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. sc iencedi rec t . com

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ava i lab le a t www

journal homepage: www.elsevier .com/ locate /b iocon

Optimizing wetland restoration and management for aviancommunities using a mixed integer programming approach

Diana Stralberga,*, David L. Applegateb, Steven J. Phillipsb,Mark P. Herzoga, Nadav Nura, Nils Warnocka,c

aPRBO Conservation Science, 3820 Cypress Drive #11, Petaluma, CA 94954, USAbAT&T Labs-Research, 180 Park Avenue, Florham Park, NJ 07932-0971, USAcWildlife Health Center, School of Veterinary Medicine, University of California, Davis, CA 95616, USA

A R T I C L E I N F O

Article history:

Received 4 June 2008

Received in revised form

23 September 2008

Accepted 1 October 2008

Keywords:

Optimization

Birds

Tidal marsh

Salt ponds

Conservation planning

San Francisco Bay

0006-3207/$ - see front matter � 2008 Elsevidoi:10.1016/j.biocon.2008.10.013

* Corresponding author. Tel.: +1 707 781 2555E-mail addresses: dstralberg@prbo.org (D.

mherzog@prbo.org (M.P. Herzog), nnur@prbo

Please cite this article in press as: StralbConserv. (2008), doi:10.1016/j.biocon.200

A B S T R A C T

Conservation planning and management decisions often present trade-offs among habi-

tats and species, generating uncertainty about the composition and configuration of habi-

tat that will best meet management goals. The public acquisition of 5471 ha of salt ponds in

San Francisco Bay for tidal-marsh restoration presents just such a challenge. Because the

existing ponds support large numbers of waterbirds, restoring the entire area to tidal

marsh could cause undesirable local declines for many species. To identify management

strategies that simultaneously maximize abundances of marsh- and pond-associated spe-

cies, we applied an integer programming approach to maximize avian abundance, compar-

ing across two objectives, two models, and five species weightings (20 runs total). For each

pond, we asked: should it be restored to a tidal marsh or kept as a managed pond, and with

what salinity and depth? We used habitat relationship models as inputs to non-linear inte-

ger programs to find optimal or near-optimal solutions. We found that a simple linear

objective, based on maximizing a weighted sum of standardized species’ abundance, led

to homogeneous solutions (all-pond or all-marsh). Maximizing a log-linear objective

yielded more heterogeneous configurations that benefit more species. Including landscape

terms in the models resulted in slightly greater habitat aggregation, but generally favored

pond-associated species. It also led to the placement of certain habitats near the bay’s edge.

Using the log-linear objective, optimal restoration configurations ranged from 9% to 60%

tidal marsh, depending on the species weighting, highlighting the importance of thought-

ful a priori consideration of priority species.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Conservation planning and reserve-design algorithms are

well-developed and have been applied widely for large land-

scapes that encompass multiple habitat types containing dif-

ferent suites of species (Csuti et al., 1997; Margules and

Pressey, 2000; Possingham et al., 2000). Measures of species

er Ltd. All rights reserved

x325; fax: +1 707 781 168Stralberg), david@researc.org (N. Nur), ndwarnock

erg, D. et al., Optimizi8.10.013

diversity, rarity, endemism, and complementarity (Vane-

Wright et al., 1991) have been used to identify conservation

configurations with the highest biodiversity conservation po-

tential (Williams et al., 1996; Kerr, 1997; Faith et al., 2004).

However, in smaller areas with similar habitat and species

composition throughout, conservation potential may depend

more upon landscape configuration and habitat management

.

5.h.att.com (D.L. Applegate), Phillips@research.att.com (S.J. Phillips),@ucdavis.edu (N. Warnock).

ng wetland restoration and management for avian ..., Biol.

2 B I O L O G I C A L C O N S E R V A T I O N x x x ( 2 0 0 8 ) x x x – x x x

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strategies than upon the protection of individual sites. This is

especially true for urbanized or otherwise transformed land-

scapes where ecological functions are compromised, distur-

bance levels are high, and active management may be

necessary to maintain biodiversity, often with trade-offs

among species. Typically, reserve design and conservation

planning research in urban and transformed areas has been

focused on small habitat ‘‘islands’’ where species are threa-

tened by isolation, extinction, and changes in predator/prey

dynamics (Soule et al., 1988; Bolger et al., 1991; Crooks and

Soule, 1999). However, opportunities for large-scale (on the or-

der of 103–104 ha) conservation and restoration also exist

within urban and transformed settings, especially for major

estuaries such as San Francisco Bay, where large expanses

of tidal marsh may be restored passively by breaching levees

and restoring tidal action to diked former wetlands (Williams

and Faber, 2001). In addition to providing conservation and

flood-control benefits, tidal marshes also sequester carbon

at high rates (Chmura et al., 2003), providing an additional

economic incentive for restoration with the emergence of glo-

bal carbon markets.

1.1. Optimization algorithms

There are many possible approaches available for conserva-

tion planning and reserve-design problems, including simple

expert opinion. Quantitative approaches can be generally

classified as those that provide guaranteed optimal or near-

optimal solutions, such as integer programming (Papadimitri-

ou and Steiglitz, 1982), and heuristic methods that attempt to

find good solutions, but provide no guarantee on their solu-

tion quality, such as simulated annealing (Possingham et al.,

2000). Both have been used in conservation planning and re-

serve design, but heuristic approaches are more widely used

(Bedward et al., 1992; Cabeza and Moilanen, 2003; Leslie

et al., 2003). In part, the popularity of heuristic approaches

has been facilitated by the development of user-friendly soft-

ware packages such as Marxan (Possingham et al., 2000), C-

Plan (Pressey et al., 2005), and Zonation (Moilanen, 2007).

Some researchers have argued that optimal solutions, espe-

cially for integer programming, are too computationally

expensive or complicated to achieve, and that heuristic algo-

rithms can provide solutions that are equally or nearly as

good (Pressey et al., 1996; Moilanen, 2008). Others maintain

that heuristic approaches can be problematic and that opti-

mization approaches are well-developed and preferable,

yielding solutions that can be measured against performance

standards (Underhill, 1994; Camm et al., 1996; Onal, 2004). De-

spite the computational complexity, optimization approaches

have been used in several reserve design (Nevo and Garcia,

1996; Hof and Raphael, 1997; Rodrigues and Gaston, 2002;

Onal and Briers, 2003; Williams et al., 2004) and other spatial

ecosystem management and land-use planning applications

(Hof and Bevers, 2000; Seppelt and Voinov, 2002; Aerts et al.,

2003).

