No-take marine reserves can enhance population

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ARTICLE No-take marine reserves can enhance population persistence and support the fishery of abalone Can. J. Fish. Aquat. Sci. Downloaded from www.nrcresearchpress.com by STANFORD UNIV. on 08/24/15 For personal use only.

Marisa Rossetto, Fiorenza Micheli, Andrea Saenz-Arroyo, Jose Antonio Espinoza Montes, and Giulio Alessandro De Leo

Abstract: A critical aspect in the design of a marine reserve (MR) network is its spatial configuration (i.e., the number, size, and spacing of the individual reserves), particularly how these features influence the effect on fisheries. Here, we derived a sizebased, spatially explicit, stochastic demographic model to explore how different spatial configurations of MR networks can affect abundance and commercial yield of the green abalone (Haliotis fulgens), taking as a reference case the abalone fishery of Isla Natividad in Baja California Sur (Mexico). Our analysis suggests that a network of MRs can have a positive effect on abalone population abundance and a slightly negative effect on fishery output with respect to traditional maximum sustainable yield (MSY; i.e., with no reserves). Simulations show that maximum catches achievable with MRs are, under the best configuration, ⬃2%–14% lower than traditional MSY depending on the total fraction of the fishing grounds protected. In the case of overexploitation, long-term yields can increase following the implementation of MRs. In addition, in the presence of MRs, abundances and yields are much less sensitive to systematic errors in the enforcement of the optimal harvesting rate compared with situations in which MRs are not present. Given the limited dispersal ability of the species, the best outcomes in terms of fishery output would be achieved with very small reserves — around 100 m wide — so to maximize larval export in the fishable areas. Our results indicate appropriately designed MR networks are an effective strategy for meeting both conservation and economic goals under uncertainty. While the size of the existing reserves in Isla Natividad seems adequate to protect the abalone stock, smaller reserves could maximize fishery benefits, although this poses challenges for enforcement. Résumé : Un aspect clé de la conception d’un réseau de réserves marines (RM) est sa configuration spatiale, soit le nombre, la taille et l’espacement des différentes réserves, en particulier l’influence de ces caractéristiques sur l’incidence du réseau sur les pêches. Nous avons mis au point un modèle démographique stochastique spatialement explicite et basé sur la taille pour explorer l’incidence de diverses configurations spatiales de réseaux de RM sur l’abondance et le rendement commercial de l’haliotide verte (Haliotis fulgens), en utilisant comme scénario de référence la pêche aux haliotides d’Isla Navidad, en BasseCalifornie du Sud (Mexique). Notre analyse donne a` penser qu’un réseau de RM peut avoir un effet positif sur l’abondance de la population d’haliotides et un effet légèrement négatif sur la production de la pêche par rapport au MSY traditionnel (c.-a`-d., sans réserve). Les simulations montrent que les prises maximums possibles en présence de MR sont, dans la meilleure configuration, d’environ 2 % a` 14 % inférieures au MSY traditionnel, tout dépendant de la proportion cumulative protégée des zones de pêche. Dans le scénario de surexploitation, les rendements a` long terme peuvent augmenter après la mise en place des RM. De plus, en présence de RM, les abondances et rendements sont beaucoup moins sensibles aux erreurs systématiques dans l’application du taux de prises optimal par rapport aux situations desquelles les RM sont absentes. Étant donné la faible capacité de dispersion de l’espèce, les meilleurs résultats en termes de production de la pêche pourraient être obtenus avec de très petites réserves — d’environ 100 m de largeur — afin de maximiser l’exportation de larves vers les zones ouvertes a` la pêche. Nos résultats indiquent que des réseaux de RM bien conçus constituent une stratégie efficace pour atteindre des objectifs économiques et de conservation dans un contexte d’incertitude. Si la taille des réserves existantes a` Isla Navidad semble adéquate pour protéger le stock d’haliotides, de plus petites réserves pourraient maximiser les bénéfices pour la pêche, mais poseraient toutefois des défis en ce qui concerne leur application. [Traduit par la Rédaction]

Introduction In the past several decades, marine reserves (MRs) have become a widely advocated approach to marine resource management. The establishment of MRs has been proven to be effective in the protection of biodiversity and ecosystem structure and function (Leslie et al. 2003; Micheli et al. 2004; Lester et al. 2009). When productive populations are protected within no-take MRs, spillover of larvae produced in the reserves may also sustain recruit-

ment in the surrounding fishing zones (Roberts et al. 2001; Gell and Roberts 2003), which may increase yields of overexploited stocks (Holland and Brazee 1996; Sladek-Nowlis and Roberts 1999; Gerber et al. 2003) and contribute to reducing catch variability (Lauck et al. 1998; Mangel 2000). The effects of protection on sustainability and yield of populations depends on the spatial configuration (i.e., on how features such as size of individual reserves and total covering interact with species’ dispersal characteristics). To best accomplish the conservation goal of self-persistence, MRs

Received 3 December 2013. Accepted 20 April 2015. Paper handled by Associate Editor Marie-Joëlle Rochet. M. Rossetto. Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, via Ponzio 34/35, I-20123 Milano, Italy. F. Micheli and G.A. De Leo. Stanford University, Hopkins Marine Station, Pacific Grove, CA 93950, USA. A. Saenz-Arroyo. El Colegio de la Frontera Sur (ECOSUR), San Cristobal de las Casas, México. J.A.E. Montes. Sociedad Cooperativa de Produccion Pesquera Buzos y Pescadores, Isla Natividad, Baja California Sur, México. Corresponding author: Marisa Rossetto (e-mail: [email protected]). Can. J. Fish. Aquat. Sci. 72: 1–15 (2015) dx.doi.org/10.1139/cjfas-2013-0623

Published at www.nrcresearchpress.com/cjfas on 8 June 2015.

