Ecology
Habitat-dependent movement rate can determine the efficacy of marine protected areas
Journal:
ECY18-0101.R1
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Manuscript ID
Ecology
Wiley - Manuscript type:
Complete List of Authors:
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Date Submitted by the Author:
Articles
Substantive Area:
Fishes < Vertebrates < Animals
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Habitat:
Population Ecology < Substantive Area, Metapopulations < Population Dynamics and Life History < Population Ecology < Substantive Area, Patch Dynamics < Community Ecology < Substantive Area, Conservation < Landscape < Substantive Area, Modeling (general) < Statistics and Modeling < Theory < Substantive Area, Spatial Statistics and Spatial Modeling < Statistics and Modeling < Theory < Substantive Area, Reserves/Protected Areas < Management < Substantive Area, Computational Biology < Methodology < Substantive Area
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Organism:
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Jiao, Jing; Michigan State University, Quantitative Fisheries Center; University of Florida, Biology Pilyugin, Sergei; University of Florida Riotte-Lambert, Louise; University of Glasgow, Institute of Biodiversity Animal Health and Comparative Medicine Osenberg, Craig; University of Georgia
Marine < Aquatic Habitat < Habitat
Geographic Area: Additional Keywords:
Abstract:
differential movement, local effect, regional abundance, fishing harvest, MPA, fishing yield Theoretical studies of marine protected areas (MPAs) suggest that more mobile species should exhibit reduced local effects (defined as the ratio of the density inside vs. outside of the MPA). However, empirical studies have not supported the expected negative relationship between the local effect and mobility. We propose that differential, habitat-dependent movement (i.e., a higher movement rate in the fishing grounds than in the MPA) might explain the disparity between theoretical expectations and empirical results. We evaluate this hypothesis by building two-patch box and stepping-stone models, and show that increasing disparity in the habitat-specific movement rates shifts the relationship between the local effect and mobility from negative (the previous theoretical results) to neutral or positive (the empirical pattern). This shift from negative to positive occurs when differential movement offsets recruitment and
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mortality differences between the two habitats. Thus, local effects of MPAs might be caused by behavioral responses via differential movement rather than by, or in addition to, reductions in mortality. In addition, the benefits of MPAs, in terms of regional abundance and fishing yields, can be altered by the magnitude of differential movement. Thus, our study points to a need for empirical investigations that disentangle the interactions among mobility, differential movement, and protection.
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Title: Habitat-dependent movement rate can determine the efficacy of marine protected areas
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Authors: Jing Jiao1, 2*, Sergei S. Pilyugin3, Louise Riotte-Lambert4 and Craig W. Osenberg5
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Author Affiliation:
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1
5
[email protected])
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2
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East Lansing, Michigan 48824, USA
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3
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(email:
[email protected])
Department of Biology, University of Florida, Gainesville, Florida 32611-8525, USA (email:
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Department of Fisheries and Wildlife, Quantitative Fisheries Center, Michigan State University,
Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105, USA
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Glasgow, G12 8QQ, United Kingdom (email:
[email protected])
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5
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[email protected])
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Type: Article
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Corresponding Author: Jing Jiao; Quantitative Fisheries Center; Michigan State University;
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Urban Planning and Landscape Architecture Building, Room 100, 375 Wilson Rd, East Lansing,
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MI 48823, USA; Tel/Fax: 1-352-226-3812; email:
[email protected]
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If the manuscript is accepted, the R code supporting the results will be archived in an appropriate
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repository such as Github and the data DOI will be included at the end of the article.
Institute of Biodiversity Animal Health and Comparative Medicine, University of Glasgow,
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Odum School of Ecology, University of Georgia, Athens, Georgia 30602-2202, USA (email:
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ABSTRACT Theoretical studies of marine protected areas (MPAs) suggest that more mobile species should
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exhibit reduced local effects (defined as the ratio of the density inside vs. outside of the MPA).
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However, empirical studies have not supported the expected negative relationship between the
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local effect and mobility. We propose that differential, habitat-dependent movement (i.e., a
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higher movement rate in the fishing grounds than in the MPA) might explain the disparity
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between theoretical expectations and empirical results. We evaluate this hypothesis by building
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two-patch box and stepping-stone models, and show that increasing disparity in the habitat-
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specific movement rates shifts the relationship between the local effect and mobility from
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negative (the previous theoretical results) to neutral or positive (the empirical pattern). This shift
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from negative to positive occurs when differential movement offsets recruitment and mortality
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differences between the two habitats. Thus, local effects of MPAs might be caused by behavioral
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responses via differential movement rather than by, or in addition to, reductions in mortality. In
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addition, the benefits of MPAs, in terms of regional abundance and fishing yields, can be altered
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by the magnitude of differential movement. Thus, our study points to a need for empirical
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investigations that disentangle the interactions among mobility, differential movement, and
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protection.