1.2. Wetland conservation planning

The restoration and management of major estuaries and

associated wetlands present a conservation planning chal-

Please cite this article in press as: Stralberg, D. et al., OptimizConserv. (2008), doi:10.1016/j.biocon.2008.10.013

lenge well-suited to optimization approaches. Often, these

wetland areas are large (thousands of hectares), fairly homo-

geneous, and management-dependent, with conflicting habi-

tat requirements among species of management interest

(Vickery et al., 1997). We focus here on bird communities of

the South San Francisco Bay, where the 2003 public acquisi-

tion of 5471 ha of salt ponds provides an unprecedented

opportunity to restore large areas of tidally influenced habi-

tat, especially tidal marsh. The conversion of these salt ponds

to tidal marsh would result in a doubling of habitat for tidal-

marsh-associated bird species, possibly increasing overall

viability of species such as the endangered California clapper

rail (Rallus longirostris obsoletus) (Foin et al., 1997). The existing

salt ponds support a high diversity and abundance of water-

birds, however, including 5–13% of the federally threatened

Pacific coast population of the snowy plover (Charadrius alex-

andrinus) (Page et al., 1991), which could experience substan-

tial declines with the loss of this habitat (Warnock et al.,

2002). The potential for conflict among conservation objec-

tives is clear.

For migratory waterbirds, which have lost a large portion

of their historical coastal intertidal stopover and wintering

habitat, artificial and managed open-water habitats (includ-

ing salt ponds) now provide a substantial portion of their for-

aging and roosting habitat (Davidson and Evans, 1986; Weber

and Haig, 1996; Stenzel et al., 2002). Depth, salinity, and con-

figuration of ponds are among the most important factors

determining habitat quality and capacity (Erwin, 1996; Elphick

and Oring, 1998; Isola et al., 2000). Other wetland species, par-

ticularly tidal marsh endemic rails and songbirds, have con-

trasting habitat requirements, in that they are dependent on

tidally-influenced vegetated wetlands (Foin et al., 1997; Spa-

utz et al., 2006). Thus, the issue becomes one of assessing pri-

orities and trade-offs among different species of conservation

concern Elphick (2004).

1.3. San Francisco Bay optimization goals

For San Francisco Bay salt ponds, the problem is one of both

site selection (which salt ponds to restore to tidal marsh)

and optimal management (how to manage remaining ponds).

In theory, any pond could be managed in any way, with differ-

ent effects on different species of conservation interest.

Assuming that habitat is a limiting factor for the species of

interest and focusing on pond salinity and depth—two vari-

ables with significant predictive value for waterbirds (Stral-

berg et al., 2006)—we developed a model defining the

relationships between pond conditions and log-transformed

species density. The ability to encapsulate these management

considerations and their importance for avian species into

linear equations made this problem suitable for the applica-

tion of linear integer programming techniques. The non-line-

arity introduced by the use of log-transformed bird densities

(as well as other non-linearities to be discussed) complicated

the problem, however, resulting in a non-convex, non-linear

integer problem (Papadimitriou and Steiglitz, 1982). Such opti-

mization problems are typically limited to heuristic optimiza-

tion methods, but in our case, because of the special structure

of the optimization model, we were able to apply integer

programming.

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We were not aware of previous applications of integer pro-

gramming optimization to a biologically driven wetland resto-

ration and management problem (but see Roise et al., 2004;

Newbold, 2005). Our goal herein was to combine powerful

computational methods with comprehensive empirical avian

and habitat data to develop realistic conservation objectives

and constraints with direct applications to restoration and

management. By varying a set of key optimization parame-

ters, we sought to answer the following questions:

(1) What is an appropriate optimization metric for multi-species

conservation?

Most multi-species reserve design exercises have focused

on finding the minimum area that achieves representation

of target species (Margules and Pressey, 2000) or maximizing

coverage of target species within a fixed area (Church et al.,

1996). This binary approach ignores the value of increasing

the representation of a species above an arbitrary threshold

value and may not provide the best solution for overall biodi-

versity (Arponen et al., 2005). Recent work has explored the

use of continuous ‘‘benefit’’ functions that allow increases

in species representation to be valued in different ways (Arpo-

nen et al., 2005; Cabeza and Moilanen, 2006). In our case,

abundance was more relevant than the occurrence of species,

so we needed an approach for standardizing our optimization

objective across multiple focal species with vastly different

population sizes. We compared two objective functions: first,

a weighted linear combination of standardized species abun-

dance, and second, a weighted linear combination of log-

abundances. We considered the latter to be more biologically

relevant, given the large variation in abundance across focal

species, and expected it to give solutions that better balance

the needs of all species.

(2) How important are the spatial configuration and landscape

context of design and management choices?

Given the mobility of our avian focal species and previous

analyses of landscape vs. local predictors (Stralberg et al.,

2006), our assumption was that management conditions

were much more important than landscape composition for

restoration solutions. However, the incorporation of sur-

rounding wetland composition creates a more biologically

realistic spatial optimization problem. Rather than assuming

a positive influence of habitat aggregation and connectivity,

as many have done (Possingham et al., 2000; Onal and Briers,

2003; Cabeza et al., 2004), we used empirically-derived rela-

tionships between bird density and adjacent wetland compo-

sition and evaluated their importance in the optimization

solutions.

(3) How much do optimized restoration solutions vary depending

on the a priori conservation criteria?

Given the inherent trade-offs in this system, we knew that

the optimal solution would be highly dependent on the spe-

cies chosen as conservation targets, and the weights given

each of those species. We also suspected that restoration

solutions benefitting all species may not exist. Thus we com-

pared optimal solutions across two different sets of weighting

schemes based on different species-specific criteria. We

developed weights based on conservation status (as has been

done in a broad-scale conservation planning context; Kremen

et al., 2008) and on habitat specialist categories, which may be

more relevant for local conservation planning.

Please cite this article in press as: Stralberg, D. et al., OptimiziConserv. (2008), doi:10.1016/j.biocon.2008.10.013

2. Materials and methods

2.1. Study area

Our focus was on the South San Francisco Bay salt pond res-

toration project area, defined as the 5471 ha of salt ponds tar-

geted for restoration and management (http://

www.southbayrestoration.org) (Fig. 1). The restoration project

area is interspersed with and surrounded by commercial salt

evaporation ponds, tidal marsh, open bay and mudflats, non-

tidal wetlands, and high-density urban development (resi-

dential, commercial and industrial).

2.2. Data collection

2.2.1. Salt-pond bird surveysSalt-pond bird densities were estimated from two sets of

avian surveys conducted by PRBO Conservation Science

(PRBO, 1999–2001) and US Geological Survey Biological

Resources Division (USGS, 2002–2004) prior to any restoration

activities. PRBO surveys covered 21 ponds, 13 of which are

now part of the project area; the remaining eight ponds are

still operated as commercial salt evaporation ponds. USGS

surveys covered all 54 ponds contained in the project area

(Fig. 1). Conditions during the survey periods were highly var-

iable, encompassing broad ranges of pond depths and salini-

ties. Monthly surveys were conducted from October 1999 to

February 2000, September 2000 to April 2001, and November

2002 to January 2004.

Because ponds are used by shorebirds primarily on high

tides when nearby mudflats are unavailable (Stenzel et al.,

2002; Warnock et al., 2002) and because we assumed that

most waterbird species are limited by foraging (and not roost-

ing) habitat availability, only foraging bird data from high-tide

surveys were used. Additional details on salt pond survey

methods are provided in Warnock et al. (2002).

