specialization (spatial separation of activities) might be socially optimal with respect to environmental pollution if there are increasing returns to scale (nonconvexities). Therefore it might not be surprising that the model exhibiting supraadditivity, where there are strong nonconvexities with respect to system-wide biological production in combining patch populations, is more likely to lead to a win win outcome. What is surprising, however, is the nature of the nonconvexities and their relationship to ecological production and network characteristics. Although we focus on marine reserves, the definition, modeling, and conclusions regarding the additivity properties are applicable to the management of both terrestrial and marine ecosystems. Based on our results, we conjecture that management regimes that can exploit the supraadditive properties of ecosystems are more likely to improve the biological and socioeconomic health of the system, everything else being equal. Finally, by demonstrating additivity properties fundamental role in the assessment of spatial policies, this paper provides analysts and policymakers with insights into the complexities of spatial modeling and the management of patchy renewable resources. The paper is organized as follows:section 2 presents a spatially explicit bioeconomic model of a commercial fishing fleet that is exploiting a metapopulation. These metapopulation models can 3 For most marine finfish and crustaceans, egg or larval production per adult (fecundity) increases at an increasing rate with the size or age of individuals, implying that there is a potential for biological increasing returns to scale. In many reef fisheries that exhibit these traits, however, larval settlement is subject to a density-dependent mechanism because of competition for the limited coral habitat [10]. Therefore, even though individuals exhibit increasing returns to scale in egg production, the environment limits aggregate biological production.

4 4 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) 1 25 also be used to design and analyze other spatially explicit policies, such as a spatially explicit quota policy or policies to manage the spread of infections. In Section 3, we define the additivity properties and illustrate the role of larval and adult dispersal in determining the properties of each model. In Section 4, we analyze the effects of reserves on aggregate catches in a two-patch setting. Section 5 concludes the paper with a discussion of its findings and prospects for future research. 2. Bioeconomic metapopulation model Most of the economics literature on marine reserves is built on the traditional view that a fishery comprises one large homogeneous habitat with a population that is perfectly mixed throughout. 4 But in reality, fisheries consist of multiple patches of stocks that are interconnected and can have different roles in sustaining the population and producing returns to fishing effort. The importance of habitat quality and heterogeneity for persistence of populations has recently gained support among ecologists [25], especially in the assessment of marine reserves [10,13,16]. This has led, for example, to such questions as:what are the potential benefits and costs of setting aside areas of high biological productivity and high cost? Because the traditional view does not provide any context in which this question can be analyzed, we consider here a bioeconomic metapopulation model that treats space explicitly in the form of discrete patches [31]. This framework illustrates how patch or habitat heterogeneity interacts with dispersal and connectivity to determine the bioeconomic trade-offs. The metapopulation model combines the features of larval-dispersal processes [8,28,38] with a model that accounts for biomass (adult) movements [31,32]. There are only a few examples of models that consider both adult and larval dispersal within a bioeconomic framework e.g., [23,30], but none focus on the role each plays in assessing the effects of marine reserves or illustrate how adult and larval dispersal can lead to different types of additivity in aggregate biological production. 5 The full bioeconomic metapopulation model is 6 _E i ¼ s i R i ðe i ; x i Þþ XN i s ij ½R i ðe i ; x i Þ R j ðe j ; x j ÞŠ; 8i ¼ 1; :::; n; ð1þ j ¼ 1 jai 4 These models investigate how different marine-reserve sizes affect measures such as spawning stock biomass, yield per recruit, catch levels, and stock and catch variability [17,23,27,28]. By assuming that reserves are a fraction of the fishery, these models are discrete spatial approximations to a continuous space-time formulation. A typical assumption in these formulations is that the scale of the open and closed area is separable from the underlying biological and economic production functions an assumption that is likely not to hold as the reserve size approaches zero or one. 5 According to Carr and Reed [9], models with larval dispersal processes are best suited to analyzing marine reserve creation for fisheries in which recruitment overfishing is occurring (i.e., the number of fish entering the harvestable stock is reduced by too much fishing on spawning stocks), and models of adult juvenile dispersal processes are better suited to investigating reserve creation in fisheries that are described by growth overfishing (i.e., the mean size of the harvestable individuals is decreasing). 6 This model is a variant of a metapopulation model in which we are interested in population levels and the nature of the linkages, and not simply whether a patch is occupied the traditional structure [18]. Brock and Xepapadeas [7] use the traditional structure, which is a reduced-form model of an ecosystem, to investigate the optimal economic management of an ecosystem.

5 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) _x i ¼ f i ðx 1 ; ::; x n Þþd ii x i þ Xn i d ij x j h i ðe i ; x i Þ; 8i ¼ 1; :::; n: ð2þ j ¼ 1 jai The fleet behavioral model depicted by Eq. (1) captures a sluggish adjustment process in which E i, the level of fishing effort in each patch, changes myopically in response to the level of rents R i ðe i ; x i Þ in each patch vis-a` -vis outside opportunities and rents across the system [31,32]. That is, vessels move from patch i to patch j if rents there are higher, all else being equal. The parameters capturing the sluggish nature of the response of fishing effort are s i in the entry exit decision, and s ij in the decisions to move to capture differences in rents across patches that are connected to patch i as defined by the set N i. The predictive power of this model has been supported empirically in the California red sea-urchin fishery [36]. Net rents are assumed to be average gross operating profits per vessel, less an opportunity cost j per vessel. Gross operating profits are assumed to be a function of E i ðtþ and x i ðtþ via a catch-rate function h i ðe i ; x i Þ and a cost function C i ðe i ; x i Þ: We assume a parametric output price p i : Opportunity costs per vessel j are assumed to reflect alternative income-earning opportunities outside the fishery, which we will assume to be constant per unit of vessel capacity and common across all patches. Thus, we can write net rents (or profits) expected in patch i as R i ðe i ; x i Þ¼ ½p i HðE i ; x i Þ2CðE i ; x i Þ2jðE i ÞŠ: For the remainder of the paper, we assume a Schaefer catch-effort production function (h i ¼ q i E i x i ; where q i is the catchability coefficient), a linear cost function with constant patch specific cost parameters (c i ), and a common exvessel (dock) price (p), resulting in R i ðe i ; x i Þ¼½pq i E i x i 2ðc i þ jþe i ÞŠ: Eq. (2) represents a biological system in which biomass levels evolve in n separate biological patches, where x i is the biomass level in patch i; f i ðx 1 ;...; xþ is the local growth function in patch i; d ij is the dispersal rate between patches i and j ðd ii p0; d ij X0 8i; jþ; and n i is the set of patches biologically connected to patch i: The d ij capture the dispersal of adult and juvenile biomass which we refer to simply as adult dispersal in the system. We capture an implicit larval-dispersal process within this lumped-parameter model by generalizing two models found in the literature that also consider implicit larval-dispersal processes. The first is based on the model used in Pezzey et al. [27], 7 and the second is employed in Tuck and Possingham [38]. In the first model, the growth function in each patch is f i ðx i ;...; x n Þ¼r i a i x i þ m n X n j¼1! ð1 a j Þx j 1 x i k i ; ð3þ where r i is the intrinsic growth rate in patch i, k i is the carrying capacity in patch i; a i is the probability that larvae in patch i remain and successfully settle in patch i; 1 a i is the probability that larvae leave patch i and enter a larval pool, and m is the probability that larvae survive the journey to and from the larval pool to resettle in the patches. We make a standard ecological assumption that all larvae that do not settle in their local area ðð1 a i Þx i Þ are perfectly mixed in a 7 Pezzey et al. [27] model a two-patch system with f i ðx i ; x j Þ¼r i ðx i þ x j Þð1 x i =k i Þ for i=1, 2 with iaj; which is equivalent to assuming that m/n is equal to one, and a i ¼ 0 in Eq. (3). According to the authors, their model captures a setting in which implicitly eggs and larvae are mobile, but adults are not.