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Keywords: differential movement; local effect; regional abundance; fishing yield; fishing
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harvest; MPA
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INTRODUCTION
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Movement strongly influences the distribution of organisms across landscapes, and
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therefore plays a critical role in determining a population's interactions with other species 2
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(Mccauley et al. 1996, Hanski 1998, Morales and Ellner 2002, Leibold et al. 2004, Jiao et al.
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2016), its response to environmental change (Damschen et al. 2008, Kininmonth et al. 2011,
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Janin et al. 2012), and the effectiveness of management actions (Starr et al. 2004, Pérez-Ruzafa
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et al. 2008, Grüss et al. 2011). Although movement usually takes place in a heterogeneous
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landscape, and organisms are known to respond to a variety of landscape features (e.g., edges,
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habitat composition, and predation risk: Haynes & Cronin 2006; Nathan et al. 2008; Reeve &
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Cronin 2010; Abrams et al. 2012), the role of habitat-dependent movement (e.g., when the
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movement rate depends on the habitat; hereinafter referred to as differential movement) is not
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often considered in ecological models or management decisions.
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Marine protected areas (MPAs) are a management tool used to increase biodiversity,
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population abundance, and/or fishing yield by reducing overfishing, habitat destruction, and by-
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catch. For example, MPAs can increase fishing yield by exporting adults and juveniles (via spill-
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over) from the MPAs into fishing grounds (Abesamis et al. 2006, Kellner et al. 2008). Movement
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of individuals directly influences spillover and thus the effectiveness of MPAs (Grüss et al.
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2011). Most theoretical studies of MPAs, which have assumed random and homogeneous
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movement rates, have shown that increasing a species' overall mobility (e.g., average movement
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rate of non-larvae) decreases fish density in MPAs but increases fish density in the fishing
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grounds. As a result, the ratio of the density inside vs. outside the MPAs (hereinafter referred to
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as the “local effect”) should decrease with increased mobility (see Gerber et al. 2003; Starr et al.
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2004; Malvadkar & Hastings 2008). This decrease in the local effect results from increased
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mixing between the two habitats. However, empirical studies have failed to document the
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expected negative relationship between the local effect and mobility. Meta-analyses have either
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found no significant relationship (Micheli et al. 2004, Lester et al. 2009) or a possible positive
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relationship (Claudet et al. 2010).
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To explain this apparent mismatch between theory and empirical results, some authors have suggested that fishing harvest, trophic interactions, and structure of fish assemblages would
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largely influence the densities inside and outside of MPAs, thus masking the expected negative
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effect of mobility on the local effect (Micheli et al. 2004, Palumbi 2004, Lester et al. 2009).
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While such explanations are valid, we suggest that the expectations from existing models might
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be misleading. Instead, we suggest that organisms might move differentially throughout an
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MPA network, rather than homogeneously as most existing models have assumed. One notable
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exception is Langebrake et al. (2012), who proposed and theoretically evaluated two hypotheses
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to explain the mismatch between existing theory and data: 1) that organisms move at different
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rates in the MPA vs. in the fishing grounds (i.e., differential movement), and 2) that organisms
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actively bias their movement towards the MPA when they are at the reserve boundary, but
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otherwise move at similar rates in the two habitats (i.e., biased movement). They rejected the
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first hypothesis as a viable mechanism to explain the mismatch, but accepted the second. Here,
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we re-evaluate the differential movement hypothesis.
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To test their differential movement hypothesis, Langebrake et al. (2012) assumed that
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fish and other harvested organisms move more in the fishing grounds where they are harvested
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and/or where the habitat has been degraded by fishing exploitation, and move less in MPAs
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where they are not harvested and/or where the habitat is in better condition. This habitat-specific
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movement pattern could result from ontogenetic shifts (Gerber et al. 2005), aggregative
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behaviour (Eggleston & Parsons 2008), and/or behavioral responses to habitat structure (e.g.,
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Tischendorf & Fahrig 2000, Baguette & Van Dyck 2007), mortality risk (Douglas-Hamilton et
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al. 2005, Eggleston and Parsons 2008, Januchowski-Hartley et al. 2012), or abiotic conditions
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(e.g., oil spill effects; see Fodrie et al. 2014). Higher mobility in the fishing grounds vs. the MPA
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could lead to net movement of individuals from the fishing grounds to the MPA, creating a spill-
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in pattern (in contrast to the classic spillover pattern; see Eggleston & Parsons 2008 for an
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empirical example of spill-in). This net movement could increase the local effect by increasing
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the density of the focal species in the MPA but decreasing its density in the fishing ground, thus
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potentially explaining the empirical results of Micheli et al. (2004), Lester et al. (2009), and
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Claudet et al. (2010). However, the above verbal argument, which motivated Langebrake et al’s
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(2012) study, was not supported by their formal theoretical investigation.