2.2.2. Salt pond site variablesPond salinity was measured on the same days that birds were

surveyed, using the temperature and specific gravity of water

samples to obtain a salinity concentration in parts per thou-

sand (ppt). We averaged 2–4 samples from different pond

locations to obtain a mean salinity value for each survey.

Ponds were classified as low (20–60 ppt), medium (60–

120 ppt), high (120–200 ppt), or very high (200+ ppt) salinity

(based on SFEI, 1998).

Salt-pond water-depth metrics were calculated using two

sources of bathymetric data: USGS boat-based depth sound-

ings (Takekawa et al., 2005) and USGS light detection and

ranging (LiDAR) data (Foxgrover and Jaffe, 2005) for dry ponds

that could not be surveyed by boat. Boat-based depth mea-

surements were interpolated at a 5-m pixel resolution across

all ponds, using an inverse distance-weighted algorithm. We

developed bathymetric surfaces at a 5-m pixel resolution for

each pond and each survey period, which were summarized

to obtain the mean depth, shallow (<15 cm) proportion, and

deep (>1 m) proportion of each pond for the month corre-

sponding to each survey. These statistics were used to classify

ponds as shallow [(at least 10% <15 cm deep and not more

than 10% >1 m deep) or (mean depth < 0.5 m)]; deep (at least

ng wetland restoration and management for avian ..., Biol.

Fig. 1 – South San Francisco Bay, California study area, with restoration project ponds depicted in a hatch pattern. 1999–2003

data from ponds and tidal marsh areas outlined in bold were used to develop bird density models. Commercial salt

evaporation ponds are shown in white. Existing tidal marshes are shaded gray.

4 B I O L O G I C A L C O N S E R V A T I O N x x x ( 2 0 0 8 ) x x x – x x x

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50% >1 m deep and not more than 10% <15 cm deep); or inter-

mediate (not shallow or deep).

2.2.3. Tidal-marsh bird surveysTidal-marsh bird densities were estimated from fall, winter,

and spring area surveys conducted by PRBO between Septem-

ber 1999 and April 2001 and from breeding season point-count

surveys (Ralph et al., 1993) conducted between March and

May from 1999 to 2004. Area surveys, following the same

method described for salt ponds, were used to survey water-

birds and all non-passerines, which generally tend to be

patchily distributed within the marsh. Due to access and

detectability issues, we did not survey entire marshes, but

sub-sampled our study sites. Because visibility within the

marsh was variable, we noted the distance of each bird to

the observer and limited our analysis to observations within

200 m of the observer’s survey route; survey areas were also

adjusted accordingly for the purpose of calculating bird den-

sities. Area surveys were conducted at 12 tidal marshes in

the South Bay (Fig. 1). Eight tidal marshes were surveyed dur-

Please cite this article in press as: Stralberg, D. et al., OptimizConserv. (2008), doi:10.1016/j.biocon.2008.10.013

ing the 1999–2000 season and nine were surveyed during the

2000–2001 season.

We conducted point count surveys (Ralph et al., 1993),

which are better suited for estimation of passerine densities,

at 102 point count stations in 14 tidal marshes in the South

Bay (Fig. 1). Survey points were placed 200 m apart along per-

mitted access routes, i.e., on peripheral levees and board-

walks. From each survey point, an observer recorded all bird

species detected by sight and sound, with observations up

to 50 m included in this analysis. Additional details on tidal-

marsh survey methods are provided in Spautz et al. (2006).

2.2.4. Tidal-marsh site variablesFor characterization of tidal-marsh habitat, we used large-

scale (1:4800), high-resolution (scanned at 0.167-m pixel reso-

lution) color-infrared photos (flown at high tide in August

2001) to map channels and natural ponds within the tidal-

marsh study sites. We used ArcInfo 8.1 (ESRI, 2001) to digitize

ponds and channels, classifying the channels by width cate-

gory. The resulting pond and channel GIS layers were used

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to calculate pond/panne proportion and channel density met-

rics by width class (<2 m and >4 m) for each survey marsh.

2.2.5. Salt-pond and tidal-marsh landscape variablesTo characterize salt-pond and tidal-marsh landscape context,

we used a composite land use GIS layer comprised of 1998

data for current and former Baylands (SFEI, 1998) and 1985

data for surrounding uplands (USGS, 1996). Using a 1-km buf-

fer (see results from Spautz et al., 2006) around each pond and

marsh survey site, we calculated the proportion of marsh, salt

pond, tidal flat, urban development, and other upland land

uses around that salt pond or tidal marsh.

2.3. Habitat models

To optimize habitat restoration for multiple pond- and marsh-

dependent species, we selected a representative and parsimo-

nious set of 29 focal species (by season) that are likely to be

Table 1 – Focal species and seasons included in South San Frangroups, optimization weight categories, and abundance estimdance = estimated species abundance in tidal marshes and poMaximum abundance = maximum achievable population (inclindividually. mps = managed pond specialist; tms = tidal mars

Group Focal species

Large shorebirds American avocet Recurvirostra americana (W)

Large shorebirds Black-necked stilt, Himantopus mexicanus (W)

Large shorebirds Greater yellowlegs, Tringa melanoleuca (W)

Large shorebirds Willet, Catoptrophorus semipalmatus (F)

Large shorebirds Willet, Catoptrophorus semipalmatus (W)

Small shorebirds Dunlin, Calidris alpina (W)

Small shorebirds Dunlin, Calidris alpina (S)

Small shorebirds Western sandpiper, Calidris mauri (F)

Small shorebirds Western sandpiper, Calidris mauri (W)

Small shorebirds Western sandpiper, Calidris mauri (S)

Small shorebirds Least sandpiper, Calidris minutilla (F)

Small shorebirds Least sandpiper, Calidris minutilla (W)

Small shorebirds Least sandpiper, Calidris minutilla (S)

Small shorebirds Semipalmated plover, Charadrius semipalmatus (W)

Phalaropes Wilson’s phalarope, Phalaropus tricolor (F)

Phalaropes Red-necked phalarope, Phalaropus lobatus (F)

Dabbling ducks Gadwall, Anas strepera (W)

Dabbling ducks Mallard, Anas platyrhynchos (W)

Dabbling ducks Northern pintail, Anas acuta (W)

Dabbling ducks Northern shoveler, Anas clypeata (W)

Diving ducks Ruddy duck, Oxyura jamaicensis (W)

Diving ducks Greater/lesser scaup, Aythya marila/A. affinis (W)

Fish-eaters American white pelican, Pelecanus erythrorhynchos (

Fish-eaters Froster’s tern, Sterna forsteri (W)

Eared grebe Eared grebe, Podiceps nigricollis (W)

Rails Clapper rail, Rallus longirostris (S)

Landbirds Common yellowthroat, Geothlypis trichas (S)

Landbirds Marsh wren, Cistothorus palustris (S)

Landbirds Song sparrow, Melospiza melodia (S)

a Outside abundance estimates are based on mean densities from a sam

b Maximum abundance estimates are based on models without landsca

c US Shorebird Conservation Plan moderate concern species (Brown et a

d US Shorebird Conservation Plan high concern species (Brown et al., 20

e National Waterfowl Management Plan region 43 high non-breeding ne

f National Waterfowl Management Plan region 43 moderately high non-

g Federal Endangered Species Act listed species.

h Bird species of special concern (Shuford and Gardali, 2008).

