Theoretical Ecology

, Volume 1, Issue 2, pp 117–127

Spillover from marine reserves related to mechanisms of population regulation

Authors

    • Department of Environmental Science and PolicyUniversity of California
  • Roger M. Nisbet
    • Department of Ecology, Evolution and Marine BiologyUniversity of California
  • Steven D. Gaines
    • Department of Ecology, Evolution and Marine BiologyUniversity of California
    • Marine Science InstituteUniversity of California
Original Paper

DOI: 10.1007/s12080-008-0012-6

Cite this article as:
Kellner, J.B., Nisbet, R.M. & Gaines, S.D. Theor Ecol (2008) 1: 117. doi:10.1007/s12080-008-0012-6

Abstract

Spillover of fish from marine reserves to adjacent harvested waters may be mediated by density-independent movement, density-dependent movement, or both. If dispersal is by random movement, populations within the reserve must be regulated by density-dependent population growth (DDG). Density-dependent movement (DDM) can also regulate the population if accelerated emigration from a reserve to the surrounding fishing grounds leads to substantially increased mortality. Using spatially explicit models, we show that stock per unit area is bounded for DDG and increases with size for DDM. With DDG, spillover rate per unit area of reserve is maximized with reserves around 50% larger in linear dimension than the minimum size for population persistence. With DDM, spillover per unit area of reserve increases with reserve size. The results highlight the need for the mechanism of population regulation to be incorporated into theoretical and empirical investigations of marine reserve ecology.

Keywords

SpilloverDensity-dependent movementPopulation regulationPersistenceMarine protected areaFishery yields

Introduction

Marine reserves aim to mitigate against high rates of fishing mortality by protecting a portion of the stock within specified boundaries. The buildup of a reserve population projects two potential benefits to the surrounding exploited areas: regional larval replenishment and local augmentation of juvenile and adult biomass to the adjacent fishing grounds. Of these, ironically larval replenishment through dispersal of larvae from females protected within reserves (“recruitment effect”) has received far greater conceptual emphasis (Botsford et al. 2001; Gaines et al. 2003), even though empirical evidence for larval replenishment is limited.

By contrast, conceptual studies of adult spillover from reserves are rare, even though emigration of post-settlement fish from reserves to adjacent areas (“spillover effect”) provides an easily documented, direct measurement of reserve effectiveness, which has been documented repeatedly through tag–recapture movement studies (Gitschlag 1986; Attwood and Bennett 1994; Johnson et al. 1999). Additional evidence supporting the spillover effect has been demonstrated by commercial fisheries, including the congregation of fishers adjacent to reserve boundaries (Shorthouse 1990; Murawski et al. 2000) and increased catch per unit effort near protected areas (Alcala and Russ 1990; Yamasaki and Kuwahara 1990; Ramos-Espla and McNeill 1994; Rakitin and Kramer 1996). Politically, local support for the implementation and continued enforcement of a marine reserve is likely to be influenced by the yield in the remaining fishing grounds. As a result, optimizing spillover of adults while maintaining large protected populations within reserves has become an important objective for many marine reserves because it may provide rapid, observable (i.e., harvestable) feedback of reserve effectiveness to the displaced fishery.

Given that the size of a marine reserve is most often limited by socioeconomic and political considerations, there has been much discussion about optimizing reserve size with respect to protected stock size, productivity, and spillover. Ecologically, population buildup within the reserve and export to adjacent areas are determined by species-specific population growth dynamics and movement strategies in response to habitat availability and quality. Optimizing reserve size for stock buildup, productivity, and spillover therefore requires understanding the density-dependent processes responsible for population regulation within the reserve.

The two mechanisms of population regulation that are likely to predominate in marine reserves are density-dependent population growth (DDG) and density-dependent (crowding-accelerated) movement (DDM) (Lizaso et al. 2000). In this paper, we use a spatially explicit model to determine how these density-dependent processes differentially influence stock size within the reserve (a measurable surrogate for recruitment effect) and the net export rate of juveniles and adults to the surrounding fished areas (the spillover effect). We show how the mode of population regulation influences the way in which standing stock and spillover rate scale with reserve size and formulate a “benefit-per-area” optimization method that characterizes the proportional benefit of expanding the protected area.