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Here, we propose that Langebrake et al. (2012) did not find a positive relationship
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between mobility and the local effect because of assumptions they made in analyzing their
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model. They developed a reaction-diffusion model that assumed that new organisms recruited
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via larval rain from outside the system, died at higher rates in the fishing grounds (due to
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harvesting), and moved at different rates inside vs. outside the MPA. They solved this model by
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constraining the density of the target organism to be continuous across the boundary between the
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MPA and the fishing grounds (see Neumann & Girvan 2004). However, fish density can abruptly
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change near boundaries (Chapman and Kramer 1999), and other theoretical studies have
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successfully modeled such discontinuities (Ovaskainen and Hanski 2003, Maciel and Lutscher
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2013). The discontinuity at the boundary could influence population dynamics in different
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patches (see Maciel and Lutscher 2013), further influencing the equilibria and the local effect.
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We therefore hypothesized that the restrictive assumption that density was continuous across the
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MPA boundary may have been responsible for the failure of Langebrake et al. (2012) to produce
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flat or positive relationships between mobility and the local effect.
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To address this conjecture and to re-evaluate the differential movement hypothesis, we studied the isolated and combined effects of differential movement and overall mobility on
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several measures of MPA efficacy, but without assuming density continuity at the boundary.
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Because MPA size can influence the strength of animal movement as well as directly affect
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MPA efficacy (Pittman et al. 2014), we also considered the effect of the MPA size (relative to
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the fishing ground). We did this by building discrete spatial models: a two-patch box model in
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which one cell represents the MPA while the other represents the fishing ground, and a stepping-
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stone model in which fish moved between discrete cells along a linear shoreline. We first
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analytically solved the two- patch box model and then used numerical simulations to solve the
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multi-cell, stepping stone model. In all cases, we examined how differential movement (i.e., the
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relative migration from one cell to the other) and overall mobility (i.e., the overall magnitude of
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the migration rates) affected three indices reflecting the performance of MPAs: 1) the local effect
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(i.e., the density in the MPA relative to the density in the fishing grounds); 2) regional abundance
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(i.e., the combined abundance in both the MPA and the fishing grounds); and 3) fishing yield
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(assumed proportional to the product of the size of the fishing grounds and the density of the
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species in the fishing grounds). We also varied the relative sizes of the MPA and the fishing
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grounds by adjusting the proportion of cells in the landscape that were in the MPA or in the
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fishing grounds. To highlight the changes in abundances and/or densities resulting from the
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MPA, we rescaled the above three indices by their values before establishment of the MPA.
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MODEL DEVELOPMENT We built a discrete, 1-dimensional spatial model (i.e. a stepping-stone model) in which fish moved between adjacent, discrete cells on a linear shoreline, along which MPAs and fishing 6
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grounds alternated periodically. The study region consisted of one of these repeating units, which
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was comprised of SM+SF discrete and equal-sized cells: the MPA consisted of SM cells and the
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fishing ground consisted of SF cells (see Fig. 1). We used a circular representation of space by
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connecting the left side of the fishing ground with the right side of the MPA.
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We developed three models structures depicting: 1) an open system (with constant larval rain); 2) a closed system (with logistic growth in each cell), and 3) a semi-closed system (with
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logistic growth but larval redistribution among cells). For the open system, we assumed that the
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density of the focal species in each cell increased via constant larval recruitment (R): i.e., we
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assumed that all reproduction came from outside the system because the MPA system was small
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relative to the dispersal ability of the organism. We then relaxed this assumption and explored a
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closed system by assuming logistic growth in a cell in which all larvae were retained locally (i.e.,
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in the same cell as their parents). We then explored an intermediate version, for a semi-closed
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system, in which there was logistic growth within a cell but larvae were redistributed equally
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among all cells. Thus, these three models represent a gradient in the scale of the MPA-fishing
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ground system relative to the scale of larval dispersal. We focus on the results of the open
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system (with constant larval rain) for its simplicity, but briefly summarize results for the two
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models with logistic growth (which are presented in more detail in the Appendix). In general,
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results regarding the effect of differential movement and mobility were similar for all three
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systems.