Please cite this article in press as: Stralberg, D. et al., OptimiziConserv. (2008), doi:10.1016/j.biocon.2008.10.013

affected by salt-pond restoration activities, due to their

dependence on salt-pond and/or tidal-marsh habitats in San

Francisco Bay for breeding, wintering, or migratory passage

(Table 1). Details on the selection of these focal species are

provided by Stralberg et al. (2006).

We constructed two sets of models (with and without

landscape terms) to address the spatial configuration and

landscape context question. For each focal species/season

and each habitat type (salt pond and tidal marsh) we identi-

fied a subset of site and natural log-transformed landscape

proportion variables (proportion + 0.01) that we believed could

influence the density of birds using an area (Table 2). Using

this list of candidate variables, we then constructed all possi-

ble models (i.e., all possible combinations of candidate vari-

ables), with and without landscape terms. We used natural

log (density + 1) as the response variable for each focal

species, with density measured in units of number of birds

ha�1, not only because densities tended to be log-normally

cisco Bay optimization runs, with corresponding functionalates. F = fall; W = winter; S = spring. Outside abun-nds outside the restoration area, based on mean densities;uding outside abundance), based on optimizing each pondh specialist; cs1-5 = conservation status 1 (low) to 5 (high).

Weights Outside abundancea Maximum abundanceb

cs2c 1544 5046

cs2c 2269 12,661

cs2c 169 310

cs2c 428 4089

cs1 519 2 921

cs1 1931 8633

cs3d 858 13,892

cs1 1554 47,366

cs1 1534 9686

cs3d 1925 15,868

cs1 2652 6679

cs1 2037 4973

cs2c 666 3121

cs1 20 223

mps, cs3d 0 329

mps, cs2c 72 968

cs1 56 286

cs1 80 248

cs3e 30 200

cs2f 6425 8069

cs1 129 1074

mps, cs3e 132 753

W) mps, cs2 1 404

mps, cs2 17 198

mps, cs1 2001 10,405

tms, cs5g 305 793

tms, cs4h 429 1017

tms, cs1 3408 16,039

tms, cs4h 15,801 37,308

ple of tidal marsh and salt pond sites.

pe terms.

l., 2001).

01).

ed (NAWMP, 2004).

breeding need (NAWMP, 2004).

ng wetland restoration and management for avian ..., Biol.

Table 2 – Candidate site and landscape variables for avian density models. MP = managed pond; TM = tidal marsh. S = site;L = landscape. F = fixed; V = Variable.

Variable Definition Type

hectares Pond size (ha) MP, S, F

salinlowa Low salinity (<60 ppt) pond MP, S, V

salinmeda Medium salinity (60–120 ppt) pond MP, S, V

salinhigha High salinity (120–180 ppt) pond MP, S, V

salinvhigha Very high salinity (>180 ppt) pond MP, S, V

depthdeepa Deepb pond MP, S, V

depthshala Shallowc pond MP, S, V

depthmeda Intermediated depth pond MP, S, V

pondprop Proportion of area surveyed that contained ponds/pannes TM, S, F

lindens Linear channel density within survey area (m/ha) TM, S, F

chann12 Linear channel density of channels less than 2 m in width TM, S, F

chann45 Linear channel density of channels greater than 4 m in width TM, S, F

logmp Proportion of area within a 1-km buffer containing managed ponds or salt evaporation pondse MP/TM, L, V

logtm Proportion of area within a 1-km buffer containing tidal marshe MP/TM, L, V

logmud Proportion of area within 1-km buffer containing tidal flatse MP/TM, L, F

logbay Proportion of area within 1-km buffer containing bay open water or tidal flatse MP/TM, L, F

logntm Proportion of area with a 1-km buffer containing non-tidal marshe MP/TM, L, F

lognatup Proportion of area within 1-km buffer containing natural uplandse MP/TM, L, F

a Boolean variable (0 or 1).

b (At least 50% >1 m deep) and (not more than 10% <15 cm deep).

c ((At least 10% <15 cm deep) and (not more than 10% >1 m deep)) or (mean depth < 0.5 m).

d Not shallow or deep.

e Natural log-transformed with + 0.01 adjustment.

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distributed, but also because we considered density to be

influenced in a multiplicative rather than an additive fashion

by the independent variables (Nur et al., 1999). Thus, we as-

sumed that a unit increase in a predictor variable resulted

in a proportionate increase or decrease in density rather than

in a fixed increase or decrease.

This analysis was performed uniquely for each habitat and

season that a focal species was of interest. The process was

automated using Proc Mixed in SAS 9.1 (SAS, 2003), using a

generalized linear model with an identity link function and

normal error distribution. All variables were considered fixed

effects. Because the tidal-marsh models were based on data

from just 12 to 14 sites, we limited the models to one site var-

iable and two landscape variables.

For each candidate model analyzed, we calculated a

weight based on the adjusted Akaike information criterion

(AICc), a measure of model suitability and parsimony. This

criterion quantifies model fit but also penalizes more complex

models, thus quantifying the trade-off between simplicity

and model fit (Burnham and Anderson, 2002). The entire suite

of models (for each species–season–habitat) was used to gen-

erate model-averaged coefficients for each variable, based on

the AICc weights. This procedure was conducted for each set

of models (with and without landscape terms).

2.4. Optimization problem

Our optimization problem consisted of identifying restoration

configurations that would provide the greatest conservation

benefit, in terms of South Bay bird abundance, across a suite

of species, weighted by a measure of conservation importance.

This meant that for a given solution, each of 55 restoration

units (current ponds) was assigned a type (tidal marsh or man-

Please cite this article in press as: Stralberg, D. et al., OptimizConserv. (2008), doi:10.1016/j.biocon.2008.10.013

aged pond), and managed ponds were assigned specific man-

aged conditions (salinity and depth categories). Each

managed pond could only have one salinity range, but was al-

lowed to contain a combination of depth categories. We made

the simplifying assumption that each pond could be manipu-

lated independently, although salinity conditions are more

easily manipulated using a chain of evaporation ponds (Siegel

and Bachand, 2002). Restored tidal marshes were assumed to

resemble existing tidal marshes in terms of their habitat val-

ues, and a fixed set of geomorphic characteristics, based on fac-

tors such as elevation and bay proximity (HT Harvey and

Associates, unpubl. data), was applied to each wetland restora-

tion unit.