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should be large enough relative to the dispersal distance of the target species to ensure sufficient larval retention (Botsford et al. 2001; Kaplan et al. 2006). On the other hand, maintaining high yields in fisheries may require a more sophisticated spatial configuration; in fact, a fishery-effective network of MRs requires that the size of, and space between, marine reserves is designed so that reproductive output within no-take areas can contribute through spillover to recruitment outside MRs (i.e., in the still fishable ground; Halpern and Warner 2003; Hastings and Botsford 2003; Neubert 2003; Gaines et al. 2010). Abalones (Haliotis spp.) are large benthic mollusks that have been intensely fished worldwide, and stocks are currently depleted in most countries (Shepherd et al. 1998a; Karpov et al. 2000; Dichmont et al. 2000). The implementation of MRs is receiving growing attention as a promising tool for protecting and possibly recovering abalone stocks. Empirical work on the effect of MRs on Haliotis spp. confirms the expectation that sedentary species with dispersal in the larval phase and a history of overexploitation might benefit from the implementation of MRs (Hastings and Botsford 1999; Botsford et al. 2003). Inside MRs, significantly greater densities (up to ten times) relative to unprotected sites have been reported for several Haliotis species (Wallace 1999; Maliao et al. 2004; Parnell et al. 2005). In addition, inside reserves abalones attain bigger size than abalones in fished areas (Edgar and Barrett 1999; Wallace 1999; Parnell et al. 2005; Maliao et al. 2004; Micheli et al. 2008, 2012). Finally, within the boundaries of protected areas, abalones are more likely to be found in aggregations relative to exploited sites (Parnell et al. 2005). As abalone fecundity increases with individual size (Tutschulte 1976) and fertilization success is enhanced when abalones are found in high densities (Button and Rogers-Bennett 2011), combined responses of abalone densities and size structure to protection can augment the reproductive output of protected populations (Micheli et al. 2012). The enhanced production of larvae could, in turn, sustain recruitment in the surrounding fished areas if MRs are properly designed to allow the export of larvae beyond the reserve boundaries (Micheli et al. 2012). Recent modeling work on the conservation and economic effectiveness of proposed networks of MRs in central California indicates that for Haliotis rufescens, small protected areas can support population persistence inside their boundaries, and conservation benefits increase with increasing total reserve coverage (White et al. 2010, 2011). The model of White et al. (2010, 2011) indicates that MRs could increase abalone yields when the resource is heavily fished, but given the species’ short dispersal ability, the expected catches under the proposed network configurations were substantially lower than the theoretical maximum sustainable yield (MSY) (White et al. 2010, 2011). The goal of the present work is to investigate under which spatial arrangement — in terms of size of individual reserves and total closed area — MRs can be part of the optimal management of abalone fisheries (i.e., to assess whether a properly designed network of MRs can foster the recovery of overexploited populations and, at the same time, sustain fishery output). To address this question we developed a size-structured demographic model of green abalone (Haliotis fulgens) and used it to explore alternative scenarios of MR implementation. While our model structure is general, we performed the analysis taking inspiration from the green abalone fishery of Isla Natividad, Baja California Sur, Mexico, where two MRs have been recently established with the aim of protecting remaining populations of abalone, allowing for their recovery and enhancing fishery yields. In the Baja California peninsula, the fishery of Haliotis corrugata and H. fulgens is still a valuable business; however, current catch levels of both species remain far below the historical landings in the middle of the

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Can. J. Fish. Aquat. Sci. Vol. 72, 2015

past century, and great concern exists regarding the sustainability of the fishery (Ponce-Díaz et al. 1998; Morales-Bojórquez et al. 2008). As other fishing communities in Baja California are establishing similar MRs (so far at two additional locations; F. Micheli and A. SaenzArroyo, unpublished data), it is crucial to explore under what reserve network configurations larval spillover from the protected areas may enhance abalone population persistence and possibly support fishery yield.

Materials and methods We developed a size-structured, spatially explicit demographic model for H. fulgens to explore the possible effect of MR implementation on abalone population and fishery yields. The demographic model was developed with reference to the green abalone fishery operating since the 1940s in Isla Natividad, a small island on the Pacific coast of Baja California Sur (Mexico). A detailed description of this fishery is presented in the section on The abalone fishery of Isla Natividad. The model incorporates quantitative information on abalone growth, size-dependent mortality, size-dependent fecundity, larval and settler survival, and dispersal in the larval phase, as described in the sections on Model description and Parameter values and specific formulation. We used the demographic model to explore how different combinations of fishing harvest and MR network configuration might influence (i) abalone population persistence in terms of long-term abundance and (ii) long-term fishery yields, as detailed in the section on Decision variables, scenarios, and model simulations. Elasticity and sensitivity analyses (see section below) were carried out to evaluate the importance of vital rates and the effect of uncertainty in demographic parameters on model outputs. The abalone fishery of Isla Natividad Isla Natividad is representative of the 22 fishing cooperatives operating in the Baja California peninsula targeting green (H. fulgens) and pink abalone (H. corrugata) (Ponce-Díaz et al. 1998). The management of abalone fishery on the island is based primarily on exclusive access granted for 20-year periods by the Mexican federal government, in addition to minimum landing sizes (MLS) and annual quotas for the target species. The MLS in Isla Natividad for H. fulgens is 155 mm in shell length (SL). In the past decades, annual quotas have been set at as much as 30% of the estimated commercial stock biomass of animals above the MLS (Shepherd et al. 1998a). Control of fishing effort is well enforced, and poaching is virtually absent thanks to the exclusive access privileges and the geographical isolation. However, errors in stock assessment can lead to overestimation of population densities, and therefore, actual harvest rates may exceed the target. Catches were high in the period 1965–1990, with a maximum harvest of 140 t·year−1 registered in the early 1970s; from 1990, green abalone harvests on the island have declined, remaining below 20 t·year−1 in the last decade (see online Supplementary material, Fig. S11). The fishing ground of the island stretches more or less uniformly around the ⬃15 km island perimeter and up to 500 m from the coast (Fig. 1), covering an area of about 750 hectares (ha). To facilitate management, abalone fishing grounds of Isla Natividad are divided into blocks of 500 m × 500 m, a practice also in place in other abalone cooperatives along Baja California (Rodríguez-Valencia et al. 2004; Morales-Bojórquez et al. 2008). In 2006, two no-take MRs, 500 and 1000 m wide, were established on Isla Natividad, covering ⬃8% of the fishing grounds (Fig. 1).

Supplementary data are available with the article through the journal Web site at http://nrcresearchpress.com/doi/suppl/10.1139/cjfas-2013-0623. Published by NRC Research Press

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Fig. 1. Location of Isla Natividad, Baja California Sur, Mexico. The fishing grounds are divided into blocks (⬃500 m × 500 m); dark grey blocks represent the two marine reserves currently established on the island.