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Organisms in all cells incurred the same intensity of natural mortality (μN ), and
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organisms in the fishing grounds incurred additional mortality due to fishing (μF ). We assumed
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that the mortality due to fishing (μF ) was constant in each cell in the fishing ground and
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independent of the MPA size (as a special case, this can be achieved when there is no
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redistribution of fishing effort in response to the establishment of an MPA).
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The dynamics within a cell were also affected by emigration to, and immigration from, the adjacent cells. We assumed that total emigration rate from each cell in the MPA was DM,
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while it was DF in the fishing ground, and that emigrants from a cell had the same chance of
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moving into either of the two adjacent cells (i.e., movement was not directed and there was no
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movement bias). Adopting the notation NM,i (i = 1, 2…SM) for the density of the focal species in
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each MPA cell, where cells 1 and SM are adjacent to the fishing grounds, and NF,i (i = -SF,…-2, -
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1) for the density in the fishing ground, with cells -SF and -1 adjacent to MPAs (see Fig. 1), these
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assertions led to the following model for the dynamics in the fishing grounds for the open
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system:
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dt
= R − (μN + μF )NF,i − DF NF,i +
dNF,−1 dt
DF 2
NF,−SF+1 +
2
dt
= R − (μN + μF )NF,−1 − DF NF,−1 +
dNM,i dt
= R − μN NM,1 − DM NM,1 + = R − μN NM,i − DM NM,i +
dNM,SM dt
DM 2
DM 2
2
NM,SM
(NF,i−1 + NF,i+1 ) for i = −SF + 1, … , −3, −2 DF 2
NF,−2 +
DM 2
NM,1
In the MPA (where there was no fishing mortality), we had: dNM,1
DM
NM,2 +
DF 2
NF,1
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dNF,i
DF
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dt
= R − (μN + μF )NF,−SF − DF NF,−SF +
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dNF,−SF
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(NM,i−1 + NM,i+1 ) for i = 2,3, … SM − 1
= R − μN NM,SM − DM NM,SM +
DM 2
NM,SM −1 +
DF 2
NF,−SF
(1) (2) (3)
(4) (5) (6)
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When the size of the MPA and the fishing ground were equal (i.e., SM=SF), we sought an
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approach that would facilitate analytic solutions, so we simplified the model to have only 2 cells
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(i.e., SM=SF=1). We refer to this model as a two-patch box model, which with constant larval
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rain becomes: 8
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dNF dt dNM dt
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= R − (μN + μF )NF − DF NF + DM NM
(7)
= R − μN NM − DM NM + DF NF
(8)
Armed with insights from the box model, we then analyzed results of the stepping-stone model (SM + SF > 2). Although the stepping-stone model (Eq. 1 to 6) can sometimes admit
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explicit analytic solutions, the resulting expressions are too cumbersome for subsequent analysis;
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hence we resorted to numerical simulations. We used a fixed landscape size (with SM + SF=10),
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although simulations with larger landscapes demonstrated that the qualitative patterns were
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unaffected by the number of cells in the study system. We first simulated the results prior to
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establishment of the MPA (i.e., when all cells were fished) to provide a baseline from which to
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evaluate effects of protection. We then simulated the responses after establishment of the MPA
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network to study the combined effects of differential movement, overall mobility, and the
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relative size of the MPA (SM /(SM + SF)) on the average density within each habitat and on the
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spatial pattern in density across the entire MPA-fishing ground system.
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For analyses of the box model and simulations of the stepping-stone model, we focused on the local effect, regional abundance and fishing yield. For ease of interpretation of figures, we
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rescaled all responses relative to their value before establishing the MPA. Note that by definition
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the local effect was equal to 1 before the MPA was established, so for these response variable
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rescaling had no effect.