Decision variables were defined as follows:

Th,i a binary variable, where Th,i = 1 if unit i is restored as

wetland type h, Th,i = 0 otherwise. For each i,

T0,i+Tl,i = 1

pi,j value of variable salt pond attribute j for restoration

unit i, where h = wetland type (MP = managed pond,

TM = tidal marsh), i = restoration unit (Fig. 1), and

j = variable managed pond attribute (e.g., salinity,

depth) (Table 2). The pi,j variables determine the

managed conditions (salinity and depth) for restora-

tion unit i, so they are only relevant if TMP,j = 1

The following constants were used:

Lj,k value of the fixed attribute k in restoration unit i

ah,j,m slope parameter for variable attribute j for species m

in wetlands of type h (necessarily 0 if h = TM)

ah,k,m slope parameter for fixed attribute k for species m in

wetlands of type h

ing wetland restoration and management for avian ..., Biol.

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ch,l,m slope parameter for the dependence of species m in

wetlands of type h on the total nearby area (within 1

km) of wetlands of type l

qh,i area (ha) of wetland type h within 1 km of restora-

tion unit i

Ji.j area (ha) of wetland j within 1 km of restoration unit

i

Ai area (ha) of restoration unit 1 i

Nmaxm maximum possible abundance of species m, calcu-

lated by maximizing its abundance in each pond

independently while ignoring other species (Table 1)

Outsidem estimated abundance of species m in the remainder

of the South Bay (Table 1)

Wm weight assigned to species m based on its conserva-

tion priority

We then maximized the following two objective functions,

based on the number Nm of individuals of each species m sup-

ported by the entire system (South Bay ponds and marshes):

Zlinear ¼X

m

WmNm

Nmaxm

ð1Þ

Zlog ¼X

m

Wm logðNmÞ ð2Þ

Subject to the following constraints:

Nm ¼ NMP;m þNTM;m þOutsidem ð3Þ

Nh;m ¼X

i

Th;iAi expX

j

ah;j;mpi;j

0@

0@

24

þX

k

ah;k;mLi;k þX

l

ch;l;mqh;i

!� 1

!#ð4Þ

qh;i ¼ logX

j

Th;iJi;j

0@

1Aþ 1

0@

1A ð5Þ

The first objective (1) maximizes the sum of standard-

ized (relative to the maximum possible abundance) pond-le-

vel abundances. Because it is a linear objective, each

additional individual of a given species adds the same

increment to the objective. In contrast, the second objective

(2) makes the marginal value of an extra individual of a gi-

ven species inversely proportional to the species’ popula-

tion. This is because the derivative of the log function isddx ðlog xÞ ¼ 1

x .

Slope parameters (4) are based on model-averaged coeffi-

cients for the two sets of habitat models described above with

and without landscape terms. The latter case has all ch,l,m = 0

so the qh,i (5) are not used. The inclusion of landscape terms

was intended to evaluate the importance of spatial configura-

tion in the optimal restoration configuration.

2.5. Integer programming formulation

The equations defining the species abundances Nh,m involve

an exponential function, while the equations defining the

log-transformed wetland areas qhi and the log objective Zlog

both involve logarithms. As a result, the model is signifi-

cantly non-linear and non-convex, so it was not immediately

clear that linear and integer programming methods were

applicable. We used two standard piece-wise linear approxi-

mations to these non-linear functions (Markowitz and

Please cite this article in press as: Stralberg, D. et al., OptimiziConserv. (2008), doi:10.1016/j.biocon.2008.10.013

Manne, 1957; Dantzig, 1963), the details of which are de-

scribed in Appendix A. For all three non-linearities in our

model, we used an approximation that consistently overesti-

mates abundance, which allowed us to obtain global bounds

on the optimum.

2.6. Species weights

To assign Wm values, we considered three types of focal-

species weighting schemes: one based on favoring tidal-

marsh and salt-pond specialists, one based on conservation

status, and a neutral weighting scheme (all species equal).

Because there are fewer tidal-marsh specialists than salt-

pond specialists, and because there are several special sta-

tus and endemic species for which San Francisco Bay tidal

marsh is particularly important, our starting point was to

assign higher weight to tidal-marsh specialist species.

Although the ecological definitions for specialist species

may vary, we defined specialists as those that were detected

in one habitat but not the other on our surveys (Table 1). We

evaluated two different weighting schemes for these spe-

cialists. The weights assigned to tidal-marsh specialists,

salt-pond specialists, and other species, respectively, were

10/4/1 (‘‘m10p4’’) and 10/1/1 (‘‘m10p1’’). The other two

weighting schemes were based on the conservation status

of our focal species. We ranked species on a scale of 1–5

based on their threat status according to the federal endan-

gered species act, the California bird species of special con-

cern list (Shuford and Gardali, 2008), the North American

Waterfowl Management Plan (NAWMP, 2004), the US Shore-

bird Conservation Plan (Brown et al., 2001), and the North

American Waterbird Plan (Kushlan et al., 2002) (Table 1).

These conservation status species (ranks from 2 to 5) were

then weighted on a scale of powers of two (2, 4, 8, 16;

‘‘cs2’’) and powers of three (3, 9, 27, 81; ‘‘cs3’’), compared

to a weight of 1 for the species with no special conservation

status.

2.7. Optimization criteria

Based on the criteria described above, we completed 20 opti-

mization runs using the CPLEX 11.0 Mixed Integer Linear Pro-

gram (MILP) solver (ILOG, 2007): 2 objective functions (linear

and log-linear) · 2 habitat models (with and without land-

scape terms) · 5 species weighting schemes (two based on

conservation status, two favoring habitat specialists, and

one neutral). This took an average of 230 min per run. The per-

formance of each run was assessed by comparing the value of

the objective function with the upper bound for the solution,

and the resulting restoration configurations were compared

in terms of habitat composition and configuration, as well as

species abundance (by functional group) and diversity.

To assist in the initial development of optimization crite-

ria, and improve the speed at which runs could be evalu-

ated, we also programmed a simple local-search heuristic

that started from the best homogeneous (i.e., the same

management option used in all wetlands) solution of the

decision variables and made changes to one variable at a

time as long as the change resulted in an improved objec-

tive value.

ng wetland restoration and management for avian ..., Biol.

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3. Results

3.1. Optimization performance

In terms of optimization performance, all solutions produced

by the integer programs were within 12% of optimal, as mea-

sured by the relative gap (percentage difference between the

solution and the upper bound) (Table 3). For most sets of opti-

mization criteria, the relative gap was actually less than 2%.

All else equal, the runs using models that included landscape

terms had somewhat larger optimality gaps, reflecting their

added complexity due to the linearized logarithms in the

landscape terms. The optimality gaps for the linear objectives

were generally (but not always) larger than those for the log-

linear objectives. This was a function of the abundance off-

sets (estimated abundance outside the restoration area and

maximum possible overall abundance) used to standardize

species within the linear objective. Because the log-linear

objective did not include this offset, the range of possible val-

ues stayed within a relatively narrow range.

For each of the runs we conducted, the solution found by

the local-search heuristic resulted in an objective value that

came within 0.02% of that found by the optimization. Since

the heuristic was used as a starting point for the optimization,

the optimization results were never worse than the heuristic

ones. Additional details can be found in on-line Appendix B.