Model formulation

Reserve

Optimally, a reserve population should be robustly self-sustainable. Even with the recent emphasis on marine reserve networking (Murray et al. 1999; Roberts et al. 2003), autonomy of each individual reserve population will be the best buffer against catastrophic environmental disturbances and intense fisheries harvest that can disrupt multiple reserve connectivity. Independence requires self-replenishment, such that the population is sustained by the retention of propagules produced solely by the reserve residents. This provides the most conservative challenge for persistence of a reserve population because population growth cannot be maintained by the arrival of young from other reserves or fished areas. Accordingly, we model a population with limited larval dispersal where population growth is governed by local conditions, an assumption that is appropriate for populations whose spatial scale of recruitment is similar to or smaller than the movement scale of adults. We assume limited larval dispersal to isolate the effects of adult movement to surrounding fished areas from the more frequently studied effects of export of young.

We assume that the reserve has absorbing boundaries, i.e., fish that leave the reserve are rapidly caught (Turchin 1998, p 312). This abstraction reasonably represents a scenario where many fishing boats congregate around the reserve edge in an effort to maximize their catch rate and yield, a situation which is prevalent near many marine reserves especially when fishers are displaced by the establishment of a new reserve (Polunin and Roberts 1993; McClanahan and Kaunda-Arara 1996; McClanahan and Mangi 2000; Kelly et al. 2002). Elsewhere, we show that this distribution of fishing effort is the expected optimal solution (Kellner et al. 2007). There is no baiting by the fishers to entice fish out of the reserve, but emigrants are immediately accessible and are assumed to be caught before they are able to reproduce or return to the reserve. This high catch efficiency near the reserve boundary is assumed to imply zero standing stock outside the reserve. While this scenario represents an extreme idealization, it can be rationalized as a conservative first step for modeling persistence of a population in the face of intense harvest. Furthermore, this assumption is best justified for populations that are close to collapse, where the likelihood of replenishment from outside a reserve may be minute.

The model reserve is sited within continuous, homogeneous habitat, which, in conjunction with the assumption of no baiting by the fishing fleet, asserts that fish cannot explicitly detect reserve boundaries (Lima and Zollner 1996). Landscape continuity enhances the likelihood of landscape-unconscious movement, which could facilitate spillover. Further support for this assumption comes from recordings of coral reef fish movements across fringing reefs and reserve boundaries by Chapman and Kramer (2000), who provide some evidence that movement may be more highly dependent on continuity and type of substrate, site attachment, and home range than explicitly on reserve boundaries. Conversely, an observational study of queen conch in a Caribbean marine reserve also suggests that movement constrained by natural substrate barriers led to a crowding effect of decreased shell length as population density recovered from fishing pressure (Béné and Tewfik 2003).

Population balance equations

Partial differential equations are used to develop a simple, deterministic model of an autonomous population in a nonfished circular reserve of diameter L. Fish population dynamics are described by the reaction–diffusion equations
$$\frac{{\partial n}}{{\partial t}} = \left[ {\frac{\partial }{{\partial x}}\left( {D\left( n \right)\frac{{\partial n}}{{\partial x}}} \right) + \frac{\partial }{{\partial y}}\left( {D\left( n \right)\frac{{\partial n}}{{\partial y}}} \right)} \right] + F\left( n \right)\;{\text{within}}\;{\text{the}}\;{\text{reserve}}$$
(1)
$$n = 0\;{\text{at}}\;{\text{the}}\;{\text{reserve}}\;{\text{edges}}\;{\text{and}}\;{\text{outside}}\;{\text{the}}\;{\text{reserve}}$$
(2)
where n = n(x, y, t) represents the fish density at position (x, y) at time t. The term in square brackets represents changes in local population density due to movement and the final term describes changes due to local fecundity and mortality. Species-specific functions D(n) and F(n) are independent of time and position, and describe the diffusion and population growth rate, respectively. Equation 2 corresponds to the absorbing boundary condition discussed previously.

