#### The species–area relationship for a West Coast fish assemblage

The tendency for the number of species to increase with area (the species–area relationship, SAR), is one of the oldest (Arrhenius 1921) and most robust (Rosenzweig 1999) empirical observations in ecology. While it has proved difficult to use the SAR to make inferences about the mechanisms structuring communities (Connor & McCoy 1979; Coleman *et al.* 1982; McGuinness 1984; Ney-Nifle & Mangel 2000), the SAR is a useful tool for estimating how species richness changes with area (Rosenzweig 1999; Williamson *et al.* 2001). Trade-offs between biodiversity in MPAs and fisheries yield depend fundamentally on how species richness changes with area. SARs therefore provide a useful starting point for understanding the conservation axis of this trade-off (Neigel 2003).

We used data on species richness from the 1999–2003 West Coast trawl survey conducted by the Northwest Fisheries Science Center, NMFS (e.g. Keller *et al.* 2007) to generate SARs for groups of species within the ‘groundfish’ assemblage (i.e. fish species typically associated with the sea floor). Trawl surveys extended from near Cape Flattery, Washington (48°27′N) to Point Conception, California (34°28′N) and ranged in depth from 35 to 1200 m. The survey was conducted during summer and consisted of 1786 tows using a 15-m-wide Aberdeen style net with a small (3·8-cm stretched measure or less) mesh liner in the cod end. Trawl duration was approximately 15 min of bottom contact at a speed of *c*. 2·2 knots. Bottom contact and acoustic instruments were attached to the nets to record aspects of mechanical performance and gear depth. Catches were sorted to species or the closest taxonomic level, counted and weighed. Complete details of the trawl survey are available from Keller *et al.* (2007). During 1786 tows, 213 species belonging to 61 families, 24 orders and 5 classes were sampled.

We generated SARs using the most common representation of the SAR, *S *= *cA*^{z}, where *S* is the species richness and *A* is the area. The constant, *c*, represents the height of the curve near the *y*-axis and the slope, *z*, is a scale-independent parameter that determines how steeply the curve rises. Traditional SAR studies often focus on *z* as a measure of the rate at which the natural logarithm of species richness increases with the natural logarithm of area (Rosenzweig 1995).

We fit the SAR to the West Coast groundfish assemblage using a simple Monte Carlo approach. We first generated 50 randomly placed boxes, each of which encompassed 5% of the latitude of the coast (4–6% of the area) and could overlap in space. We then estimated the total species richness within each of the boxes (see below). This process was repeated for boxes of increasing size (5%, 10%, 15%, etc., up to 95% of the coast). The 19 unique box sizes, each with 50 replicates, yielded a total of 950 points. We natural logarithm transformed both the number of species and the area, and fit a linear regression to the 950 data points generated by this process. The parameter *z* was the slope of the regression and constant *c* was the exponential of the intercept of the regression.

This method required estimation of total species richness in each box. Simply counting the species present ignores potential sampling effects; larger boxes could display higher species richness solely because they are likely to contain more trawls. To control for this effect, species accumulation curves (Colwell & Coddington 1994) were constructed for each box by plotting species richness vs. number of trawls (e.g. Fig. 1) and estimating the true species richness as the asymptote of this curve, essentially asking the question, what would observed species richness be if there was unlimited sampling effort (Gotelli & Colwell 2001; Colwell *et al.* 2004)? We estimated the asymptote (true species richness) using the Michaelis Menten equation (cf. Colwell 2005). The main effect of this estimation procedure was for the ‘true’ species richness for small boxes (<20% of the coast) to be larger than observed in the data. For most boxes that covered more than 20% of the coast, the asymptote was approximately equal to the maximum number of species observed in the data. The maximum number of species and slope were estimated by minimizing the sums of squares using the Nelder–Mead simplex method in matlab v. 6.1 (The Mathworks, Natick, MA, USA). One caveat of this approach is that it may slightly overestimate *r*^{2} and underestimate the confidence intervals of *c* and *z* because the SAR was fit to point estimates of the asymptotes of the species-accumulation curves, ignoring uncertainty in these estimates.