6 6 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) 1 25 larval pool and evenly redistributed among all the patches (e.g., [14,19]). 8 This implies that the distribution of larvae is so widespread as to cover the entire range of patches in the fishery [19]. The model captures larval production that is density independent and larval settlement that is subject to local density-dependent mechanisms [19]. 9 This can be seen by considering the case in which all dispersal enters the common pool (a i ¼ 0). Here, the larvae that are produced and enter the common pool are a function of the population level but the overall growth in the population is driven by the local density-dependent effect ð12x i =k i Þ; all else being equal. This model predicts higher marginal and total biological growth than if the patches were not connected via larval dispersal, and it predicts that the effect of the population in patch j on patch i s production is greatest when patch i s population is low. At the other extreme, if patch i is close to its carrying capacity, the effect of other patches population levels decreases because of competition for limited space and food in patch i. Because these characteristics are commonly referred to in the biological literature as postdispersal density dependence [14], we employ the mnemonic POSTDIS to refer to this model throughout the remainder of the paper. In the second model, which follows Tuck and Possingham [38], 10 the patch-specific growth functions are f ðx i ; :::; x n Þ¼ a i r i x i 1 x i k i þ m X n ð1 a j Þr j x j n j¼1 1 x! j ; ð4þ k i where the parameters have the same interpretation as in POSTDIS. 11 Eq. (4) assumes that the production of larvae in the system is the result of density-dependent mechanisms in each patch. That is, the closer each patch s population is to its carrying capacity, the less each patch contributes to the overall biological productivity in the system. For example, suppose that a i ¼ 0 8 The assumption of perfect mixing can be relaxed to account for larval-dispersal patterns that result from prevailing currents or seasonal upwelling along a coastline. To see this, we can write equation (3) in matrix notation as follows, ða þ MðI AÞGÞFx:where A is an n n diagonal matrix of the a i parameters, F is an n n diagonal matrix of patch growth functions (e.g., in the logistic growth curve f ðx i Þ¼r i ð1 x i =k i ÞÞ; M is an n n diagonal matrix of larvalmortality parameters, and G is a connectivity matrix that provides information on which patch is connected to which other patches. Under the perfect mixing assumption, the elements of G are all equal to 1/n. The same formulation can be derived for Eq. (4). 9 Ecologists are careful to consider the interactions that introduce density dependence interactions that in a continuous time framework become difficult to disentangle. In this setting, the density dependence could result from competition between the settlers and the existing set of residents, and not from competition among settlers. 10 Tuck and Possingham [38] model a two-patch system in discrete time, where the change in population growth from one period to the next is assumed to be x i;tþ1 ¼ d i x i;t þ p ii f ðx i Þþp ij f ðx j Þ; where d i x i;t is the proportion of adults surviving from one period to the next, p ij is the proportion of larvae produced by local population i that recruit to local population j in each period, and f ðx i Þ¼r i x i ð1 x i =k i Þ for i=1,2 with iaj; and r i and k i as defined above. 11 The implicit nature of larval production within a lumped-parameter model can be defined as follows:suppose that larvae produced in each patch and instant are a function of the patch biomass level; for example, L i ¼ gðx i Þ; where L i is the level of larvae in patch i: Further, using the parameter definitions from Eq. (3), the fraction of larvae that successfully settle in the patch is equal to ðm=nþl i ; where we continue to assume that all larvae mix in a common pool. Then ðm=nþgðx i Þ is the amount of larvae that successfully settle in patch i as a function of the population level in the patch. In POSTDIS, we have gðx i Þ¼gx i and in PREDIS gðx i Þ¼gð1 x i =k i Þx i ; where g is a scaling factor, which is accounted for above in the rescaling of the probability of survival (m). Brown and Roughgarden [8], who explicitly include a larval state equation, model larval production as proportional to stock biomass ðgðx i Þ¼gx i Þ; and they include a more structural but qualitatively similar process as in Eq. (3) for larval mortality in the dispersal process.

7 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) (for all i). Then from Eq. (4), it is obvious that the biological-productivity effects of larval connectivity are greatest when all populations are at maximum sustainable yield (MSY), which is not necessarily the case when there is postdispersal density-dependence, as in Eq. (3). This model is similar to predispersal density dependence formulations in the biological literature [14], and we use the mnemonic PREDIS to refer to it throughout the paper. In Eqs. (3) and (4), when all larvae are retained locally (a i ¼ 1 for all i), the growth function in each patch reduces to the standard logistic growth function. Both frameworks, therefore, nest the model used in Sanchirico and Wilen [32]. Because their model is nested in both models, we use it as the benchmark by which to compare the different assumptions regarding larval and adult dispersal. We denote this model LARLOC, a mnemonic that stands for all larvae retained locally. All three models are capable of depicting a variety of behavioral characteristics of a metapopulation as well as connectivity patterns stemming from typical oceanographic features. For example, a sedentary adult species such as sea urchins, whose larvae perfectly mix in a larval pool and are redistributed equally across the patches, would be connected by larval dispersal only, with no adult migration (a i =0 and d ij =0 for all i, j). Depending on the species, perhaps half the larvae produced in a patch remain there and the other half drift via ocean currents and coastal upwellings into larval pools (a i =0.5 for all i), where only a fraction survive to settle (mo1). Or both larval and adult-dispersal processes may connect the patches. For example, certain pelagic species, such as bluefin tuna and swordfish, and some reef fish, such as grouper and snapper, have both adult and larval-dispersal phases; in many cases, the scale of the larval and adult-dispersal processes need not be identical a characteristic that can be modeled in our formulation. A linear system representing a coastline, for instance, could capture a species whose larval dispersal is uniformly distributed across the entire coastline, but adult biomass moves only between neighboring patches. A discrete model of this type can also depict a range of productivity assumptions in a system of individual patches. Some patches may have higher biological productivity than others, while others, like larval pools that receive and disperse larvae from other patches, may have no inherent productivity. This important feature of the model enables us to investigate how the selection of sites for reserves can affect dispersal benefits and opportunity costs. It is important to note that the lumped-parameter representation is highly stylized. It ignores important aspects of real population growth and dispersal dynamics, including age- and size-specific mechanisms, selectivity issues, egg production per recruit, and more complicated recruitment processes. However, it is analytically tractable under simplifying assumptions, whereas richer models must be analyzed using simulation methods. By also embedding larval dispersal within a lumped-parameter model, we capture many of the essential features of more complex larval processes without adding the complexity of another set of n state variables. Of course, the ad hoc reduced-form representation limits our ability to map these results directly into more complex systems, but it is our conjecture that the qualitative properties illustrated are robust. 3. Additivity properties in metapopulation production In this section, we define the supraadditive, additive, and subadditive properties of metapopulation models that characterize aggregate biological production at any point in