Model description The fishing ground was schematically modeled as a closed circular belt of 150 contiguous patches of 100 m width, extending 500 m from the coastline, with the first and last patch being connected as in a torus. All patches were assumed to be equally suitable for abalone so that the demographic processes characterizing abalone life cycle were identical in each patch. Although it is possible that some patches could support higher growth rates or carrying capacity owing to local favorable conditions of food and (or) habitat, no data were available to this regard; therefore, we simulated a homogeneous habitat without incorporating spatial patterns in abalone’s life traits. The demographic model described yearly changes in the population abundance of green abalone in each patch as resulting from (i) reproduction and planktonic larval dispersal; (ii) harvesting; (iii) natural mortality; and (iv) somatic growth. What differentiated one patch from another was whether the patch was exploited

at a harvesting rate h or whether it was set aside as no-take reserve (h = 0). The structure of abalone population in each patch was represented by nine size classes of 25 mm width: three juvenile classes (5–30, 30–55, 55–80 mm) below the size at first reproduction (i.e., ⬃80 mm; Shepherd et al. 1991); three classes including unfished adults (80–105, 105–130, 130–155 mm); and three classes including fished individuals (155–180, 180–205, 205–230 mm), whose sizes are comprised between the MLS and the maximum size of green abalone registered in Baja California (⬃230 mm; Rossetto et al. 2013; Rodríguez-Valencia et al. 2004). Accordingly, the state variables of the model were Ni,z,t (i.e., the abalone density (individuals·ha−1) in size class i, patch z at time t. The basic dynamic equation is (1)

Nz,t⫹1 ⫽ MtNz,t ⫹ Rz,t Published by NRC Research Press



where Nz,t = [N1,z,t,N2,z,t,…,N9,z,t]T is the vector of population densities in each size class at time t in patch z, M is a 9 × 9 transition matrix (Caswell 2001) including growth, adult


(2)

冢

g1,1s1 g1,2s1 g1,3s1 g1,4s1 g1,5s1 g1,6s1 g1,7s1 g1,8s1 g1,9s1

0 g2,2s2 g2,3s2 g2,4s2 g2,5s2 g2,6s2 g2,7s2 g2,8s2 g2,9s2

0 0 g3,3s3 g3,4s3 g3,5s3 g3,6s3 g3,7s3 g3,8s3 g3,9s3

0 0 0 g4,4s4 g4,5s4 g4,6s4 g4,7s4 g4,8s4 g4,9s4

0 0 0 0 g5,5s5 g5,6s5 g5,7s5 g5,8s5 g5,9s5

0 0 0 0 0 g6,6s6 g6,7s6 g6,8s6 g6,9s6

0 0 0 0 0 0 g7,7s7 (1 ⫺ h) g7,8s7 (1 ⫺ h) g7,9s7 (1 ⫺ h)

0 0 0 0 0 0 0 g8,8s8 (1 ⫺ h) g8,9s8 (1 ⫺ h)

where gi,j is the probability of individuals of class i to growth to class j the following year, si is the size-dependent fraction of individuals in class i in a given year that survive to the following year, and ht is the fraction of individuals yearly harvested by fishermen in year t. The number of eggs produced in each patch z at time t was calculated as follows: (3)

Ez,t ⫽

兺0.5ew ␰ N

where e is the number of eggs per unit mass (g−1) produced by a green abalone female; wi is the mass (g) of abalone in size class i; ␰i is the fraction of sexually mature individuals of class i; and the factor 0.5 accounts for a 1:1 sex ratio. Our model assumes dispersal in the larval stage; therefore, fertilized eggs successfully developing in larvae were assumed to be partially retained in the source patch and partially exported to contiguous patches. In particular, probability of dispersal at distance x (m) from the center of the patch was described by a Gaussian distribution, namely x2

(4)

p(x, t) ⫽

兹2␲␴2d(t)

2 e 2␴d(t)

␳(k) ⫽

冕

1

兹2␲␴2d(t) 100冉k⫺ 冊 2

Sz,t ⫽ ␴E

⫺

e

x2 2␴d2(t) dx

1

兺 E ␳(|j ⫺ z|) j,t

J

(7)

Rz,t ⫽ ␴S Sz,t

(8)

Ct ⫽ 106

兺A兺 w h N i t

Zf

i,z,t

If

where A is the area of each patch (5 ha), Zf is the set of fishable patches (i.e., those that are not set aside as no-take zone), If identifies the size classes above MLS, and ht is the fraction of commercial size abalone that are harvested from each fishable patch in year t. Model variables and parameters are summarized in Table 1, and a detailed description of specific model equations and parameters is presented below.

Body mass Body mass wi (g) was computed from mean shell length li of an individual in class i (mm) using the length–mass relationship for green abalone reported in Shepherd et al. (1998a): wi ⫽ 2.24 × 10⫺5 li3.36

1

where “100” in the integration interval is the size of any of the 150 patches representing the coastline. Accordingly, ␳(0) is the retention rate (i.e., the fraction of locally produced larvae that remains in the source patch). The number of settlers in patch z at time t was calculated summing the contribution of all patches j as follows: (6)

where J is the set of patches, and ␴E represents the survival from eggs to settlers, which accounts also for the fertilization success. Recruitment in patch z at time t was assumed to be equal to number of settlers S in patch z at time t that survive to the first size class:

(9)

冉 2冊

(5)

冣

Parameter values and specific formulation

with standard deviation ␴d(t) set so that 99% of larvae are retained at a distance ±d(t) from the source. The fraction of larvae ␳ dispersing k patches away from the source patch, with k = ±1, ±2, …, was hence computed as follows: 100 k⫹

0 0 0 0 0 0 0 0 s9 (1 ⫺ h)

with ␴S being the survival of settlers. Total yearly catch on the island in biomass (t) was computed as follows:

i i i,z,t

i

1

survival, and harvest, and R = [Rz,t,0,0..0]T is the recruitment vector. Matrix M takes the following form:

Natural mortality in adults In abalone, natural mortality rates of settled individuals have been shown to be size-dependent, with survival probabilities increasing with size (Rossetto et al. 2012). Accordingly, size-specific mortality rates ␮i (year−1) were calculated from mean body mass in each size class wi (g) using the empirical allometric relationship between instantaneous mortality rates and body mass reported for abalones in natural environments (Rossetto et al. 2012): (10)

ln ␮i ⫽ ␪ ⫹ ␣ · ln wi

where ␣ was assumed to be −0.317 (SD = 0.027), and ␪ was 0.635 (SD = 0.102). These are the estimated means and standard deviations of the parameters of the log–log relationship obtained for H. fulgens, as in Rossetto et al. (2012). Annual survival si (i.e., the Published by NRC Research Press


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Table 1. Parameters and variables of the demographic model for Haliotis fulgens.