Holmes et al. (1994), Turchin (1991, 1998), and many other authors suggest that diffusion is an appropriate approximate description of movement for a variety of populations. Numerous empirical and theoretical studies support this: Examples include turbot (Sparrevohn et al. 2002), chinook salmon (Zabel 2002), green crab (Grosholz 1996), and marine microorganisms (Okubo 1980), as well as many terrestrial organisms. Further explanation of the characteristics, assumptions, and appropriateness of diffusion to describe population movement can be found in Turchin (1998), Holmes et al. (1994), Lima and Zollner (1996), and Okubo and Levin (2001).

Integrating the local population density, n, over the entire reserve area yields the standing stock size for the reserve (Nisbet and Gurney 1982). Similarly, integrating with respect to the outward flux along the reserve boundary generates a total spillover rate for the reserve (see, for example, Cantrell and Cosner (2003), p 23).

Our model focuses on standing stock and spillover rate at steady state, a reasonable assumption given that protected populations reach average long-term values of mean density and biomass within 1 to 3 years of reserve establishment, and are sustained thereafter (Halpern and Warner 2002). This rapid approach to steady state indicates that density-dependent processes are likely to increase in intensity as population stock recovers (Lizaso et al. 2000; St. Mary et al. 2000). We explore two forms of population regulation in the following sections, density-dependent population growth (DDG), and density-dependent movement (DDM).

Density-dependent population growth model (DDG)

A compensatory (negative) relationship between population growth and density has been observed in a number of marine populations including fish, lobster, and abalone (Shepherd 1990; Pollock 1993; Koslow et al. 1995; Hixon and Carr 1997; Anderson 2001). Density-dependent demographic rates such as survival and fecundity arise from (1) resource limitation (food, settlement sites, refuge sites, etc.), (2) direct interference among conspecifics, and/or (3) increases in the per capita predation rate in response to prey aggregation (Lizaso et al. 2000; Rose et al. 2001; Holbrook and Schmitt 2002).

To describe populations regulated solely by fecundity and mortality, we assume logistic population growth and random diffusion. Thus
$$F(n) = rn\left( {1 - \frac{n}{K}} \right)\;{\text{and}}\;D\left( n \right) = D_0 = {\text{constant}}$$
(3)
where r and K represent, respectively, the intrinsic growth rate and the carrying capacity, both of which are assumed constant within the reserve. The constant density-independent diffusion coefficient implements random-walk dispersal with infinitesimally small steps whose size and frequency are independent of population density or location in space. Substituting from Eq. 3 in Eq. 1 yields Fisher’s (1937) model
$$\frac{{\partial n}}{{\partial t}} = \left[ {\frac{\partial }{{\partial x}}\left( {D_0 \frac{{\partial n}}{{\partial x}}} \right) + \frac{\partial }{{\partial y}}\left( {D_0 \frac{{\partial n}}{{dy}}} \right)} \right] + rn\left( {1 - \frac{n}{K}} \right).$$
(4)

Equations combining diffusive movement with logistic growth (such as the Verhulst–Pearl reaction term used here) have been developed extensively for models of population dynamics (e.g., Skellam 1951; Holmes et al. 1994; Cantrell and Cosner 2003). Various forms of these types of models have also been explored in a number of recent papers related to fisheries management and harvest refugia (Gaines et al. 2003; Neubert 2003; Fryxell et al. 2006; Baskett et al. 2007).