8 8 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) 1 25 time. 12 We then show how the type of additivity in the model depends on the nature of larval production and settlement, whether larvae are retained locally or disperse to other patches in the system, and the survival rate of larvae in the dispersal process. And given that the existence of additive properties in metapopulation models of renewable resources has not been previously discussed in the literature, definitions of each of the additivity properties are presented. Additivity properties are employed in many areas of scientific inquiry, from economics and network engineering to probability and statistics, as well as in theoretical and applied mathematics. Basically, additivity properties are a means of summarizing how the overall system (a multivariate function) relates to the individual components (separate functions of each variable). Economists developed additivity properties to describe the costs of producing joint outputs relative to producing each output separately e.g., [1 3,11] as a way to understand the existence of multiproduct firms and natural monopolies. 13 Recently, Weitzman [39] defined additivity, supraadditivity, and subadditivity properties for induced utility functions in order to illustrate how diversity indices for species may be mapped into utility functions. In Weitzman s analysis, supraadditivity corresponds to the case in which species are complements in utility and there are increasing returns to utility by combining species. Subadditivity occurs when species are substitutes and there exists decreasing returns to combining species. The biological additivity properties developed here are similar to these other examples, but our goal is to define a set of properties that classify how spatial linkages affect aggregate biological production. Let Fðx i ;...; x n Þ be the system-wide biological production function that is defined to equal the sum of patch production functions ðf i ðx i ;...; x n ÞÞ across all patches in the system, and f 0 i ðx iþ is the production function in patch i when the patch is independent of the others (i.e., patches are not biologically connected). We measure the additivity properties from the case in which the patches are not biologically connected. There are likely other baselines against which to measure aggregate production, but we believe that none are as intuitive; it makes sense to measure the implications of including space in models of renewable-resource extraction against the case of no explicit spatial processes (patches are not connected by adult or larval dispersal). With these functional representations, we formally define the three types of additivity that can exist in metapopulation models with more than one patch population (n41). It is important to mention that these properties are based on aggregate production and that it is very likely that some patches play more important (more productive) roles in overall biological production than others. Having said that, we formally present only the global properties of the system in this paper, analogous definitions can be written down for the local additivity properties between any combinations of patches. 12 Although our focus is on spatially explicit populations, these properties can analogously be defined for multispecies models, where connectivity is defined by trophic levels. 13 In particular, subadditive costs occur when there are lower costs from joint production than from producing each output separately [11]. A subadditive cost function is a necessary and sufficient condition for a natural monopoly when all firms have access to a common technology and when market coordination among distinct firms, because of intrafirm social networks [2,11], cannot attain the economies of a single firm. Although subadditive costs can lead to multiproduct industries, firms with supraadditive costs could save money by breaking apart, and firms with additive costs most likely have determined their optimal configuration [11].

9 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) The definitions are as follows: 8 C Pn f 9 i ðx i ; :::; x n Þ Pn f 0 i ðx iþ40; Supraadditivity >= >< Additivity exists if and only if C Pn f i ðx i ; :::; x n Þ Pn f 0 i ðx iþ¼0; >; Subadditivity >: C Pn f i ðx i ; :::; x n Þ Pn f 0 i ðx iþo0: Supraadditivity describes the situation in which aggregate biological production in the system is greater with linkages than without. If all patches are identical, supraadditivity implies that if a patch in the system is removed (disconnected), we remove more than its contribution because we also lose the positive effects of an element in the ecological network. With patch heterogeneity, it is possible that some patches contribute more to system-wide production than others, and if these patches were removed (by habitat degradation, for example), then the properties of the system could change dramatically. 14 Supraadditivity characterizes an ecosystem in which there are increasing returns to overall biological productivity from combining patch populations. Additivity can characterize the situation in which the biomass-dispersal process sums to zero across the system and therefore washes out in the aggregate. This would be the case, for example, if there were no mortality during dispersal:what leaves one patch arrives in another. One could equate additivity with a system-wide production function that exhibits constant returns to combining patch populations. Subadditivity corresponds to the situation in which the connectivity in the system results in lower production than if the patches were not connected. Intuitively, if the patch populations contribute most of their larvae to a common pool and there is a low survival rate, then we would expect to find subadditivity properties in this system. 15 Subadditivity characterizes a system in which there are decreasing returns to patch combinations. Two points are worth mentioning about the applicability of these properties. First, they apply to the instantaneous growth rates of the aggregate biomass in the system without harvesting, and therefore apply for all t: To illustrate this point, let X to be the population biomass of the entire fishery (system); then X is equal to the sum of the population biomass in the patches. If we take the derivative of X with respect to t; we find that the instantaneous rate of change of population biomass in the system is equal to the sum of the instantaneous rates in each patch throughout the fishery. 16 By substituting for each patch i the patch-specific growth function, the above definitions apply directly to the instantaneous growth rate of system-wide biomass. 14 With patch heterogeneity, supraadditivity holds when for all combinations of patches there exists at least one pair of patches in which supraadditivity holds strictly (4), and for all other combinations the relation is greater than or equal to (X) zero. Of course, it is also possible that some patch combinations exhibit local subadditivity, and in that case the property of the system will depend on the relative sizes of the effects. 15 As with supraadditivity and patch heterogeneity, this condition holds if there exists at least one combination of patches where it holds strictly (o), and for all other combinations of patches it is less than or equal to zero(p). 16 That is, X P n x i ) dx=dt ¼ P n dx i=dt:

10 10 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) 1 25 Second, in a bioeconomic steady-state equilibrium, the additive properties map directly into equilibrium catch levels. Therefore the properties also describe how larval and adult dispersal affect aggregate equilibrium catches in the system. For example, a system with supraadditivity properties will have greater sustainable catch levels than a system with subadditivity, all else being equal. We return to this latter point in the analysis, later in this paper, of the effects of marine reserves on aggregate catches Additivity properties of LARLOC, PREDIS, and POSTDIS Next, using the three metapopulation models, we determine how the type of additivity in a model depends on the nature of larval production and settlement, whether larvae are retained locally or disperse to other patches in the system, and the probability of larval survival in the dispersal process. We begin with LARLOC because it is easier to demonstrate the analysis without larval dispersal. In LARLOC, aggregate production is Fðx i ; :::; x n Þ¼ Xn r i x i ð1 x i =k i Þþd ii x i þ Xn i j¼1 d ij x j!: ð5þ To determine the nature of the additivity, we need to compare (5) with the production in the system when the patches are not connected (d ij ¼ 0 for all i, j). Formally, the comparison is between!! X n r i x i ð1 x i =k i Þþ d ii x i þ Xn i d ij x j X Xn r i x i ð1 x i =k i Þ : ð6þ j¼1 It is clear from Eq. (6) that the additivity property depends on the assumptions regarding the dispersal process and, in particular, the dispersal parameters. This is formally illustrated in Proposition 1. Proposition 1. In LARLOC, when there is no mortality in the adult-dispersal process, the model exhibits the additivity property for all t. No mortality in adult dispersal is satisfied when a set of adding-up restrictions on the dispersal process (d ij ) is imposed, ensuring that whatever leaves patch i for j also arrives in j from i: In particular, we require that a symmetry condition d ij ¼ d ji and that S n i k¼1 d ki ¼ 0; i ¼ 1; 2;...; n be satisfied (recall that d ii o0 and d ij 40). 17 To prove Proposition 1, we can rearrange the left-hand side of Eq. (6) to X n r i x i ð1 x i =k i Þþ Xn X n i k¼1 d ki!x i : ð7þ With the adding-up restriction imposed, it obvious from Eq. (7) that the left-hand and right-hand sides of Eq. (6) are equal. The dispersal process zeroes out in the aggregate because whatever shows up in a patch (positive effect) came from another patch in the system (negative effect). 17 We assume that the spatial response rates of the fleet (s ij ¼ s ji ) are symmetric; this guarantees that vessels that leave patch i for patch j arrive in patch j.

11 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) Therefore in LARLOC, where there is no larval dispersal, aggregate production increases in proportion to the biological production in each patch; and this holds for all levels of patch populations and for n subpopulations. If there is mortality during the adult-dispersal process, then the adding-up restrictions would ensure that what arrives in one patch from another is less than the amount that left. In this case, LARLOC is characterized by subadditivity properties. A similar analysis can be done for PREDIS, when larval production and settlement are density dependent. If we continue to impose the zero-mortality restrictions on adult dispersal, aggregate biological production in this system with linkages is equal to Fðx 1 ; x 2 ; ::; x n Þ Xn r i x i ð1 x i =k i Þða i þ mð1 a i ÞÞ: ð8þ Consider first the case in which all larvae are dispersed uniformly across the system (a i =0 for all i) and the probability that the larvae survive the journey is one (m=1). Under these assumptions, aggregate production is equal to the sum of the local patch growth functions. In this case, as in LARLOC, adding or removing a patch simply changes the aggregate production in an amount equal to the growth in the patch. 18 But what if the probability of survival is less than one (mo1) and a i a1 for all i? Then we find that the model exhibits subadditivity for all interior levels of the population. This is illustrated in Proposition 2. Proposition 2. In PREDIS, when a fraction of the larvae enter the common pool (0pa i o1 8i) and the probability of survival is less than one (mo1), then subadditivity exists for all interior levels of the population (x i 2ð0; k i Þ 8i). Formally, we have C ¼ Xn r i x i ð1 x i =k i ðai þ mð1 a i ÞÞ Xn r i x i ð1 x i =k i Þp0: ð9þ Under these assumptions, it is straightforward to show that a i þ mð12a i Þo1 (recall that m 2 ½0; 1Š and a i 2½0; 1Š), and the inequality in Eq. (9) follows directly. Because each patch is losing some larvae to the common pool and not all the larvae that leave return to redistribute among the patches, it is not surprising that aggregate production exhibits diminishing returns to combining populations. It is interesting to note that with mp1; if all the population levels in the system equal their carrying capacities (or zero), the model exhibits additivity. This would be the case at the unexploited biological equilibrium. Therefore, the subadditive properties can be thought of as a transitory phenomenon. In an exploited system, the metapopulation or ecosystem will continue to exhibit subadditivity (catches keep levels below their carrying capacities), even if additive properties are ecologically more advantageous over the long run. We now turn to POSTDIS when larval production is density-independent but settlement is subject to local density dependent processes. In this model, assuming that adding-up restrictions 18 In Eq. (4), we explicitly impose that the larval dispersal component also satisfies an adding-up restriction; that is, larvae cannot be contributing simultaneously to local and distant patches production.

12 12 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) 1 25 on the adult-dispersal process continue to hold, aggregate production with linkages is! Fðx 1 ; x 2 ;...; x n Þ Xn r i a i x i þ m X n ð1 a j Þx j!ð1 x i =k i Þ n j¼1 and it can be rearranged into local (1st term) and spatial components (2nd term) as such 0 1 Fðx 1 ; x 2 ;...; x n Þ Xn r i x i ð1 x i =k i Þ a i þ m n ð1 a iþ þ m X n X n r i ð1 x i =k i Þ ð1 a j Þx j n B C : j ¼ 1 A iaj As Eq. (11) illustrates, the nature of the additivity properties in POSTDIS depends not only on the particular assumptions regarding the mixing of larvae and the probability of their survival, but also on the biological heterogeneity in the system (r i ; k i ) and the relative population levels across the patches (x i relative to k i ). Because of the additional complexity in POSTDIS, we are not able to make statements regarding the nature of aggregate production unless we make some restrictive assumptions. For example, if all larvae are retained locally (a i ¼ 1), then Eq. (11) reduces to Eq. (5) and the model is characterized by additivity. As with PREDIS, we also can show that if all patch populations are equal to either extreme (0 or k i ), then the model exhibits additivity. Furthermore, if we assume that both the carrying capacities (k i ¼ k) and patch biomass levels (x i ¼ x) are equal and a i ¼ 0; then with some algebra it can be shown that C ¼ xð12x=kþðm21þ P n r i: Therefore, with mo1 the system is subadditive, and with m=1 the system is additive. We can further build our intuition on the additive properties of this model by considering the case of two patches, where we measure biomass in density units to reduce the parameter space 19 and assume that all biomass enter a common pool (a i ¼ 0). Under these assumptions, C is equal to C ¼ð1=2Þ½r 2 ð1 x 2 Þðmx 1 þðm 2Þx 2 Þþr 1 ð1 x 1 Þðmx 2 þðm 2Þx 1 ÞŠ: ð12þ Because it is difficult to intuit from this expression what is going on for example, what conditions lead to its being negative or positive we make the additional assumption that m=1. 20 Under these assumptions, Eq. (12) simplifies to C ¼ð1=2Þðx 1 x 2 Þ½r 2 ð1 x 2 Þ r 1 ð1 x 1 ÞŠ: ð13þ We can learn about the sign of C in Eq. (13) by considering special cases. First, POSTDIS exhibits additivity when the density level in each patch is equal or when both patches are at 0 or This eliminates the carrying-capacity parameters, but all remaining parameters must be rescaled to reflect the new units of measurement. It is important to point out that this does not necessarily imply that the carrying capacities are equal across the system; rather, any differences in the carrying capacities are now reflected in the rescaling of the remaining parameters. For the remainder of the paper, the results are derived in terms of density levels, but the conclusions are directly applicable to other measures, such as population levels. 20 Although these assumptions may appear restrictive, they are commonly employed in the literature on marine reserves. For example, Pezzey et al. [27] make the same assumptions and allow larvae to be in same place at the same instant, which removes the 1/2 above. ð10þ