3.2. Abundance solutions

A large source of variation in abundance for the optimal solu-

tion across our 20 optimization runs was the weighting

scheme (Fig. 2). Tidal-marsh specialist groups (rails and land-

Table 3 – Performance statistics for 20 sets of optimization critesolution (log-linear and linear), the relative gap between the soto managed ponds. lin = linear objective function; log = log-linnone = model without landscape terms; neutral = no species wof two); consp3 = conservation status weighting (powers of thrpond species ratio); m10p4 = specialist weight (10:4 tidal mars

Optimization criteria Log-linear objective Linea

neutral-lin-land 209.8 0.

neutral-lin-none 210.8 0.

neutral-log-land 211.4 0.

neutral-log-none 211.4 0.

consp2-lin-land 574.4 1.

consp2-lin-none 573.3 1.

consp2-log-land 580.3 1.

consp2-log-none 580.7 1.

consp3-lin-land 1480.4 4.

consp3-lin-none 1250.0 5.

consp3-log-land 1524.1 4.

consp3-log-none 1532.5 4.

m10p1-lin-land 453.8 1.

m10p1-lin-none 462.4 1.

m10p1-log-land 492.3 1.

m10p1-log-none 497.3 1.

m10p4-lin-land 599.7 1.

m10p4-lin-none 474.5 1.

m10p4-log-land 616.0 1.

m10p4-log-none 619.3 1.

Please cite this article in press as: Stralberg, D. et al., OptimizConserv. (2008), doi:10.1016/j.biocon.2008.10.013

birds) were completely absent from all solutions based on a

neutral weighting scheme, as well as from two of the pow-

ers-of-two conservation status (cs2) solutions. They were

most abundant using the 10:1 (marsh-to-pond) specialist

weighting scheme (m10p1) and the powers-of-three conser-

vation status weighting scheme (cs3). Pond specialist groups

(fish-eaters, eared grebes and phalaropes) were most abun-

dant in the solutions based on a neutral weighting scheme

and were absent from some that were based on the 10:1 spe-

cialist weighting scheme (m10p1) or the powers-of-three con-

servation status weighting scheme (cs3). For other species

groups, the differences among weighting schemes were gen-

erally less important than differences within weighting

schemes (i.e., due to objective function or model). Dabbling

ducks, as a group, had the lowest variation in abundance

across optimization criteria. Results for individual species (to-

tal and by pond) can be found in on-line Appendix B.

For all species groups, the effect of the type of objective

function (linear vs. log-linear) depended on the weighting

schemes. That is, there were no groups that were consistently

higher or lower using the log-linear vs. linear objective func-

tion, although individual species with small ‘‘outside’’ abun-

dances relative to their overall maximum abundance (e.g.,

fall-season Wilson’s phalarope, Phalaropus tricolor) were fa-

vored by the linear objective (Table 1). As expected, the log-

linear objective function resulted in greater species represen-

tation, ensuring the inclusion of lower-weight species and

‘‘minority’’ species, including some tidal-marsh specialists

(Fig. 2).

All else equal, the inclusion of landscape terms had differ-

ent effects, depending on the landscape sensitivities of indi-

vidual species. Eared grebes, large shorebirds, and small

ria. Performance statistics include objective values for eachlution and the upper bound, and the ratio of tidal marshesear objective function; land = model with landscape terms;eighting; consp2 = conservation status weighting (powersee); m10p1 = specialist weighting (10:1 tidal marsh to salth to salt pond species ratio).

r objective Relative gap (%) Marsh:pond ratio

606 8.78 0:55

682 0.19 0:55

603 2.20 0:55

680 0.59 0:55

631 8.23 0:55

755 1.59 0:55

596 2.17 5:50

706 1.04 14:41

839 2.70 49:6

349 0.04 55:0

599 1.55 22:33

814 0.80 30:25

542 0.10 55:0

658 0.00 55:0

427 1.50 26:29

495 0.80 33:22

639 11.78 37:18

745 1.31 55:0

561 2.87 11:44

627 1.76 20:35

ing wetland restoration and management for avian ..., Biol.

Fig. 2 – Abundance (log-transformed) solutions by species group for 20 sets of optimization criteria. lin = linear objective

function; log = log-linear objective function; land = model with landscape terms; none = model without landscape terms;

neutral = no species weighting; m10pX = specialist weighting scheme, where 10:X = marsh specialist weight relative to pond

specialists; csX = conservation status weighting scheme, where X = power used to differentiate weights. See Table 1 for

species weights.

B I O L O G I C A L C O N S E R V A T I O N x x x ( 2 0 0 8 ) x x x – x x x 9

ARTICLE IN PRESS

shorebirds generally had higher abundance in the solutions

that were based on models that included landscape terms

(Fig. 2). Landbirds and rails had lower abundance in the land-

scape-model solutions.

With respect to total abundance (all species combined),

there was little discernable pattern across optimization crite-

ria (Fig. 3). Total abundance was highest for the powers-of-

two-weighted conservation status (cs2) solutions, and lowest

for the solution based on the powers-of-three conservation

status weighting scheme (cs3). Species richness and Shan-

non diversity were generally higher in the solutions that

were based on log-linear objective functions, all else equal.

The highest species diversity was achieved using the 10:4

(marsh to pond) specialist weighting scheme (m10p4), and

a log-linear objective function.

3.3. Habitat solutions

The largest overall differences in habitat composition were

driven by the objective function. The use of a linear objec-

Please cite this article in press as: Stralberg, D. et al., OptimiziConserv. (2008), doi:10.1016/j.biocon.2008.10.013

tive function resulted in habitat configurations that were

quite homogeneous, usually dominated by either managed

pond or tidal marsh, depending on the weighting scheme

(Table 3, Fig. 4). Managed pond salinities and depths also

tended to be fairly uniform within the pond-dominated

solutions. With neutral weighting and no landscape terms

included, low-salinity shallow ponds dominated. Other

weighting schemes resulted in managed ponds that were

all shallow and mostly high-salinity.

Using the log-linear objective function, pond-marsh ra-

tios of our solutions were still quite variable, depending

on the weighting scheme used (Table 3, Fig. 4). With a

completely neutral weighting scheme, both solutions (with

and without landscape terms) were comprised entirely of

shallow managed ponds. Weighting schemes that greatly

favored tidal-marsh specialists (m10p1) resulted in solutions

that were comprised of 47–60% tidal marsh, while higher

weightings for salt pond specialists (m10p4) produced con-

figurations comprised of 20–36% tidal marsh. Using the con-

servation status weighting schemes, there was a big

ng wetland restoration and management for avian ..., Biol.

Fig. 3 – Diversity and abundance solutions for 20 sets of optimization criteria. Open triangles represent neutral weighting;

closed triangles represent the m10p1 (10:1 marsh-to-pond) specialist weighting scheme; open squares represent the m10p4

(10:4 marsh-to-pond) specialist weighting scheme; closed squares represent the cs2 (powers-of-two) conservation status

weighting scheme; closed circles represent the cs3 (powers-of-three) conservation status weighting scheme. lin = linear

objective function; log = log-linear objective function; land = model with landscape terms; none = model without landscape

terms. See Table 1 for species weights.