Density-dependent movement model (DDM)

Accelerated movement in response to increasing population density has been documented in a number of marine organisms including fish, echinoderms, and seals (Rosenberg et al. 1997; Hixon 1998; Gaggiotti et al. 2002). As population density accumulates inside the reserve, increased local competition for resources or interference between conspecifics may cause organisms to move preferentially to areas of lower density (Rakitin and Kramer 1996; Travis et al. 1999; Lizaso et al. 2000; Abesamis and Russ 2005). A DDM strategy “arises either because individuals of a population can directly obtain information about the spatial distribution of their comrades or because the direct response of individuals to local population density can cause their average behavior to reflect gradients in population density” (Okubo 1980). When accelerated movement leads to changes in the mortality or fecundity rates (e.g., increasing export to the lethal fished surroundings), positive density-dependent movement acts as a self-regulating compensatory mechanism that stabilizes the population below the resource carrying capacity (Gurney and Nisbet 1975; Lizaso et al. 2000; Rose et al. 2001).

To describe nonterritorial populations regulated by crowding-accelerated movement, we assume density-independent growth and “biased-random motion” (Gurney and Nisbet 1975). The model functions take the form
$$F\left( n \right) = rn\;{\text{and}}\;D\left( n \right) = D_0 + D_n n{\text{.}}$$
(5)
In this equation, r represents the density-independent per-capita growth rate (births minus deaths) of the local population at any location within the reserve. The species-specific constants, D0 and Dn, describe, respectively, random diffusivity in the absence of conspecifics and the increase in the rate of random movement per additional individual added to the local population. The population dynamics are then described by the equation (Gurney and Nisbet 1975)
$$\frac{{\partial n}}{{\partial t}} = \left[ {\frac{\partial }{{\partial x}}\left( {\left( {D_0 + D_n n} \right)\frac{{\partial n}}{{\partial x}}} \right) + \frac{\partial }{{\partial y}}\left( {\left( {D_0 + D_n n} \right)\frac{{\partial n}}{{\partial y}}} \right)} \right] + rn.$$
(6)

Density-dependent dispersal across reserve boundaries has also been incorporated into spatially-implicit models initiated by Hannesson (1998), Conrad (1999), and Sanchirico and Wilen (2001).

Conditions for population persistence

The dynamics of the DDG and DDM models are identical if populations are sufficiently small everywhere in the system such that the nonlinear terms in Eqs. 4 and 6 become negligible. The condition for population persistence is well known and was originally applied to determine the size of nutrient patches needed to sustain phytoplankton blooms (Kierstead and Slobodkin 1953). Persistence is only possible if the diameter of the system, L, is large enough that on average, an individual resides in the reserve long enough to reproduce before emigrating. With some tedious algebra (e.g., Kot 2001, chapter 16), this requirement can be shown to imply
$$L \geqslant L_{\text{c}} \;{\text{where}}\;L_{\text{c}} = 4.81\sqrt {\frac{{D_0 }}{r}} .$$
(7)

Analysis and numerical solution of the model equations

Some analysis of a 1D version of the model (detailed in part 2 of the Appendix) is possible and provides insight into our 2D model behavior. Most of the proofs rely on a special mathematical property of the 1D models that does not carry over to higher dimensions; consequently, results for our 2D model relied on numerical solutions of the model equations. However, a comparison of model behavior in 1D and 2D (Table 1) demonstrates that most of the qualitative dynamics are independent of dimensionality.
Table 1

Dependence of reserve population size and spillover rate for increasing values of model parameters. Results are analogous for 1D and 2D unless otherwise indicated

Model parameter

Conditions for persistence (Kot 2001, p 293) For 1D, m = π For 2D, m = 4.81

DDG

DDM

Total population size (TPS)

Spillover rate

Total population size (TPS) and spillover rate

L

Small reserve L close to Lc

LLc where \(L_{\text{c}} = m\sqrt {\frac{{D_0 }}{r}} \)

Proportional to LLc

Equal to r × TPS

1D, increases asymptotically as L3 for L→∞ 2D, increases asymptotically as L4 for L→∞

Large reserve LLc

1D, proportional to length 2D, proportional to area

1D, has finite limit as L→∞ 2D, proportional to L as L→∞

 

r

rrmin where \(r_{\min } = D_0 \left( {\frac{m}{L}} \right)^2 \)

Approaches reserve capacity (KL) as L→∞

Increases with r

1D, increases approximately linearly with respect to r when r is large) 2D, increases linearly with respect to r