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After exploring the behavior of the open system (with constant larval rain), we then
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briefly explored results for the two other systems: closed and semi-closed. For the closed system
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(with logistic growth, but without larval redistribution), we simply replaced the recruitment term,
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R, in Eqs 1-6 with a logistic term representing gains via recruitment and losses via density-
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dependent mortality, rNij-rN2ij/K, where r is per capita growth rate, and K is the carrying capacity 9
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of a cell (we assumed r and K were the same in the MPA and fishing grounds). For the semi-
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closed system, we assumed that larvae were well mixed and redistributed uniformly among all
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cells in the system, rather than being retained locally. Thus, for the semi-closed system, we
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replaced the recruitment term in the logistic (i.e., rN) with S
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For the closed system, we analytically solved the two-patch box model and then simulated
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results for the stepping-stone model. However, the semi-closed system was more complex, so
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we restricted our analyses to simulations of the two-patch box model as well as the stepping-
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stone model.
1 M +SF
S
M (rF ∑−1 i=−SF NF,i + rM ∑i=1 NM,i ).
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RESULTS
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The open system (constant larval rain)
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The two-patch box model. – This model (Eqs 7 and 8) has the following analytical solutions for
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the densities of organisms in the MPA and the fishing ground:
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NF∗ = (μ
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∗ NM = (μ
R(μN +2DM ) N +μF +DF )μN +(μN +μF )DM (μN +μF +2DF )R N +μF +DF )μN +(μN +μF )DM
By setting
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rearranged as:
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NF∗ = (μ
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∗ NM = (μ
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DF DM
(9) (10)
= β, the strength of differential movement, the above solutions can be
R(μN +2DM )
(11)
N +μF +βDM )μN +(μN +μF )DM
(μN +μF +2βDM )R
(12)
N +μF +βDM )μN +(μN +μF )DM
From Eqs 11 and 12, we derive the influence of differential movement on the densities of organisms in each cell by differentiating Eqs 11 and 12 with respect to β: 10
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d(N∗M ) dβ d(N∗F ) dβ
= [(μ
RDM (μN +μF )(μN +2DM )
N +μF +βDM )μN +(μN +μF )DM ]
= [(μ
−RμN DM (μN +2DM )
2 N +μF +βDM )μ+(μN +μF )DM ]
2
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>0
(13)
0 μN μF +2FDM
2 N +μF +βDM )μN +(μN +μF )DM ]
dβ d(μF SF N∗F ) dβ
2DM
N +2DM
= [(μ
−μF SF RDM μN (μN +2DM )
N +μF +βDM )μN +(μN +μF )DM ]
2
(15)
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NF
dβ
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∗
N d( M ∗)
>0
(16)
0): see the relative locations of the three lines at β=1 in Fig. 2 and the solid line in Fig.
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3. These results reiterate those of past theoretical studies (e.g., Gerber et al. 2003; Starr et al.
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2004; Malvadkar and Hastings 2008). However, in the presence of differential movement (β >
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1), the sign of these derivatives (Eqs 18 and 20) depends on the relative mortality rates in the
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fishing grounds (μN + μF) vs. the MPA (μN ). When the differential movement parameter is
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smaller than the relative mortality rates (i.e., 1 < β < (μN +μF )/ μN ), the qualitative patterns are
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identical to the results when there is no differential movement (i.e., when β=1). When the
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differential movement parameter is equal to the relative mortality rates in the two patches
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(i.e., β = (μN + μF )/μN; represented by the blue line in Fig. 2), the spatial patterns are
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independent of the overall movement rate (note the crossing of the three lines in Fig. 2 and the
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horizontal dashed line in Fig. 3). However, when the differential movement parameter exceeds
d(N∗ ) M
dDM
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the relative mortality rates (i.e., β > (μN + μF )/μN ), increasing mobility increases the local
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effect, and increases regional abundance, but decreases fishing yield (see the relative locations of
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the three lines on the right of the blue line in Fig. 2 and the dotted line in Fig. 3). Thus, if
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differential movement is sufficiently strong (i.e. if the movement rates in the two habitats are at
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least as dissimilar as their respective mortality rates), it can explain the discrepancy between
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previous theoretical results (which assumed equal movement rates) and the empirical studies that
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either found no significant relationship (Micheli et al. 2004, Lester et al. 2009), or a possible
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positive relationship (Claudet et al. 2010), between the local effect and mobility.
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The stepping-stone model.– As differential movement increased, the local effect also increased
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(Figs. 4 and S5 in Appendix). Although regional abundance increased and fishing yield
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decreased, these responses were negligible (Figs. 4 and S5 in Appendix). Overall, these
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qualitative results are consistent with our analyses of the two-patch box model (Eq. 15-17).