10 B I O L O G I C A L C O N S E R V A T I O N x x x ( 2 0 0 8 ) x x x – x x x

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difference between powers of two (cs2), which resulted in

solutions with 9–25% tidal marsh, and powers of three

(cs3), which yielded 40–55% tidal-marsh solutions. Across

this spectrum, there was a fairly consistent mix of manage-

ment conditions, with low- and high-salinity shallow ponds

most prevalent. Deep ponds occurred in only two solutions

(m10p4), and intermediate and very high-salinity ponds

were not included in any solutions.

In terms of the spatial configuration of our solutions,

there was also high variability, given the similar conserva-

tion potential of all ponds (Fig. 4). No single restoration

unit had the same outcome across all 20 optimization runs

(or even across the 10 log-linear runs). The models that in-

cluded landscape terms resulted in solutions with tidal

marshes and low-salinity ponds near the bay edge, and

high-salinity ponds farther landward. The landscape

models also resulted in slightly greater, but barely notice-

able, spatial aggregation of marsh and managed pond hab-

itats. Of the different habitat types, high-salinity ponds

appeared most aggregated in the landscape model

solutions.

4. Discussion and conclusions

4.1. Optimization benefits

This work represents a novel application of integer pro-

gramming techniques to wetland restoration planning, facil-

itated by the availability of comprehensive avian and

habitat datasets. Although optimization techniques have

Please cite this article in press as: Stralberg, D. et al., OptimizConserv. (2008), doi:10.1016/j.biocon.2008.10.013

been applied to other problems of species conservation

and reserve design (Nevo and Garcia, 1996; Hof and Ra-

phael, 1997; Onal and Briers, 2003) and to the prioritization

of restoration sites based on economic, hydrologic, water

quality, and habitat connectivity factors (Roise et al., 2004;

Newbold, 2005; Crossman and Bryan, 2006), we have applied

the approach to wetland restoration using biological (avian

abundance) criteria. The optimization problem that we for-

mulated also integrates reserve design and habitat manage-

ment considerations, and applies empirical habitat-

abundance relationships to a multi-species conservation

optimization problem.

Our use of a log-linear objective function allowed us to de-

velop a biologically-meaningful optimization problem: one

that valued the proportional change in a species’ abundance

in response to habitat and landscape conditions over the

absolute change. The incorporation of log-transformed bird

densities and landscape variables in our models also allowed

a more realistic representation of species’ non-linear habitat

responses. Although the incorporation of these biological

realities resulted in a non-convex, non-linear integer model,

we solved it with an integer program by using an appropriate

piece-wise linear approximation to the non-linear elements,

taking advantage of the considerable machinery that has

been developed for integer programming. Ensuring that our

approximations were conservative (consistently over-esti-

mating the predicted abundance) allowed us to obtain global

bounds on the optimum, resulting in reliable optimization

performance measures. Typically such global bounds are only

available for linear (or at least convex non-linear) models.

ing wetland restoration and management for avian ..., Biol.

Fig. 4 – Habitat solutions for 20 sets of optimization criteria. Restored tidal marsh areas are depicted in green; low salinity

shallow and medium depth managed ponds are shown in light and dark blue, respectively; high salinity shallow depth

managed ponds are shown in orange. lin = linear objective function; log = loglinear objective function; land = model with

landscape terms; none = model without landscape terms; neutral = no species weighting; m10pX = specialist weighting

scheme, where 10:X = marsh specialist weight relative to pond specialists; csX = conservation status weighting scheme,

where X = power used to differentiate weights. See Table 1 for species weights.

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However, because integer programming resulted in only neg-

ligible improvements in objective values over the simple lo-

cal-search heuristic, the primary benefit of the optimization

in this particular case was to provide information about the

quality of the solutions. Although computation time was

not a major issue for our problem, our results support the

assertion that, in some cases, heuristic approaches may, more

quickly and easily, achieve solutions as good as those yielded

by integer programming (Pressey et al., 1996; Moilanen, 2008).

However, given that performance is highly dependent on the

heuristic algorithm used (Csuti et al., 1997; Vanderkam et al.,

2007), as well as the size and complexity of the problem (Onal,

2004), we do not think this result should be broadly

generalized.

4.2. Objective functions

Habitat heterogeneity and species diversity in the restoration

solutions were achieved with the implementation of a log-

linear objective function. Although our linear objective func-

tion was standardized so that less abundant species had the

same importance as more abundant species, it favored spe-

Please cite this article in press as: Stralberg, D. et al., OptimiziConserv. (2008), doi:10.1016/j.biocon.2008.10.013

cies with a small ‘‘outside’’ abundance relative to the maxi-

mum possible abundance within the restoration area.

Furthermore, using a linear objective function, the ‘‘major-

ity’’ species (those with habitat requirements that were sim-

ilar to the greatest number of other species) tended to drive

the optimization, which sometimes resulted in solutions

sacrificing some species. In contrast, the log-linear objective

served as an equalizer, providing greater representation for

‘‘minority’’ species and resulting in solutions that contained

all focal species; this is a fundamental property of the log-

linear objective. We suggest that the use of a log-linear

objective function may be more appropriate for optimization

of multi-species conservation. Not only are log-transforma-

tions often appropriate for modeling, especially for high

abundance flocking species such as shorebirds, but a log-lin-

ear objective function can provide greater species represen-

tation and reduce the variability of solutions, thereby

providing more useful information for the selection of actual

conservation configurations. If the primary goal is to maxi-

mize the overall increase in abundance of specific species,

however, a linear objective function would be more

appropriate.

ng wetland restoration and management for avian ..., Biol.

12 B I O L O G I C A L C O N S E R V A T I O N x x x ( 2 0 0 8 ) x x x – x x x

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4.3. Weighting schemes

Our optimization results were heavily influenced by the

weights assigned to individual species, and undoubtedly by

our initial selection of focal species. Because there are fewer

bird species that use tidal marsh in high numbers relative

to managed ponds, our list of focal species was biased toward

pond-associated species. A neutral weighting scheme there-

fore tended to favor salt-pond specialists such as eared gre-

bes, fish-eaters, and phalaropes as well as species that are

more abundant in salt ponds (shorebirds and diving ducks).

Because tidal-marsh species (rails and landbirds) generally

have higher levels of conservation concern and because resto-

ration of tidal-marsh habitat is the primary goal of the South

Bay restoration project, we assigned higher weights to these

species to achieve substantial tidal-marsh representation in

the restoration solutions. Another alternative would have

been to include minimum abundance thresholds for certain

species as constraints in the optimization, but in a system

like this one, with such dramatic trade-offs among species,

it can be difficult to find feasible solutions when too many

constraints are imposed. Thus, careful a priori consideration

of focal species and weighting systems is a critical component

of any conservation-oriented optimization exercise (Kremen

et al., 2008). Furthermore, initial weightings may need to be

modified if diversity of species and habitats is a desired out-

come. Otherwise the habitat type that meets the needs of

the greatest number of species (in our case, shallow managed

ponds) may dominate the solution. Alternatively, one could

explicitly optimize species diversity, for which computation-

ally convenient (linear) metrics have been proposed (Onal,

1997).