 

D0

D0 ≤ D0,max where \(D_{0,\max } = r\left( {\frac{L}{m}} \right)^2 \)

Approaches 0 as D0D0,max

Unimodal convex relationship

Approaches 0 as D0D0,max

 

K or Dn

N/A

Proportional to K

Proportional to K

Approaches 0 as Dn increases

Model equations were solved numerically for population distribution and reserve population size under time-dependent and steady-state conditions using FEMLAB v2.3 (Comsol, Burlington, Massachusetts, USA), a MATLAB (MathWorks, Natick, Massachusetts, USA) toolbox for solving partial differential equations using the finite element method (FEM). FEM is an approximation solution technique for solving partial differential equations that have complicated geometry, boundary conditions, and/or spatio-temporal dynamics. All simulations, results, and figures reported here are for the 2D DDG and DDM models and used a minimum mesh size of 5,000 points for the spatial domain.

Results

Dependence of steady-state standing stock size and spillover rate on model parameters

The density dependence in both the DDG and DDM models leads to a nonzero, globally stable, steady state, provided the reserve diameter exceeds the critical value in Eq. 7. Table 1 summarizes how the steady-state values of standing stock size and spillover rate vary with model parameters for each mode of regulation.

For both models, increasing reserve size provides more area for the population to occupy, resulting in a greater standing stock. Moreover, as the circumference-to-area ratio declines with increasing reserve size, the influence of edge effects declines, leading to a higher average density across the spatial domain. For the DDG model, standing stock scales asymptotically with reserve area, whereas standing stock increases much more rapidly for the DDM model – at a rate proportional to area squared (Table 1).

In both models, the export rate also increases as reserve size increases. The additional regulatory movement mechanism in the DDM model imposes substantially greater emigration as reserve size increases, as the spillover rate due to density-dependent movement scales with population density. Accordingly, spillover rate for populations regulated by density-dependent population growth scales with reserve diameter, whereas populations regulated by density-dependent movement have a substantially higher spillover rate that scales with squared reserve area.

Dependence of standing stock and spillover rate on other model parameters can be inferred from the model analysis with respect to reserve length (Table 1, Appendix).

Spatio-temporal dynamics: approach to steady state and population distribution

The dynamics of population recovery after reserve establishment can be explored by evaluating the approach to steady state. In the absence of space or in homogeneous spatial models with impervious (reflecting) boundaries, a logistically growing population monotonically approaches the local carrying capacity, K, at each spatial locale for positive growth rates. For configurations with absorbing boundaries such as the DDG model utilized here, the population distribution is additionally influenced by the emigration rate of individuals from the reserve. Thus, the standing stock at steady state is lower than the reserve capacity, K multiplied by the reserve area (Fig. 1a).
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Fig. 1

Time development of standing stock (a, b) and spillover rate (c, d). Solid and dashed lines represent two reserve sizes, of diameter lengths 1.5 and 5 times the minimum length for population persistence, respectively. For all figures, D0 = 1, r = 1, Lc = 4.81, and n(x, y, 0) = 0.1. For the DDG model, K = 1 such that the reserve capacity (K × reserve area) is 41 fish for the smaller reserve and 454 fish for the larger reserve. Based on these capacity values, the parameter Dn in the DDM model was set to 4.929 and 45.99 \(\frac{{{\text{distance}}^{\text{3}} }}{{{\text{fish}} \times {\text{time}}}}\) corresponding to L = 1.5Lc and L = 5Lc

It is interesting to note that the approach to steady state is almost indistinguishable between the DDG and DDM models for parameters of K and Dn that yield the same steady-state standing stock (Fig. 1a and b). By contrast, the two models exhibit very different spillover rates, and the DDM spillover rate at all times is greater than that of the DDG model (Fig. 1c and d).