Symbol

Description (units and values when applicable)

1 1 1 3 3 3 1, 3, 8 3, 6 6, 7, 12 1, 7

Harvest MLS h Cw,t

— 2, 8 8

Minimum landing size in shell length (155 mm) Fraction of commercial size abalone that are harvested from each fishable patch Total yearly catch on the island in year t (tonnes)

Length and mass wi Mean body mass of individuals of class i (g) li Mean shell length of individuals of class i (mm) Natural mortality ␮i Instantaneous annual mortality rate of individuals in class i (year−1) si Fraction of individuals in class i that survive from year t to year t + 1 ␪ Intercept of the allometric relationship between instantaneous mortality rates and body mass (mean = 0.631, SD = 0.102) ␣ Slope of the allometric relationship between instantaneous mortality rates and body mass (mean = −0.371, SD = 0.027)

3, 8, 9, 10 9 10, 11 2, 11 10 10

Somatic growth gi,j Growth transition from class i to class j G Growth parameter of the probabilistic Gompertz growth curve (year−1) Ln Mean asymptotic length (mm) ␴L2 Variance of the mean asymptotic length (mm2) ␤ Phenomenological parameter of the probabilistic Gompertz growth curve ␥ Phenomenological parameter of the probabilistic Gompertz growth curve

2, A6 A1, A2 A3, A4 A3, A4 A3, A4 A3, A4

Fecundity e Number of eggs produced per gram of individual (eggs·g−1) (mean = 3772, SD = 330) ␰i Fraction of sexually mature individuals of class i

3 3

Larval and settler survival ␴E Survival from eggs to settlers (mean = 3.09×10−3, SD = 0.33×10−3) ␴S Survival of settlers during the first year of life ␴0 Survival of settlers at low densities (0.01) K Carrying capacity of settlers (107 individuals·ha−1)

6 12 7, 12 12

Larval dispersal d Larval dispersal distance (m); variable according to a gamma distribution with maximum probability of dispersal around 1000 m p(x,t) Probability of dispersal at distance x (m) from the center of the patch ␳(k) Fraction of larvae dispersing k patches away from the source patch

fraction of individuals in class i that survive to the following year) was then computed as (11)

Equation

Index and spatial variables t Time index (year) i Size class index z Patch index If Set of size classes above minimum landing sizes (MLS) Zf Set of fishable patches A Patch area (5 ha) Abalone density in size class i, patch z, at time t (individuals·ha−1) Ni,z,t Ez,t Number of eggs produced in patch z at time t (eggs·ha−1) Sz,t Number of settlers arriving in patch z at time t (individuals·ha−1) Rz,t Number of new recruits in patch z at time t (individuals·ha−1)

si ⫽ e⫺␮i

Somatic growth Growth transitions represent the fractions of individuals in a given class that remain in the same size class or grow to the following ones after a year. To model the somatic growth of H. fulgens, we assembled a dataset of 53 observations consisting of annual growth rates in shell length registered in the field and in laboratory conditions. Specifically, we retrieved growth data from tagging experiments conducted in Bahía Tortugas and in Bahía Asunción (Baja California Sur, Mexico) by Shepherd et al. (1991) and by Guzmán del Próo and Lopez-Salas (1993) and from laboratory experiments by McCormick et al. (1992), Aviles and Shepherd (1996), and Durazo-Beltrán et al. (2003). Laboratory data were in-

— 4 5, 6

cluded as they provide information on the growth of juveniles, not represented in the field samples. A summary of the data reporting the range of size at capture, time at liberty, and length increment is reported in Table 2. To estimate growth transitions from green abalone mark–recapture data, we used a probabilistic Gompertz model for non-negative growth, developed by Bardos (2005) to describe the somatic growth of another abalone species (Haliotis rubra), which consists of a probability distribution of size increments given the size at capture in which the asymptotic length is conditional on the initial length. The somatic growth model, described in more detail in Appendix A, included three biologically significant parameters — the growth parameter G, the mean asymptotic length Ln, and its variance ␴L2 — and two additional, phenomenological parameters, ␤ and ␥ (Bardos 2005). The model was fitted to the tag–recapture data using maximum likelihood estimation. The estimated growth parameters were Published by NRC Research Press




Table 2. Summary of growth data used for estimation of growth transitions of H. fulgens. Length increment (mm·year−1)

Source

N

Location

Shepherd et al. 1991 Guzmán del Próo and Lopez-Salas 1993 Durazo-Beltrán et al. 2003 Aviles and Shepherd 1996 McCormick et al. 1992

32 13

Bahía Tortugas Bahía Asunción

59.4–140.3 68–143

1 0.98

11–43.6 13–50

Laboratory Ocean barrel culture Land and ocean culture

5.7–6.1 14.8 35.5

0.9 0.9 0.73

14.6–15.9 16.3 19.1–21.9

3 1 4

then used with eq. A6 in Appendix A to derive the transition elements gi,j (i.e., the fraction of individuals in size class i in year t that grow into class j the following year). Fecundity The number of eggs e (g−1) produced by a green abalone female was assumed to be equal to 3772 with a standard deviation of 330 (Tutschulte 1976). ␰i (i.e., the fraction of sexually mature individuals of class i) was assumed to be a sigmoidal increasing function of size li, as described in Rossetto et al. (2013). Larvae and settlers survival The survival from eggs to settlers ␴E was assumed to be equal to 3.09 × 10−3 (SD = 0.33 × 10−3) as estimated by Rossetto et al. (2013) on stock–recruitment data from Isla Natividad. The survival of settlers during the first period of life can be negatively affected by settler density because of competition for food, space, or refuges (Connell 1985; McShane 1992). Indeed, density-dependent survival in the postsettlement phase has been observed for several Haliotis species, both in laboratory (Daume et al. 2004; Day et al. 2004) and natural conditions (McShane 1991; Shepherd et al. 1998b), with instantaneous mortality of settlers increasing linearly with settlers’ density (Shepherd 1990). Accordingly, recruitment was modeled as a Ricker function with settlers’ survival (␴S) being a negative exponential function of their density: (12)

Time at liberty–rearing (years)

Initial length (mm)