13 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) Second, if we suppose that x 1 4x 2 ; then for Eq. (13) to be positive it must be that r 2 =r 1 4ð12x 1 Þ=ð12x 2 Þ; where the right-hand side of the inequality is less than A sufficient condition for this to hold is that r 2 4r 1 : How can we interpret this condition? With x 1 contributing more larvae to the common pool (x 1 4x 2 ), the biological-dispersal gradient is tilted toward the ecologically more productive patch (r 2 4r 1 ) and total productivity is increased. If, however, patch 1 has significantly greater ecological value to the system (r 1 r 2 ) and x 1 is still contributing relatively more to the common pool, then it is possible that the system exhibits subadditivity, as the biological gradient is tilted away from the ecologically more valuable patch. Even if the relative levels do not change (x 1 4x 2 ), it is likely that the size of ð12x 1 Þ=ð12x 2 Þ will vary with changes in the patch densities, and therefore it is possible that the properties of the system could change over time. We illustrate this possibility below. In Proposition 3, we employ an additional assumption that there is no biological heterogeneity in the system (r 1 ¼ r 2 ). This additional simplification reduces Eq. (13) and highlights rather clearly the conditions for supraadditive properties. Proposition 3. In POSTDIS with two patches, if a i ¼ 0, r i ¼ r ði ¼ 1; 2Þ and, m ¼ 1, then supraadditivity is present when the patch densities are unequal and fall within the interior of their range ðx i 2ð0; 1Þ 8i ¼ 1; 2Þ for all i ¼ 1; 2). The proof of Proposition 3 follows directly from Eq. (13), which reduces to Eq. (14) with the appropriate substitutions C ¼ r 2 ðx 1 x 2 Þ 2 X0: ð14þ Under the current assumptions, we find that POSTDIS exhibits supraadditivity. It is also the case that the greater the difference in patch-density levels, the greater the difference between the aggregate and patch production. Furthermore, the productivity gains due to the linkages are increasing at an increasing rate, as the differential in patch densities grows. This simple example, therefore, shows that the gains from linkages are more likely to occur when there are significant differences in patch populations a divergence that is created when one patch is closed to fishing. Eq. (13) and (14) are derived with the condition that m=1. This is a very optimistic scenario for marine systems because the considerable variability in oceanographic conditions affects larval survival. Therefore, it is interesting to consider the effect of mo1 on the additivity properties of POSTDIS. (Recall that Proposition 2 shows that with mo1, PREDIS exhibits subadditivity.) To simplify the analysis on m, we assume that there is no biological heterogeneity; this allows us to rewrite Eq. (12) as follows: C ¼ðr=2Þ½2ðx 2 1 þ x2 2 Þ ðx 1 þ x 2 Þðmðx 1 þ x 2 Þþ2ð1 mþþš: ð15þ Because Eq. (15) is positive with m=1 and negative with m=0, it is evident that as the probability of survival decreases (mk), the magnitude of the Eq. (15) decreases (C #). Putting this together with Eq. (14), we conjecture that when mo1 and there is no significant difference in biomass levels, Eq. (15) can be negative. But C is likely to switch signs as the difference in patch 21 Symmetrical conditions hold when x 2 4x 1 :

14 14 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) 1 25 population levels increases. In other words, it is possible for the additivity properties of the model to differ, based on which region of the biomass-density space we are considering. We illustrate this phenomenon in Fig. 1 by mapping C in Eq. (15) as a function of the twopatch density levels with m=0.95. Because C is a function of two variables, this is a threedimensional map. But for expositional purposes, Fig. 1 is drawn as a contour map of the surface, where the lines represent different levels of C and the two axes correspond to patch-density levels. We see that there are three regions in the graph, two of which exhibit supraadditivity. The nature of the subadditive region follows our intuition regarding the levels at which Eq. (14) is likely to be smallest, all else being equal. We also find that the greater the difference in patch populations, the larger is C; as shown by the contour lines. What is also clear is that this surface exhibits strong nonconvexities that depend critically on the relative levels of the patch densities. That is, the returns to combining patch populations depend not on the absolute level of the population but on the differential in the patch populations. Although it would have been nice to develop the properties of all three models under the same conditions, the complexity of POSTDIS and PREDIS prohibited such conclusions. We were able to show, however, that LARLOC exhibited additivity for any number of patches. And PREDIS was shown to be at best additive, though it exhibited subadditive properties when mo1. But for POSTDIS, general conclusions for any number of patches are difficult, which led us to investigate special cases regarding m and biological heterogeneity with only two patches. Fig. 1. Additivity properties in POSTDIS. Note:The regions in which the additivity properties of the model hold are labeled on each graph, and the bold line is the combination of patch-biomass densities such that additivity properties hold. The magnitude of the condition increases as the difference in the patch biomass levels increase, which is represented by the contour lines. This graph was drawn with m=0.95.