#### A model for habitat, reserves and species–area effects

The relationship between species richness and area provides an important tool that can be used to examine trade-offs between a non-consumptive value (e.g. biodiversity) of no-take MPAs and the potential cost in fisheries yield of such areas. We develop a model to illuminate the costs and benefits of an MPA, the number of species potentially protected by the MPA and the change in fisheries yield as a result of implementing the MPA. Although we use species richness as a metric of biodiversity, the approach we describe could use any biodiversity attribute that varies with area. The model we use is stylistic and simplified but is motivated by the US West Coast fish assemblage. In this system, a number of species are currently or have been individually targeted by fisheries (e.g. Pacific hake *Merluccius productus*, sablefish *Anoplopoma fimbria*, petrale sole *Eopsetta jordani*, widow rockfish *Sebastes entomelas*, market squid *Loligo opalescens* and Dungeness crab *Cancer magister*). At the same time, a number of fishery target (e.g. MacCall 2007; Stewart 2007) and non-target species (e.g., Levin *et al.* 2006; Mangel *et al.* 2006) have experienced precipitous declines in recent years and concern about this system has led to calls for the establishment of MPAs (Lubchenco *et al.* 2003). Thus, below we develop an example that focuses on how the fisheries yield of one species may be affected by MPAs established to conserve an entire suite of species.

We envision a domain that is divided into two areas, one open and the other closed, to fishing. In addition, we consider a category called ‘untrawlable’, which is open to fishing but experiences little fishing pressure due to bathymetry that makes the area nearly inaccessible to the fishery.

We model untrawlable habitat (indexed by 0) with area, *A*_{0}, logistic population growth, and fishing removing a small fraction, *u*_{0}, of the fish. Trawlable habitat (indexed by 1) has area *A*_{1}, logistic population growth, and proportion *u*_{1} of the fish population removed by fishing. Fishing also reduces habitat quality by an amount *q* in the trawlable habitat. For most model runs, we assumed that the untrawlable area was 30% of the total area (NOAA 2006). However, we also explored the consequence of different levels of untrawlable area. We assumed that a fraction *α* of the trawlable habitat is set aside in an MPA.

In year *t*, biomass in the entire region is *B*(*t*), partitioned into the untrawlable and trawlable habitats:

- ( eqn 1)

Assuming an ideal free distribution (Fretwell 1972) of fish between the two habitats:

- ( eqn 2)

where *r*_{i} is the maximum per capita reproduction in habitat *i*, *K*_{i} is the carrying capacity and *q* scales the trawlable area to account for lower habitat quality. The maximum per capita reproduction *r*_{i} and carrying capacity *K*_{i} are independent of area.

An ideal free distribution is one in which individuals are distributed among habitats such that their fitness is maximized. Thus, individuals choose habitats based on the quality of the habitat and the density of conspecifics in the habitat. Therefore, individuals are distributed such that the ratio of density between any two patches will equal the ratio of resource levels in those patches. This pattern of density-dependent habitat selection has been used to explain patterns of habitat use by a wide range of marine fishes (Myers & Stokes 1989; MacCall 1990; Levin *et al.* 2000; Lindberg *et al.* 2006; Swain & Benoit 2006) and is a useful starting point for our work.

We solve eqn 2 to partition initial biomass at the start of year *t* (*B*_{init}) into *B*_{0}(init) and B_{1}(init):

- ( eqn 3)

- ( eqn 4)

We then define *B′*(t), the biomass remaining in the trawlable habitat after fishing, as:

- ( eqn 5)

Biomass in the next year is then:

- ( eqn 6)

after which partitioning of biomass across the habitats occurs.

The yield from the fishery during year *t*, *Y*(*t*|*α*), depends on the fraction of habitat in the MPA:

- ( eqn 7)

The state yield was computed as a function of reserve fraction . We also calculated the effective area of the entire domain (*A*) given the reserve fraction:

- ( eqn 8)

so that the species richness is predicted to be:

- ( eqn 9)

Our approach makes the assumption that the slopes of the SARs are similar between trawlable and untrawlable habitat. While it is possible that this assumption is invalid, no data are available to test this assumption. Thus, for simplicity, we use a single SAR. However, as more data become available, total species richness could easily be generated as the sum of the number of species estimated by two (or more) different species–area relationships.