␴S ⫽ ␴0e⫺Sz,t/K

where ␴0 represents the survival of settlers in low-density conditions, and K is a measure of carrying capacity (Sladek-Nowlis and Roberts 1999). Direct measures of postlarval survival during the first year of life are not available for H. fulgens in the natural environment. However, experimental studies conducted in the wild for several abalone species (Shepherd et al. 1998b) suggest an instantaneous mortality rate of about 2 (months−1) in the first month of life and 0.2 (months−1) from 2 to 12 months at lowdensity conditions. By integrating these settler mortality rates over the year, we assumed a value of ␴0 of the order of 1%. The value of K was assumed to be 107 individuals·ha−1, given the high mortality rates observed at postlarvae densities above this threshold (⬃1000 individuals·m−2; McShane 1991; Daume et al. 2004). Abalone larval dispersal Available studies suggest that abalone larvae — which spend 5–10 days in the water column before attaching to the substratum (Guzmán del Próo et al. 2000) — are mostly retained in areas close to parental reefs, because of the short larval duration and because extensive kelp beds and small-scale eddies substantially attenuate the larval flux (Shepherd et al. 1992; Guzmán del Próo et al. 2000). A majority of studies indicates that abalone larval dispersal distance is limited to hundreds of metres (Prince et al. 1988; McShane et al. 1988; Shepherd et al. 1992; Guzmán del Próo et al. 2000; Shanks et al. 2003; Temby et al. 2007). However, strong currents or storms may occasionally drive abalone larvae several kilometres

away (Tegner and Butler 1985; Sasaki and Shepherd 1995). In Isla Natividad, an experimental study on abalone larvae spillover suggested dispersal distances of ⬃300 m (Micheli et al. 2012). In the present study, we assumed that the larval dispersal distance d(t) of abalone varies stochastically from year to year according to a gamma distribution with shape = 3 and rate = 0.007 (Fig. S21). This right-skewed distribution has a mode around 300 m and assumes that dispersal on the scale of hundreds of metres is more plausible than dispersal over longer distances (>1 km). Decision variables, scenarios, and model simulations Reserve effect on long-term abalone abundance and yield We used the model to explore the effects on stock and fisheries yield of a combination of different levels of harvest rates h (fraction of abalones over MLS removed each year from the patches open to fishing), protection level (PL, the overall share of fishing ground set aside in no-take reserves), and size (m) of the individual reserves in the MR network. Mean harvest rate ranged between 0 and 0.9; PL was set to 0% (no reserve), 10%, 20%, 30%, 40%, or 50% of the overall fishing ground; size of individual reserves was 100, 500 (the actual size of the existing no-takes zones in Isla de Natividad), or 1500 m, with the first and the last size being smaller and greater, respectively, than the mean dispersal distance of abalone larvae. Individual reserves were assumed to be uniformly spaced along the coastline so as to cover the assigned PL of the fishing ground; an example of a 30% PL achieved with no-take areas of different sizes is depicted in Fig. 2, and the number of protected areas for each combination of PL and size is reported in Table 3. Each combination of the harvest rate, PL, and individual reserve size identified a management scenario. To initialize the model, we reproduced the plausible exploitation history of green abalone (see Shepherd et al. 1998a) by simulating 50 years of fishing in which 30% of individuals above the MLS are removed each year in each patch. The resulting population abundance and structure was used as initial conditions for each management scenario. To account for year-to-year variability in larval dispersal, we used Monte Carlo simulations and randomly drew the value of d (dispersal distance) at each time step from the gamma distribution as explained in the section on Model description. Fishery performance and stock size for each scenario were analyzed over 150 years, a time frame that was sufficient to reach stochastic stationarity. For each scenario we replicated simulations a large number of times and used the corresponding output to compute mean and standard deviations of stock abundance and fishery yield. Preliminary tests showed that both means and standard deviations of model outputs were already stable after 100 replicates, and the confidence intervals remained substantially constant for the number of replicates ranging between 1000 and 10 000; therefore, we set the number of replicates to 1000, as this appeared to be a reasonable choice to derive robust statistics without excessively increasing computing time. All simulations were run using R (version 3.1.0). Published by NRC Research Press


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Fig. 2. Example of three alternative spatial arrangements of marine reserves to achieve an overall protection of 30% of the fishing grounds on an island coastline divided in 150 patches. Shaded areas indicated protected blocks, while white areas indicated fished blocks. A 30% protection of the fishing grounds could be achieved with networks of 45 no-take zones of 100 m, 9 zones of 500 m, or 3 zones of 1500 m. The larval dispersal kernel corresponding to d = 300 m is also depicted. For sake of simplicity, the coastline around the island is depicted as a linear array, although in the model the first and the last patches are connected (see Materials and methods). Reserve size: 100 m


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Elasticity and sensitivity analyses Sensitivity analyses are crucial to determine which parameters are most influential on model results and how uncertainty propagates on model outputs. In classical population matrix models of the form Nz,t+1 = AtNz,t, sensitivity is defined as the partial derivative of a population’s finite growth rate (␭) to changes in the projection matrix elements (aij) (Caswell 2001). Elasticities (i.e., proportional sensitivities) are frequently used instead of sensitiv-

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Here, we rewrote eq. 1 in the form Nz,t+1 = AtNz,t, and following the approach of Caswell et al. (2004) for models with density dependence, we evaluated elasticities of the projection matrix at the long-term equilibrium, when ␭ = 1, in the absence of fishing and under the scenario of no MRs. Additionally, we assess how uncertainty in the estimation of demographic parameters affects long-term population abundances and yields following the approach proposed by McCarthy et al. (1995). Specifically, we were interested to assess model sensitivity to highly uncertain parameters (i.e., mortality parameters (␪ and ␣), eggs produced per gram of individual (e) and survival from eggs to settlers (␴E)). We proceeded as follows: (i) A set of model parameters was randomly sampled from normal distributions defined according to the mean and standard deviation of their estimates (see Table 1). Published by NRC Research Press



Fig. 3. Predictions of the probabilistic Gompertz growth curve fitted on green abalone length-increment data. (a) Comparison between observed and simulated length increments ⌬l (mm) versus initial length l1 (mm). Larger black dots are observed length increments (see Table 2). Small points are the result of 20 stochastic simulations of the Gompertz growth curve. (b) Predicted size-at-age based on 20 stochastic simulations of length increments, assuming that individuals are 1 mm shell length at age zero.

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l 1 (mm) (ii) A simulation of 150 years each was run by keeping the model parameters drawn at point (i) fixed for the entire simulation time. For each simulation, we retained the value of long-term population abundance and yield (at time t = 150). (iii) We went back to point (i), randomly drew a new set of model parameters, and replicated the process 100 times. Sensitivity was assessed for the scenario with no MR (PL = 0) and over harvest rates from h = 0 to h = 0.9. We explored the relationship between the 100 values of model parameters (independent variables) drawn at point (i) and the values of population size and long-term yield (response variables) using multiple ordinary regression with normal error structure, using the “src” function of R package “Sensitivity” (version 3.1.0). The relative importance of each model parameter was indicated by its standardized regression coefficient, that is, the coefficient value divided by its standard error (Cross and Beissinger 2001). The absolute value of the standardized regression coefficient represents the importance of the parameter in determining the status of the population, while the sign represents the direction of the contribution. For each set of random parameters, we also retrieved the harvest rates that maximize catches to evaluate their sensitivity to parameter uncertainty.