15 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) One common theme across all models is that the long-run ecological unexploited equilibrium is characterized by additivity properties. Because we know that exploitation maintains population levels below this equilibrium, these properties are especially relevant for a bioeconomic analysis of managed ecosystems. To recap, we find that POSTDIS is the only model considered that could lead to supraadditivity (when larval production is density independent). PREDIS has the greatest potential for subadditivity (when larval production is density dependent). And the predictions of LARLOC reside between the optimistic predictions of POSTDIS and the pessimistic predictions of PREDIS (when aggregate production is equal to the biological production in each patch). In the next section, we show that nature of the additivity properties is a critical determinant of whether and with what likelihood certain models predict that marine reserves can yield a win win outcome. 4. Equilibrium analysis of marine reserves The bioeconomic system outlined here is particularly useful for examining the implications of additivity properties for the assessment of marine reserves. It also allows us to conduct many more analyses of the roles of various ecological and economic structures than can be done empirically and to investigate the effects of habitat quality on the reserve site-selection problem. In addition, the model can be used to analyze other spatial instruments, such as gear restrictions and individual transferable quotas, and to compare these instruments with reserves. For example, Sanchirico and Wilen [33] use LARLOC to investigate the optimal spatial distribution of fishing effort where reserves are a potential boundary solution (see also [8]). We focus on finding conditions in which marine reserves will increase aggregate catches, and we consider how these conditions depend on assumptions about larval and adult dispersal. Aggregate catches are considered for two reasons:previous studies have shown that closing an area to fishing increases the biomass in the patch and, most likely, throughout the system; and because we assume that the fishery is operating under open-access conditions, aggregate catch is one potential measure of importance to fishermen. To derive the aggregate condition, we first establish the base equilibrium condition and catch levels that exist in the fishery before the reserve is created. Although solving for closed-form solutions is feasible, comparisons across the models are difficult because of the number of parameters. We therefore make the following simplifying assumptions throughout the remainder of the paper:first, we measure biomass in terms of density levels. Second, we model an adult-dispersal mechanism that depends on relative density levels:the amount of biomass flowing between patches depends on the densities of each patch. We also assume that the simplest representation of biomass dispersal between patch 1 and patch 2 is d 11 x 1 +d 12 x 2 b(x 2 x 1 ), and between patch 2 and patch 1 is d 22 x 2 +d 21 x 1 b(x 1 x 2 ), where b is the common dispersal rate. Third, we consider only the limiting cases in which all larvae either enter a common larval pool or are retained locally and m is set equal to 1; under these conditions, the model provides an upper-bound prediction on the potential for larvae from any one patch to contribute to biological growth throughout the system. It will also be the case that supraadditivity can exist in POSTDIS but not in the other models.

16 16 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) Prereserve creation In the prereserve (steady-state) equilibrium, the level of biological growth in each patch is exactly offset by net dispersal and the catch in the patch in question. 22 Although the biomass levels in each patch are constant, the levels in each patch are maintained in part by biological dispersal; hence there is some biomass movement across space, matching outflows to inflows in equilibrium. In addition, net rents will be identically equal to zero in each patch, leading to an economic equilibrium over time and space. Because of the assumption that net rents are multiplicatively separable with respect to effort, we can derive closed-form solutions for the equilibrium effort and biomass levels. 23 The result is equilibrium levels of biomass in each patch that depend on its own bioeconomic parameters, and not on assumptions regarding dispersal. The closed-form solutions for the equilibrium levels of biomass and effort, solved for by setting Eqs. (1) and (2) equal to zero, are illustrated in Table 1. An interesting question is how the assumptions regarding biological productivity affect the distribution of effort and the total amount of effort in the system relative to LARLOC. It is easy to see that the effort level in patch i is greater in PREDIS than in LARLOC if r j w j ð12w j Þ4r i w i ð12w i Þ: This condition will hold if population levels are below maximum sustainable yield and patch j is the high-cost patch. We also find that the total level of effort in the system is identical in the two models. This implies that if effort is greater in one area, it must be less in the other. When the condition is w j 4w i ; POSTDIS also results in a more skewed distribution of effort toward the lower-cost patches than LARLOC predicts. The reason for this skewness is simple. The lower-cost patch essentially has a higher biological productivity in PREDIS and POSTDIS than in LARLOC, all else being equal. A simple comparative static exercise will show that higher biological productivity (i.e., greater r) leads to greater equilibrium levels of effort. There is, however, a difference between aggregate effort levels in PREDIS and POSTDIS and those in LARLOC. Without further specification of relative parameters, it is not clear whether the aggregate level of effort in POSTDIS is less or greater than that in LARLOC. 24 Why the divergence between the models with respect to total effort? Larval production increases proportionally with stock or biomass level, and settlement of larvae is subject to the densitydependent mechanisms in POSTDIS. Therefore, for low populations the returns from larval dispersal are increasing, and it is possible depending on the patch population levels that gains (or losses) in one area might outweigh the potential losses (or gains) in the other. The effect is that 22 In POSTDIS and LARLOC, each patch converges to its respective carrying capacity in an unexploited system. In PREDIS, multiple equilibria are possible in the unexploited system, one of which is the respective carrying capacity. Another feasible equilibrium is that the population of Patch 1 exceeds its carrying capacity and the population of Patch 2 is below it. This latter equilibrium illustrates that the concept of carrying capacity in a spatial setting can be endogenous. A similar result is found with respect to the equilibrium level in a source-sink system when there is no exploitation [32]. 23 With rents nonlinear in effort, the equilibrium is fully integrated and simultaneous, assuming that the equilibrium levels of biomass and effort in each patch depend upon biological and economic parameters (except response rates) in all other patches. 24 The condition for aggregate effort levels to be higher in POSTDIS than in PREDIS and LARLOC is ðw 1 w 2 Þðq 2 ðw 1 1Þw 2 r 1 q 1 w 1 ðw 2 1Þr 2 Þ40:

17 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) Table 1 Open-access equilibrium levels of effort and biomass Biomass levels POSTDIS x i ¼ðc i þ jþ=pq i w i E i ¼ 1 q i w i PREDIS x i ¼ðc i þ jþ=pq i w i E i ¼ 1 1 q i w i 2 LARLOC x i ¼ðc i þ jþ=pq i w i E i ¼ 1 q i w i Effort levels "! # P 2 w j Þ ð1 w i Þ bðw i w j Þ r i 2 j¼1 "! # r j w j ð1 w j Þ bðw i w j Þ P 2 j¼1 r i w i ð1 w i Þ bðw i w j Þ aggregate effort levels could be higher (or lower) than in LARLOC, depending on the particular configuration Postreserve creation Simulating the effect of a marine reserve in this system is rather straightforward. We can solve Eqs. (1) and (2) with the added constraint that the effort level in the reserve is equal to zero. The parameters in the system are patch-specific and are assumed not to change with reserve creation. Under the current set of assumptions, the equilibrium levels of effort and biomass when patch 1 is closed are illustrated in Table 2. It is evident from this table that the density level in the reserve depends on the bioeconomic conditions in the open area and that the effort level in the open area depends on the spillover (both larval and adult) from the reserve. The simplest case for illustrating this is LARLOC, in which the greater the population outside the reserve (lower effort level), the greater the density level inside the reserve ð@x r 1 =@w 2 ¼ br 2 1 =ððb2r 1Þ 2 þ 4br 1 w 2 Þ 1=2 40Þ: This results from the fact that spillover depends on relative density levels. In general, the effect of the dispersal rate on the reserve density level is important in understanding how dispersal benefits can differ across species. In LARLOC, if b is equal to zero, then the reserve population is equal to its carrying capacity. As the dispersal rate increases, the reserve population density decreases because growth is now offset by emigration. 25 In PREDIS and POSTDIS, larval mixing is another way in which the reserve can contribute to population growth in the open area. In these models, the effect of increasing b is similar to that in LARLOC, though not surprisingly the degree of the effect is now somewhat muted because in both models the reserves are not only receivers but contributors of larvae in the system. As previously noted, the condition on aggregate catch levels is best expressed as simply whether the dispersal benefits are greater than the opportunity costs of closing the area. In this paper, the 25 This result is consistent with Hastings and Botsford [20], who demonstrate that if the goal of a reserve is conservation of biodiversity (as measured by greater population levels), then one should set aside an area large enough to minimize the spillover, which is analogous to closing an area in a closed system. It is interesting to note that by creating reserves of certain sizes, regulators could be giving comparative advantage to species with dispersal ranges less than the dimensions of the reserve:these species are then more likely to be protected than species with greater dispersal ranges [5].