Spillover of individuals at the reserve boundary leads to a decay in the population distribution away from the center of the reserve (Fig. 2a and b). As the reserve size increases above the minimum area required for persistence, the DDG reserve population fills to capacity at more spatial locations and takes on a platykurtic shaped distribution, while the DDM reserve population maintains a more sharply peaked distribution (Fig. 2a and b). Heterogeneity in the density distribution across space is, in both cases, not due to deliberate aggregation, simply a consequence of the balance between production and net migration rate.
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Fig. 2

Spatial distribution of population density at equilibrium for two reserve sizes, aL = 1.5Lc and bL = 5Lc. Solid lines represent populations regulated by density-dependent growth; dotted lines represent populations regulated by density-dependent movement. Parameters are consistent with Fig. 1

Reserve size considerations for optimizing standing stock and spillover rate

Determining the importance of reserve size versus other reserve design criteria (such as shape and habitat quality and continuity) requires understanding the proportional benefit of adding more reserve area. We performed a “benefit-per-area” analysis that quantifies the incremental gain in standing stock and spillover rate per each additional area of reserve added.

Total standing stock and spillover rate increase as reserve size expands for populations that exhibit either density-independent or density-dependent movement (Table 1), but the proportional benefit of protecting more area is contingent upon the population regulatory mechanism (Fig. 3). For both forms of population regulation, the average population density per unit area of reserve is maximized by protecting the largest reserve possible (Fig. 3a and b). This results because the proportional export of individuals becomes less influential in determining standing stock size as the edge-to-area ratio decreases.
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Fig. 3

“Benefit-per-area” analysis for total population size per unit reserve area (a, b) and spillover rate per unit reserve area (c, d) for populations regulated according to the DDG and DDM models. For a and b, the y-axes are non-dimensionalized values of population density divided by reserve area. Complete details of the non-dimensionalization are provided in part one of the Appendix; the focal point is the scaling of the population density to the density-dependent parameters as \(\tilde n = \frac{n}{K}\) and \(\tilde n = \frac{{D_n }}{{D_0 }}n\) for the DDG and DDM models, respectively

For populations regulated by movement (DDM model), the spillover rate per unit area increases rapidly with increasing reserve area (Fig. 3d). In contrast, the DDG model spillover rate per unit area initially increases as the population builds from a minimal size to a peak at approximately L = 1.5Lc when the reserve occupancy is around 40%, and tapers off as the circumference-to-area ratio declines (Fig. 3c). By the time the DDG reserve is five times larger than the critical length, the resource usage efficiency is ~80%, and the spillover rate per unit area has significantly declined below the maximum (Fig. 3a and c).

Discussion

Consideration of movement dynamics relative to reserve size is critically important for population persistence within marine protected areas, as well as their ability to provide larval replenishment through the buildup of a standing stock and spillover of adults to adjacent harvested populations.

Self-sustainability of a reserve population with mobile adults that is regulated through local processes requires that the area protected from fishing is large enough that individuals reproduce before emigrating to the surrounding area. This requirement is what sets the minimum reserve size, Lc. Analogous results have been deduced for open populations of sedentary adults where population persistence is dependent upon the fraction of natural larval settlement that occurs within the protected borders of reserves, such that sustainability of an isolated reserve population requires that the reserve is larger than the mean larval dispersal distance (Botsford et al. 2001).

For populations regulated by density-dependent movement, both reserve standing stock and spillover rate are maximized by protecting the largest reserve area as possible. In the case of populations with random movement that are regulated by compensatory density dependence, there is a trade-off between optimizing standing stock size (larger reserves are better) and spillover rate (“intermediate-sized” reserves are best). This latter trade-off between standing stock buildup and post-settlement export has been suggested in previous discussions of marine reserve design (National Research Council 2001; Halpern and Warner 2003), but the results of the DDG model presented here are the first attempt at relating the optimal “intermediate” reserve size relative to the rates of population growth and diffusion. The DDG model suggests that a reserve size approximately 50% larger than the critical diameter required for population persistence will provide the best return in terms of spillover per unit area. However, the distribution of payoff is very asymmetric, and a somewhat larger reserve would be prudent to enhance the likelihood of a higher return, given the limitations of the model, uncertainties in parameters, and other biotic or abiotic factors that may shift the steep incline on the optimized “benefit-per-area” curve.