Results Somatic growth The maximum-likelihood estimation of the Gompertz model on length-increment data provides G = 0.56 ± 0.10 year−1, Ln = 150.39 ± 49.29 mm, ␴L2 = 55.96 ± 28.67 mm2, ␤ = 1.47 ± 0.63, and ␥ = 1.52 ± 0.85. The Gompertz model fitted to mark–recapture data provides length increments that closely resemble the observed data (Fig. 3a). The growth model efficiently captures the high growth variability of the small size classes and the acceleration of growth rates of the juvenile stages, which can exceed 50 mm·year−1 (Fig. 3a). Although tagged individuals were all below 150 mm SL (Fig. 3a), the model predicts a growth trajectory eventually approaching ⬃250 mm (Fig. 3b), which is consistent with maximum sizes of green abalone observed in California (Parnell et al. 2005). The growth model indicates that H. fulgens reaches the size of sexual maturity (136 mm SL) at an age between 4 and 5 years and enters the fishery (155 mm SL) at an age between 5 and 7 years. Model evaluation The model predicted that under pristine conditions, densities of individuals >10 cm SL — those generally sampled by underwa-

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age (y) ter visual censuses (Parnell et al. 2005; Rossetto et al. 2013) — were 2456 individuals·ha−1, a value comparable to baseline abundances and unfished densities estimated for several Californian abalones (Rogers-Bennett et al. 2002; Micheli et al. 2008). After 50 years of fishing at h = 0.3, the model predicted that densities of abalones above 10 cm were reduced to 410 individuals·ha−1. In the absence of MRs (i.e., under the hypothesis of spatially homogeneous fishing effort), long-term abundances and yields were insensitive to the magnitude and variability of abalone dispersal distance. Estimates of H. fulgens densities at Isla Natividad in 2008–2009 in fished sites ranged from 62 ± 164 to 135 ± 260 individuals·ha−1 (Rossetto et al. 2013). Our model results for abalone densities after exploitation were higher than observed values, but they were close to the upper range of the field estimates. The model predicted that fishing at h = 0.3 for 50 years would initially deliver a maximum catch from the entire island of 149 t·year−1 that gradually decreased to 12 t·year−1 (mean over 50 years: 32 t·year−1). These figures are comparable to real catch data recorded in Isla Natividad, where, in the past decades, yearly catches reached a maximum of 143 t in 1973 and declined to 10 t·year−1 in 2008, with a mean value over 50 years of 53 t·year−1 (Fig. 1). Reserve effect on long-term abundance In the absence of MRs, the harvest rate in each patch needed to be substantially smaller than 0.3 to prevent abalone population abundance to drop below 4 million individuals (i.e., ⬃10% of its undisturbed level of approximately 32 million individuals; Fig. 4d, Fig. S31). Establishment of MRs had a positive effect on the green abalone population of Isla Natividad by offering protection to a fraction of the stock; the larger the PL, the greater the long-term abalone population size (Fig. 4, Fig. S31). Long-term population abundance depended on the combination of harvest rate and PL; the larger the harvest rates in the fishable patches, the larger the fraction of habitat that needed to be protected to prevent population decline (Fig. 4, Fig. S31). However, when reserves covered more than 40% of the habitat, mean abalone abundance never dropped below 15 million individuals (almost half of its undisturbed conditions) for any level of harvest rate applied outside MRs (Fig. 4f). Under current management (h = 0.3, PL = 10%), abalone population was predicted to stabilize around 9.26 ± 0.43 million individuals, corresponding to 28% of the undisturbed conditions (Fig. 4b, Fig. S31). Overall, mean long-term abalone stock size tended to be greater with small or medium reserves (i.e., 100 and 500 m; Fig. 4) than with larger reserves (i.e., 1500 m), except for the combinations of high harvest rates and low PL (Figs. 4c, 4e). Published by NRC Research Press

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Fig. 4. Long-term abalone abundance for the whole island (millions of individuals, mean ± SD of the 1000 replicates under the assumption of year-to-year variability in larval dispersal distance) as a function of protection level (a–c) and harvest rate (d–f) for small reserves of 100 m width (black symbols), intermediate reserves of 500 m width (gray symbols), or with networks of large reserves of 1500 m width (white symbols).



Variability in model output due to the assumed year-to-year variation in dispersal distance was limited and appeared to be slightly greater for intermediate reserve size (Fig. 4). Mean coefficients of variation of long-term abundances were 3.09%, 3.99%, and 2.97% for small, intermediate, and large reserves, respectively. Reserve effect on long-term fishery yield In the absence of reserves (PL = 0), the maximum long-term yield (or MSY) stabilized at 25 t of fresh mass (without shell) per year, and the corresponding mean harvest rate (hMSY) was equal to 0.1 of abalones above the 155 mm MLS (Fig. 5d, Fig. S41). When MRs were present, the mean long-term yields were determined by the combination of PL and the harvest rates in the fished patches. When harvest rates in the fishable patches were maintained at sustainable levels (h ⬵ 0.1), expanding the reserve network entailed a reduction of fishery yields compared with traditional management (Fig. 5a). On the contrary, when harvest rates in the fishable areas exceeded the sustainable ones, increasing PL could have a positive effect on fishery yields (Figs. 5b–5c). The relationship between the mean long-term fishery yield and the harvest rate h was unimodal but with substantial differences depending upon the PL (Fig. 5, lower panels); in the absence of MRs, the yield–h relationship was characterized by a very narrow pick around h ⬵ 0.1, with the yield sharply declining to very low values for harvest rates above 0.2 (Fig. 5a). In the presence of MRs covering a limited fraction of the grounds such as a 10% closure, the shape of the relationship between fishing mortality and fishery yields was similar to that in the absence of MR (Fig. 5e). On the contrary, for PL equal to 30% or larger, the yield–h relationship remained high and remarkably flatter at the right side of the curve, providing high yields for a wide range of harvesting rates (Fig. 5f). In general, small reserves sizes (100 m) had better fishery performances than intermediate (500 m) or large (1500 m) reserve sizes (Figs. 5, Fig. S41). In scenarios with small reserves, however, yield was predicted to be low under low PL and high harvest rates (Figs. 5c, 5e). In scenarios with large reserves, sustainable yield was substantially lower than MSY, although less sensitive to changes in h and in PL. We calculated that with the current extension of MRs (PL = 10%, h = 0.3), the maximum yield could not exceed 9.6 ± 0.5 t·year−1 (Fig. 5, Fig. S41). Year-to-year variation in yield resulting from variability in dispersal distance appeared to be limited and slightly greater for intermediate reserve size (Fig. 5), with mean coefficients of variation on long-term yields being 4.07%, 8.84%, and 7.05% for small, intermediate, and large reserves, respectively. Maximum catches achievable with networks of small reserves were 2%–14% lower than MSY under optimal traditional management depending on the total fraction of the fishing grounds protected (Fig. 6). With medium and large reserves, maximum yields were between 8% and 27% and between 9% and 44% lower than MSY in absence of reserves, respectively (Fig. 6). Optimal management with a network of MRs generally requires harvesting rate to be larger than the harvest rate providing MSY under traditional management (Fig. 6). Elasticity and sensitivity analyses The elasticity analysis of the green abalone projection matrix suggests that the population at the long-term equilibrium is most sensitive to survival in the largest size class (205–230 mm; Fig. S51), reflecting the fact that fecundity increases with body size in abalone, and thus the largest abalones are the most important spawners. All the coefficients on the main diagonal of the projection matrix are characterized by high elasticity as well as the coefficient of the first row corresponding to the product of fecundity and settlement survival. The sensitivity analysis performed over a wide range of variation of the parameters of the size-dependent mortality function, settlement survival, and fecundity showed that across all harvest rates scenarios, the demographic parameters that most affect