18 18 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) 1 25 Table 2 Equilibrium levels of effort and biomass when patch 1 is a reserve Biomass level qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi POSTDIS x r 1 ¼ 1 2 ð1 w 2Þ b r þ 1 1 2r ðr 1 ðw 2 1Þþ2bÞ 2 þ 4w 2 r 1 ð2b þ r 1 Þ 1 x 2 ¼ðc 2 þ jþ=pq 2 w 2 PREDIS q x r 1 ¼ 1 2 b r þ 1 1 2r 1 x 2 ¼ðc 2 þ jþ=pq 2 w 2 LARLOC x r 1 ¼ 1 2 b r 1 þ 1 2r 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr 1 2bÞ 2 þ 4r 1 ð2bw 2 þ r 2 w 2 ð1 w 2 ÞÞ q x 2 ¼ðc 2 þ jþ=pq 2 w 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb r 1 Þ 2 þ 4bw 2 r 1 Effort level E 2 ¼ 1 r 22 q 2 w ðw 2 þ x r 2 1 Þð1 w 2Þþbðx r 1 w 2Þ E 2 ¼ 1 1 q 2 w 2 2 ðr 2w 2 ð1 w 2 Þþr 1 x r 1 ð1 xr 1 ÞÞ þ bðxr 1 w 2Þ E 2 ¼ 1 q 2 w 2 ½r 2 w 2 ð1 w 2 Þþbðx r 1 w 2ÞŠ opportunity cost is the forgone catch, and dispersal benefits include both larval and adult components. Because we are interested in comparing results across the models, we present the condition in terms of the state variables rather than substituting the equilibrium levels directly. Of course, the equilibrium level of reserve biomass can be substituted into the conditions to yield a condition expressed in parameters alone. In Table 3, we illustrate for all three models the condition of increased aggregate catch when the inequality is satisfied. Starting with LARLOC, it is easy to see that when all larvae are retained locally, the dispersal benefits are equal to the amount of adult spillover from the reserve ðbðx r 1 w 2ÞÞ; which depends on the dispersal rate and the relative density levels in the reserve and the open area. The opportunity cost is the lost catch from patch 1 (r 1 x 1 r (1 x 1 r )). In PREDIS, when larval production is densitydependent, the opportunity costs are the same as in LARLOC, but the dispersal benefits include an additional component because of the integrated growth function in the system. In this case, the dispersal benefits can be less than in LARLOC when r 1 x 1 r (1 x 1 r )or 1 w 2 (1 w 2 ). The introduction of larvae can reduce the potential dispersal benefits not because the additional avenue for spillovers has decreased the gross benefit but because the contribution of one patch s growth (or larvae) to another patch is greatest when the density is close to maximum sustainable yield levels. This effect decreases as the density level in either patch approaches one, as it would if a reserve were created in one area. In POSTDIS, both the dispersal benefits and the opportunity costs are greater than in the other models. With both sides of the expression larger, the net result is not evident. Unlike the situation in PREDIS, creating a reserve can only increase the dispersal benefits, and the lower the density in the open area, the greater the effect. This result is due to the density-independent larval production assumption, along with the density dependency in each local patch. It is also the case that the larger the differentials in patch densities, the greater the aggregate biological production. This reflects the assertions of biologists that closing an area will increase the number of larvae in the system, thereby improving population growth throughout the system and not just in the reserve. Next, we investigate different bioeconomic habitat conditions (parameter configurations) and the likelihood that reserves increase aggregate catches.

19 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) Table 3 Aggregate catch condition for all 3 models Aggregate catch increases after Patch 1 becomes a reserve if POSTDIS r 2 ðw 2 þ x r 1 Þð1 w 2Þþ2bðx r 1 w 2Þ4r 1 ðw 1 þ w 2 Þð1 w 1 Þ PREDIS 1 2 ðr 1x r 1 ð1 xr 1 Þ r 2w 2 ð1 w 2 ÞÞ þ bðx r 1 w 2Þ4r 1 w 1 ð1 w 1 Þ LARLOC bðx r 1 w 2Þ4r 1 w 1 ð1 w 1 Þ 4.3. Site selection The current precedent for site selection, as set in the creation of reserves in the Channel Islands off southern California and in the Tortugas off the Florida Keys, is that biologists first propose a list of areas, and then the fishing industry and other groups choose from the biologists list and suggest other areas more suitable to their needs. This iterative process has been successful in reaching agreements in the two locales, but no one would describe it as the path of least resistance [34]. The decision frameworks used in these cases required, moreover, that the process be sequential rather than simultaneous. But what if regulators could predict ex ante which sites would encounter the least resistance from the fishing industry and at the same time satisfy the stated biological goals that is, which sites and which fisheries would most likely benefit all stakeholders? Using the current framework, we illustrate some of the trade-offs involved in site selection that could help regulators reduce the set of feasible sites and therefore potentially reduce transaction costs in the negotiation process. For example, with all larvae retained locally, the setting aside of low-cost patches more likely leads to win win cases [32]. This is because low-cost patches under open-access conditions are the most heavily exploited (and thus have low density levels). With prereserve catches already depressed from overfishing, the opportunity cost of setting aside such an area is low. At the same time, closing the low-cost patch will yield a greater net increase in the reserve density levels, and a higher reserve density could yield higher dispersal benefits. There is, however, a trade-off involved in choosing the low-cost patch. In this case, the net dispersal benefits are potentially lower when the high-cost patch remains open because the greater the openarea density level, the smaller the difference between it and the reserve density. Whether this effect is large enough to outweigh the higher reserve population when the low-cost patches are set aside is unclear. What is clear, however, is that setting aside the low-cost patch unambiguously decreases the potential opportunity costs under open-access conditions. 26 POSTDIS presents a very different story, because the lower the density in the open area, the greater the increase in biological production as larvae spill over from the reserve. Although the opportunity costs of the forgone catch are higher when a high-cost patch is closed, the potentially larger dispersal benefits imply that closing a high-cost patch might be better. This is because the differential in density levels between the open area and the reserve will be greatest when the 26 That fishermen might support setting aside low-cost areas is counterintuitive and depends critically on the openaccess nature of the fishery.