A further argument for larger reserves comes from the DDM model for which the benefit increases without limit as reserve size increases. Although this model is ecologically implausible for large systems (in view of the prediction of very large central population densities), it provides a pointer to an appropriate strategy for a population that is regulated by both movement and growth. Such a population will also exhibit an optimal spillover rate driven by density-dependent growth, but the peak will flatten and may not appear until very large reserve sizes due to density-dependent movement (Fig. 4). Given the disparity of optimal reserve sizes for populations regulated by DDM vs. DDG, these results highlight the importance of incorporating the underlying mechanisms of movement and population control into theoretical and empirical investigations of source–sink populations (also see Armsworth and Roughgarden 2005).
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Fig. 4

“Benefit-per-area” analysis for spillover rate per unit reserve area for populations regulated by both density-dependent growth and movement

For simplicity, we made extreme assumptions regarding fishing mortality in the area surrounding the reserve and population dynamics. Our “absorbing” boundary condition is a defensible representation of situations where harvesters are heavily “fishing-the-line” (Polunin and Roberts 1993; Murawski et al. 2000; Kelly et al. 2002). While the quantitative predictions will certainly change with a different distribution of harvesters and through other environmental factors (e.g., currents) and life history traits (e.g., which life stages disperse), the qualitative results are likely to prove robust, as long as the reserve represents a region of positive population growth surrounded by a harvested region where deaths outweigh births.

When populations are not locally regulated (e.g., uncoupled adult-recruit dynamics), the model predictions may not hold. For example, an open population with decoupled recruitment and randomly dispersing mobile adults will experience the greatest spillover rate per unit area from a small marine reserve. Truly independent recruitment implies that the protected population is self-sustainable for any reserve size; therefore, the spillover rate per area rises as the ratio of edge to area increases.

The model also lends predictions about the spatial distribution of populations across reserves. Hence, adopting monitoring protocols that incorporate comprehensive spatial measurements of population density can potentially provide information on life history processes that may not be derived from pooled inside/outside comparisons.

Our analysis was based on single-species population models. Given that reserves aim to protect an ensemble of species, one challenge is how to relate our conclusions to a multispecies fishery, whose species have different growth and movement rates. At minimum, the reserve size must be large enough to enable population persistence of the most highly mobile species, although this may be at the expense of reduced spillover for highly productive, less mobile species, which would benefit more from numerous smaller reserves. We address some of these multispecies trade-offs for populations mediated by density-dependent growth in Kellner et al. (2007).

The recent emphasis on the establishment of marine reserve networks has focused attention on the spacing of individual reserves with respect to the dispersal distance of propagules as a means of maintaining population connectivity and regional recruitment (Kaplan and Botsford 2005; Halpern et al. 2006). Similarly, decisions must be made regarding the size of each individual reserve in a network. These choices will be heavily weighted by export to adjacent areas if spillover is central to gaining the support of the local fishing community. In conjunction, these point to a pathway for the design of marine reserve networks whereby the size of an individual reserve should be guided by the optimal spillover rate of adults, while the spacing between reserves should be set by larval dispersal distances. These two criteria alone allow for a calculation of the recommended total area to be protected.

Acknowledgments

Support was provided by the Coastal Toxicology Lead Campus Component of the University of California Toxic Substances Research and Teaching Program, the University of California – Santa Barbara, US NSF (grant DEB01-08450), Santa Barbara Coastal Long Term Ecological Research, the Pew Charitable Trusts, the Andrew W. Mellon Foundation, and the Partnership for Interdisciplinary Studies of Coastal Oceans (PISCO) funded primarily by the Gordon and Betty Moore Foundation and the David and Lucile Packard Foundation. This is PISCO contribution number 282. The authors thank Bruce Kendall, Bob Warner, Shaun Belward, and Elizaveta Pachepsky for helpful discussions.

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© Springer Science+Business Media B.V. 2008