both population size and catches were mortality parameters ␪ and ␣ (Fig. 7). Specifically, an increase in ␪ (entailing an increase in mortality of all size classes) and ␣ (entailing an increase in mortality of adults) reduces both the stock and the long-term yield. Conversely, an increase in egg per unit mass (e) and survival from egg to larvae (␴E) positively affected both population size and catches. However, their effect was negligible, as the recruitment (the actual number of settlers recruiting each year) was modulated by the demographic bottleneck provided by the densitydependent survival. Overall, simultaneously accounting for parameter uncertainties translated to a high variability in model outputs, with mean coefficients of variation up to 138% and 188% for abundances and catches, respectively. Importantly, the harvest rates predicted to maximize catches (hMSY) were quite robust to parameter uncertainty, remaining equal to 0.1 for 56% of cases, 0.2 for 36% of cases, and 0.3 for 8% of cases.

Discussion Our modeling analysis shows that an appropriately designed MR network can meet both conservation and economic goals in the management of the green abalone fishery. In particular, our results suggest that a network of MRs can increase population abundance and make the system more robust to errors in the determination and enforcement of catch limits, while entailing only minor losses in yields comparable to MSY under traditional management (e.g., Hastings and Botsford 1999). Thus, MR networks can be part of management strategies aimed at maintaining abalone catches while addressing unavoidable uncertainties in the management system and the environment. Our model suggests that the implementation of MRs network is expected to have a positive effect on abalone population persistence, providing an effective tool for augmenting the abundance of the stock. The modeling results indicate that, for abalone, conservation benefits are maximized for medium reserves (500 m wide), which are predicted to ensure high long-term population abundance and a negligible risk of collapse even when harvest rates in the remaining fishing areas are intensive. The predicted positive effect of protection in preserving high abalone population abundance is consistent with empirical studies of existing MRs conducted on several species of the genus Haliotis (Wallace 1999; Maliao et al. 2004; Parnell et al. 2005) and on other marine species (Roberts et al. 2001; Micheli et al. 2004). Our results are also in agreement with analytical solutions of mathematical population models showing that the establishment of MRs can support high levels of biomass (Bensenane et al. 2013) and that for sedentary species with dispersal in the larval phase, the best outcomes in terms of conservation benefits should be met by reserves that are large relative to the dispersal distance (Hastings and Botsford 2003; Gerber et al. 2003; Kaplan et al. 2006). Our model suggests that reserves of medium size could efficiently protect abalone populations inside their boundaries while also ensuring export of larvae to subsidize recruitment in fished areas and hence would be preferable to larger reserves if the objective is to maximize abundance on the whole coastline. In addition, our modeling exercise indicates MR networks could be established without dramatic losses in fishery performance, provided that the size of individual reserves is adjusted to the species’ dispersal ability. In particular, our analysis suggests that the best outcomes for H. fulgens, in terms of fishery output, would be achieved with small reserves ⬃100 m wide. Under these management scenarios, reproductive individuals protected inside the reserves can substantially contribute to recruitment in the fishable areas, thereby supporting yields comparable to MSY without MRs. With small reserves covering less than 40% of the fishing grounds, optimal catches are only 2%–8% lower than MSY under traditional management (i.e., in the absence of the reserves). These results confirm and extend to previous findings for abalone obtained with simplified but analytically tractable mathematical models, suggesting that fishery benefits Published by NRC Research Press

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Fig. 5. Long-term abalone yield for the whole island (tonnes fresh mass per year, mean ± SD of the 1000 replicates under the assumption of year-to-year variability in larval dispersal distance) as a function of protection level (a–c) and harvest rate (d–f) for small reserves of 100 m width (black symbols), intermediate reserves of 500 m width (gray symbols), or with networks of large reserves of 1500 m width (white symbols). Dotted line indicates the maximum sustainable yield under traditional management.



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are best met by small reserves that maximize larval export outside of reserves (Hastings and Botsford 2003; Neubert 2003). In the case of large reserves (1500 m), in contrast, yields tended to be much smaller than MSY under traditional management, as individuals in the middle of the reserves cannot contribute to recruitment outside the reserves given the abalone’s limited larval dispersal, here assumed to be of the scale of hundreds of metres. This is consistent with the results of previous models on the effect of reserves on abalone yields, which suggested that MRs wider than the species’ larval dispersal

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ability would produce catches significantly lower than MSY (White et al. 2010, 2011). Regarding the effect of reserves on long-term yield, our model suggests that the strongest benefit of incorporating networks of MRs in a fishery management strategy consists in increased robustness to management errors, such as overestimates of sustainable fishing pressure. Our analyses show that the production curve (i.e., the relationship between harvest rate h and yield) is flatter around its peak with MRs than in the case of traditional Published by NRC Research Press



Rossetto et al.