20 20 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) 1 25 overexploited (low-cost) patch remains open. Therefore, in a fishery characterized by POSTDIS, it might be feasible to search out the higher-cost areas. To further explore the differences between the three models, we rescale the parameters and make the following simplifying assumptions:first, the growth rates are made equal across the system, though we still allow the cost price ratios to differ. Then we scale the dispersal rate based on the intrinsic growth rate (b ¼ r r) with rp1: With these simplifications, we can express the aggregate-catch condition in terms of three parameters, r; w 1 ; and w 2 : 27 By holding one parameter constant, we can investigate, in the parameter space of the other two free parameters, the range of possible conditions that yield win win outcomes across the three models. It is important to point out that in Figs. 2 and 3 we present the full range of solutions to the problem, holding one of the three parameter levels fixed at a numerical value. 28 The line where the aggregate catch is the same (zero line), with and without the reserve, is drawn in bold. We also black out shade the region of the parameter space where effort levels are negative. For expositional purposes, the aggregate condition as a function of two parameters is represented in two-dimensional space with contour lines. The contour lines increase (or decrease) as one moves from one line to the next in the direction away from the zero line, depicting the region of the parameter space where aggregate catches are greatest (or lowest). In addition, the contour lines are drawn for the same levels across the three models, allowing comparisons in the rates of change and levels across the models. In Fig. 2, we illustrate the effects of the dispersal-to-growth rate ratio on the aggregate condition when the economic and biological conditions across the two patches are identical. We have also investigated the cases with economic and biological heterogeneity, and the results are qualitatively the same. Using LARLOC as the benchmark, we see that if there is no dispersal, the condition is never satisfied. Although the potential increase in aggregate catches is greater for dispersal-to-growth rate ratios around 1/2, higher ratios can increase catches, though not to the same magnitude illustrated by the contour lines. PREDIS gives similar predictions, but the nature of the surface does differ, as shown by the contour lines. In particular, we find that without adult dispersal, larval spillover is insufficient to outweigh the forgone catch. This is consistent with the results presented in Tuck and Possingham [38]. The POSTDIS predictions, however, are qualitatively different. First, it is evident that even without adult dispersal (r=0-b=0), reserves can increase aggregate catches because of the increase in larval spillover a result not found in PREDIS. Second, the number of contour lines to the left of the zero line indicates that the magnitude of the increase is potentially greater. Third, adult dispersal has a somewhat muted effect on the system:higher adult-dispersal rates lower the reserve population density and therefore reduce the number of larvae produced in the system. 27 Under these assumptions, the intrinsic growth rate has only a scaling effect on the aggregate-catch condition and thus does not affect whether the condition is positive or negative. It is also interesting to note that although the reserve literature has focused on the importance of the dispersal rate per se, this parameterization illustrates that for the aggregate catch condition it is the ratio of the dispersal rate to the growth rate that matters. 28 The analysis considers the two extreme cases:either all larvae are retained locally or all larvae enter the larval pool. In PREDIS and POSTDIS, as the fraction of locally retained dispersal approaches one, the models converge to LARLOC, implying that when we consider intermediate cases the results of PREDIS and POSTDIS will approach those of LARLOC. These predictions are also best-case scenarios, because we are assuming that all larvae survive the trip to and from the larval pool and that the larvae are uniformly distributed across the system.

21 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) Fig. 2. Reserve creation in a homogeneous economic and ecological system. Note:The values associated with the contour lines, consistent across the models, are increasing to the left of the zero line (H R ¼ H) and decreasing to the right. The region labeled H R 4H is where aggregate catches within the reserve are greater than when both patches are open to fishing; and the region of H R oh is where aggregate catches are lower. The black area is the infeasible region of the parameter space. Finally, it is evident that the range of possible parameter combinations is much larger in PREDIS than in the other models. Fisheries that are best described by such a setting, therefore, are more likely to yield win win conditions than when there is just adult dispersal or when larval production is density dependent, or both. Holding the dispersal-to-growth rate ratio fixed (r ¼ :2), we can ask how patch-specific heterogeneity affects aggregate catches. The results are illustrated in Fig. 3. For LARLOC, we recreate the results presented in the Sanchirico and Wilen [32] model of a fully integrated system, in which the feasible region where the win win condition is satisfied lies below the zero line. It is evident that closing the low-cost patch is more likely to lead to win win than closing the high-cost patch, all else being equal. In addition, the area remaining open does not necessarily have to be overfished for aggregate catches to increase in the system. Whereas in the previous results PREDIS was essentially similar to LARLOC, there are some differences in the bioeconomic parameter space. First, the area of the feasible region is slightly smaller. Second, both the future reserve and the open area need to be overexploited (density levels below 0.4) for aggregate catches to likely increase after a reserve is created. POSTDIS again leads to very different predictions. As discussed earlier, it is possible under these conditions that closing the high-cost patch may lead to greater catches. This result is evident in Fig. 3, where the feasible region for the condition to be met is larger and includes a substantial

22 22 J.N. Sanchirico / Journal of Environmental Economics and Management 49 (2005) 1 25 Fig. 3. Reserve creation in a heterogeneous economic system. Note:The values associated with the contour lines, consistent across the models, are increasing below the zero line (H R ¼ H) and decreasing above it. The black area is the infeasible region of the parameter space. The results shown are for the case in which the dispersal-to-growth rate ratio is 0.2. subregion perhaps as much as 75% of the unexploited patch where the reserve is the high-cost patch. Therefore, even if an area is not overfished, closing it could increase aggregate catches in the fishery. If technological advances in fishing have opened areas that previously acted as de facto reserves and if these are the high-cost patches of today, we can see how they were able to support higher aggregate catches. Not surprisingly, the loss of these de facto reserves is one reason for the recent surge of support for creating reserves. 5. Discussion Because economists have only recently begun to analyze the implications of including space in models of renewable resources, there are many biological and economic aspects of these models that are not well understood. To help fill that gap, we illustrate the critical roles that fundamental ecological processes such as density dependence and independence, together with the nature of ecological networks, play in economic ecological system interactions. In our examples, density