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Increasing the relative MPA size led to a marked increase in regional abundance and
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decrease in fishing yield (Fig. 4b,c). However, the local effect showed a more complex pattern
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with respect to relative MPA size. The local effect increased more as differential movement
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increased when the MPA was either relatively large or relatively small; the effect of increasing
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differential movement was smallest at intermediate MPA sizes (Fig. 4a). When differential
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movement was equal to the critical value (i.e., β = (μN +μF )/μN ), the local effect did not change
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with the relative MPA size (see the dashed line in Fig. 4a). However, when differential
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movement was smaller than this value (1 ≤ β < (μN +μF )/ μN : on the left side of the dashed line in
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Fig. 4a), the local effect initially increased (as the relative MPA size increased from 0 to 0.5), but
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then decreased (as the relative MPA size increased from 0.5 to 1.0) — although the small change
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is hard to discern in Fig. 4a. In contrast, when differential movement was larger than this critical
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value (β > (μN +μF )/ μN ), the local effect first decreased then increased as the relative size of the
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MPA increased. The magnitude of differential movement determined the relationship between mobility
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and the local effect: when β < (μN +μF )/ μN , the spillover pattern existed, leading to a negative
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relationship between mobility and the local effect (Eq. 18 (μN +μF )/ μN , the spill-in
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pattern occurred, leading to a positive relationship between mobility and the local effect (Fig.
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5c). These qualitative effects on the local effect held across MPA sizes (Fig. 5).
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We also explored the spatial patterns that arose within a habitat and across the MPA-
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fishing ground boundary to gain insights about differential movement and its relationship to past
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theoretical results. We simulated the density in each cell across the entire study region under
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three levels of differential movement: i.e., when β was smaller than, equal to, or larger than the
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relative mortality rates (μN + μF )/μN). When β = (μN +μF )/ μN , densities within each habitat
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were homogeneous: i.e., there was no spatial heterogeneity within the MPA or the fishing
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grounds, but there was a sharp density difference at the boundary of the MPA and fishing
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grounds (Fig. 6b). However, when β ≠ (μN +μF )/ μN , densities were heterogeneous within
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habitats, with density in a cell depending on the cell's distance from the MPA boundary.
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Importantly, the density difference across the boundary of the MPA shifted from smooth (when β
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< (μN +μF )/μN ; Fig. 6a) to abrupt (when β ≥ (μN +μF )/μN ; Fig. 6b,c). Specifically, when
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differential movement was small (Fig. 6a: β < (μN +μF )/μN ), densities in the fishing grounds were
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greatest near the MPA boundary, but lowest in the center. In contrast, densities in the MPA were
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lowest at the boundary and greatest at the center. These smooth transitions are consistent with the
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spillover pattern classically described in models of MPA systems (see Gerber et al. 2003; Starr et
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al. 2004; Malvadkar and Hastings 2008). However, when differential movement was large (Fig.
310
6c: β > (μN +μF )/ μN ), the pattern reversed. Densities in the fishing grounds were depressed near
311
the MPA border and greatest at the center of the fishing grounds. Densities in the MPA were
312
greatest near the boundary and depressed in the center of the MPA. These patterns are indicative
313
of spill-in dynamics. These qualitative patterns were observed across all relative MPA sizes (see
314
Fig. S1 in Appendix).
Fo
315
The closed system (logistic growth without larval redistribution)
317
The two-patch box model.– Changing the form of the recruitment term from larval rain to logistic
318
growth without larval redistribution (Eqs S1-S6 in Appendix) had relatively little effect on the
319
qualitative responses regarding the local effect, and further highlights the important role played
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by differential movement. For example, the relationship between mobility and the local effect
321
depended on the magnitude of differential movement; however the transition from a negative to a
322
positive relationship occurred when β = r−μ
323
when 1 ≤ β < r−μ
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β > r−μ
iew
ev r−μN N −μF
N −μF
r−μN N −μF
instead of =(μN + μF )/μN. In other words,
ly
r−μN
On
325
rR
316
, the typical spillover pattern occurred (Eq. S19 in Appendix); and when
, the spill-in pattern occurred (Eq. S20): see Fig. S4d in Appendix). r−μN
This critical value of differential movement (β = r−μ
N −μF
; in this case, β=2) also defined
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the conditions under which the regional abundance and fishing yield did not change with
327
increased mobility (Fig. S4e,f). In comparison to the open system, the qualitative pattern of
328
change in regional abundance was a bit more complex: with an increase in , the regional
329
abundance decreased (compare the three lines in Fig. 4e) whereas in the open system, the 15
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Ecology
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regional abundance increased (Fig. S4b). In addition, the relationship between regional
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abundance and mobility was not always monotonic: e.g., when