management (Fig. 5). Without MRs or with low PLs, the range of values of fishing mortality that provide yields comparable to MSY is narrow (i.e., h = 0.1–0.2), and the yield sharply drops outside this range. On the contrary, this range is remarkably larger (up to h = 0.8) when more than a third of the fishing ground is protected inside reserves. This outcome is consistent with the results of other fishery models, highlighting the ability of MRs to prevent catch decline also when harvest rates are high (Quinn et al. 1993; Sladek-Nowlis and Roberts 1999). This result is important because it indicates that exerting excessive harvesting pressure is less likely to have deleterious effects on fishery performance and population persistence when MRs are part of the management strategy. The constant protection of a portion of the stock granted by MRs could in fact prevent the total extirpation of the population in case of overharvesting. The multiple abalone stock collapses that have occurred worldwide suggest that recruitment overfishing is frequent in the management of this marine mollusk (Sluczanowski 1984; Shepherd et al. 1998a; Leaf et al. 2008). A risk adverse strategy, such as that provided by MR networks, could thus be desirable for managing remaining abalone stocks and to guard against increasing fishing effort. The elasticity analysis of the abalone matrix shows that individuals in the largest size class have the largest impact on population growth, a result consistent with elasticity analyses conducted on other Haliotis species (Rogers-Bennett and Leaf 2006). Results of the sensitivity analysis over the observed range of variation of model parameters confirm that mortality parameters have the greatest influence on long-term population and catches. Taken together, these results suggest that (i) conservation efforts should be focused on limiting mortality of large, fecund individuals to maximize population recovery and (ii) events of increased mortality, such as those recently observed on the island due to adverse environmental conditions (hypoxia) (Micheli et al. 2012), could have a strong deleterious effect on stocks and on related harvests. Our results confirm previous findings that for sedentary species with limited larval dispersal, the goal of maintaining catches is best achieved by many small reserves covering a large fraction of the coast, alternated with fished areas (Hastings and Botsford 2003; Neubert 2003). In Isla Natividad, the fishing cooperative has decided to maintain the existing reserves and is currently in the process of deciding whether to expand protection through additional reserves. Additional communities to the north and south of Isla Natividad have also established MRs. According to our results, the size of the two existing reserves in Isla Natividad (⬃500 and 1000 m) seems adequate to protect the abalone stock inside their boundaries, but should be reduced to maximize fishery benefits. In addition, the overall level of protection (⬃8%) is probably too small; the number of reserves should be increased to successfully ensure population persistence and to increase long-term yield. Clearly, our results are based on the underlying assumption that depleted populations of H. fulgens could recover if exploitation is reduced; however, additional stressors such as climate change and diseases can also negatively affect abalone persistence (e.g., Miner et al. 2006), and the recovery of abalone populations inside MRs cannot be guaranteed. Results of the present model are clearly sensitive to assumptions about the larval dispersal ability of abalone that we based on both existing literature (see above) and our recruitment studies (Micheli et al. 2012). Dispersal distance is a critical variable that is difficult to measure in the natural environment and is therefore affected by great uncertainty. Additional studies on the larval dispersal ability of marine organisms can help ameliorate the confidence in the quantitative results of population dynamic simulations. In addition, in our modeling study, we assumed complete homogeneity in habitat and symmetry in dispersal. In reality, however, marine habitats are heterogeneous; some areas may function as larval sources and other as sinks. In addition, currents and other coastal oceanographic features may drive the dispersal in a given direction and create barriers to dispersal in other direc-

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tions. Understanding if there are major spatial patterns in habitat productivity and in dominant oceanographic currents could further guide the improvement of an MRs network. Along the Baja California and California coasts, the implementation of networks of small MRs is expected to be beneficial not only for abalones, but also for other marine invertebrates with similar life history characteristics (long-lived sedentary adults and dispersing larval stages), such as sea urchins, turban snails, and sea cucumbers, that are currently important fisheries in Mexico (McCay et al. 2014) and are expected to become increasingly important fisheries in California (Rogers-Bennett et al. 2007). While such a network has been established in California through the Marine Life Protection Act, it does not exist in Baja California. Extending the existing US network to the south, along the coast of Baja California, through the establishment of opportunely spaced MRs, is expected to improve the population status, yields, and sustainability of the benthic invertebrate fisheries in the area.

Acknowledgements Marisa Rossetto was supported through the EU grant “Towards COast to COast NETworks of marine protected areas”, Fiorenza Micheli by the Walton Family Foundation and the US NSF-CNH program (award No. DEB-1212124), and Giulio De Leo by the US NSF-OA program (agreement No. OCE-1416934). We thank members and staff of the fishing cooperative Buzos y Pescadores for their support and advice, Marino Gatto for his friendly review, and three anonymous referees for their constructive criticism on early versions of the manuscript.

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␳⫽

冋

(A1)

冉

ⱍ

冊

1 e⫺G⫺1

冉冊/ 冉冊

1⫹ ␤

⫺log(L) ⫽

According to Bardos (2005), the probability of a length increment ⌬ly conditional on initial length ly in the yth tag–recapture record is ⌬ly ⫹ ly ␭␳ 1 p(⌬ly ly) ⫽ (L ⫺ ly)␳⫺1e⫺␭(L∞⫺ly) ⫺G ⌫(␳) ∞ ly 1⫺e

(A3)

ly Ln

3

关

␭⫽

兴 1⫺e

⫺G

L∞ ⫽ (ly ⫹ ⌬ly)ly⫺e Ln

冉冊/

ly 1⫹ ␤ Ln

3

冋

␴L

冉冊

ly 1⫹ ␥ Ln

3

册

册

y

y

The growth transition gi,j from initial class j in the interval (r1,r2) to final class i in the interval (r3,r4) was then calculated as a double integral of p(⌬l | l1) over initial and final lengths across size classes i and j, respectively (Bardos 2005):

(A6)

⫺G

ly Ln

2

y

冕冕

r4⫺ly

r2

1

1⫹ ␥

3

兺⫺log关p(⌬l ⱍl )兴

where (A2)

␴L

Ln

The five parameters of the Gompertz growth function (G, Ln, ␴L2, ␤, and ␥) were estimated by minimizing the following negative log-likelihood function deriving from eq. A1: (A5)

Appendix A Can. J. Fish. Aquat. Sci. Downloaded from www.nrcresearchpress.com by STANFORD UNIV. on 08/24/15 For personal use only.

(A4)

1 gi,j ⫽ r2 ⫺ r1

dly

r1

ⱍ

p(⌬ly ly)d⌬ly

r3⫺ly

2

The resulting matrix of growth transitions gi,j is shown in Table A1.

Table A1. Growth transition gi,j from initial class j to final class i. Size class j (mm) Size class i (mm)

5–30

30–55

55–80

80–105

105–130

130–155

155–180

180–205

205–230

5–30 30–55 55–80 80–105 105–130 130–155 155–180 180–205 205–230

0.166 0.590 0.242 0.003 0 0 0 0 0

0 0.02 0.533 0.429 0.018 0 0 0 0

0 0 0.014 0.553 0.424 0.008 0 0 0

0 0 0 0.028 0.77 0.202 0 0 0

0 0 0 0 0.102 0.878 0.02 0 0

0 0 0 0 0 0.317 0.683 0 0

0 0 0 0 0 0 0.549 0.451 0

0 0 0 0 0 0 0 0.694 0.306

0 0 0 0 0 0 0 0 1