The specific weights that we used were somewhat arbi-

trary, but the ranking systems were based on objective,

repeatable criteria. Assuming that our weighting schemes

represented a sufficiently broad range of assumptions about

species’ relative importance, we were able to identify an

envelope of optimal restoration outcomes. The optimization

solutions that were based on the log-linear objective func-

tion and non-neutral weighting ranged from 9% to 60% tidal

marsh, suggesting that, if the goal is to manage for high

avian species diversity and habitat heterogeneity, in addition

to overall bird abundance, at least 40% of ponds should be

retained and managed for waterbirds. If the primary goal is

to maximize abundance of high-conservation-status or ti-

dal-marsh-specialist species, however, our results suggest

that at least 40–47% of the ponds should be restored to tidal

marsh.

4.4. Spatial configuration

Based on the physical setting of individual ponds (e.g., tidal

influence, salinity, and elevation) that determined the antici-

pated restored marsh characteristics (i.e., channel density

and pond/panne proportion), certain ones were more fre-

quently selected as restored tidal marshes than others. Over-

all, however, there was considerable variation in the outcome

of individual restoration units across optimization runs.

When landscape terms were included in the habitat models

(and using the log-linear objective), we saw greater consis-

Please cite this article in press as: Stralberg, D. et al., OptimizConserv. (2008), doi:10.1016/j.biocon.2008.10.013

tency among restoration outcomes for specific areas, based

on the modeled importance of surrounding landscape condi-

tions. Specifically, the apparent importance of surrounding

bay and mudflat proportion for birds in tidal marshes and

low-salinity shallow ponds led to solutions in which these

habitats were found primarily near the bay edge. The variabil-

ity of outcomes for individual ponds, however, suggests a

high level of flexibility in restoration planning at the land-

scape level.

Although solutions tended to have greater aggregation of

similar habitat types when landscape models were used,

the effects were subtle and not necessarily intuitive. We

did not observe the positive relationships between habitat

area and species abundance that have been demonstrated

in numerous habitat fragmentation studies (Flather and

Sauer, 1996; Bolger et al., 1997; Bender et al., 1998). In some

cases, our results may have been driven by artifacts of cur-

rent landscape configuration, because our landscape rela-

tionships were based on empirical data rather than

idealized reserve-design scenarios. For example, species

associated with high-salinity salt ponds exhibited the stron-

gest landscape relationships (positive with surrounding

ponds), but that may be due to constraints on the configu-

ration of salt-evaporation ponds, such that high-salinity

ponds tend to be clustered together for management pur-

poses (Siegel and Bachand, 2002). Conversely, landscape

relationships were not as strong for many tidal-marsh spe-

cies, perhaps because large expanses of tidal marsh no

longer exist in this area due to historical modifications such

as diking, dredging and filling (Josselyn, 1983). Thus, we

may be limited by our application of current conditions to

future restoration scenarios.

4.5. Conservation implications

For the South Bay, although shallow managed ponds provide

maximum benefits for the largest number of bird species,

habitat restoration goals and sensitive species concerns call

for a more heterogeneous wetland landscape. When we

prioritized the conservation of tidal-marsh-specialist and

high-conservation-status species, our optimization results

suggested that at least half of the ponds should be restored

to tidal marsh habitat. To achieve high species representation

and diversity, however, we found that at least 40% of the

ponds should be retained and managed for waterbirds, result-

ing in a habitat mosaic. We did not find great benefits to birds

in the spatial aggregation of habitats, but when landscape

context was considered, optimal configurations were those

in which low-salinity ponds and restored tidal marsh were

located near the bay’s edge and high-salinity ponds were

located farther landward.

More generally, the approach that we have described pro-

vides an important proof-of-concept for the application of

integer programming optimization to conservation and resto-

ration problems at different spatial scales. From the design of

individual restoration projects to the management of large

wildlife refuges to the application of landscape treatments

(e.g., grazing, timber harvest, controlled burns), our approach

can be applied to any conservation problem that involves the

optimization of habitat types and management conditions for

ing wetland restoration and management for avian ..., Biol.

�

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ARTICLE IN PRESS

multiple species. Our analysis benefitted from comprehensive

data on species’ densities and habitat relationships, but even

without such extensive empirical data, expert knowledge of

habitat preferences and relative densities could be used to

parameterize an optimization model. This is especially the

case using a log-linear objective function, which is less

dependent on actual abundance estimates. In addition, the

utility of our approach could be improved by combining bio-

logical restoration objectives (and costs) with socio-economic

costs and benefits, and physical restoration constraints (i.e.,

salinity and levee management).

Acknowledgments

Preparation of this manuscript was funded in part by dona-

tions from Carolyn Johnson, Rick Theis, and AT&T Labs-Re-

search. The data collection and analysis that made this

work possible were funded by grants from the California

Coastal Conservancy, the Gabilan Foundation, the Tides

Foundation, Rintels Charitable Trust, the Bernard Osher

Foundation, the Richard Grand Foundation, the Mary A.

Crocker Trust, and ESRI. Access to tidal-marsh and salt-

pond study sites was provided by the Don Edwards San

Francisco Bay National Wildlife Refuge, Cargill Salt, East

Bay Regional Parks, Hayward Regional Shoreline, and Palo

Alto Baylands. Access to avian survey data and input on

modeling efforts from collaborators at the USGS Biological

Research Division, particularly Nicole Athearn and John

Takekawa, and the San Francisco Bay Bird Observatory is

greatly appreciated. Salt-pond bathymetry data obtained

from USGS were also useful for this project. Model results

and interpretation benefited from comments and input

from the South Bay Restoration science team, management

team, and consultants. Earlier drafts of this manuscript

were greatly improved by comments from John Wiens,

Nathaniel Seavy, and two anonymous reviewers. Finally,

we are grateful for assistance with field surveys, data entry,

and GIS analysis by Parvaneh Abbaspour, Sue Abbott, Mat-

thew Anderson, Elizabeth Brusati, Yvonne Chan, Jim DeSta-

bler, Maria DiAngelo, Jeanne Hammond, Leonard Liu, Sue

Macias, Cheryl Millett, Hanna Mounce, Gary Page, Chris Rin-

toul, Miko Ruhlen, Amanda Shults, Hildie Spautz, Samuel

Valdez, and Julian Wood. This is contribution number 1639

of PRBO Conservation Science.

Appendix A

For the logarithmic objective, each species m involves the

term log(Nm), which we approximated as follows. Let

b0 < b1 < � � � < bk be an increasing sequence of positive real

numbers, which we term breakpoints. We replaced log(Nm)

in the objective with a new variable Lm, constrained by:

Lm � logðbiÞ þ ðNm � biÞ=bi for each i

This constrains Lm to be below a piece-wise linear curve

that is tangent to the log function at the breakpoints. Since

the objective function maximizes the Lm, their values will

be upper bounds on log(Nm), with equality if the optimum is

reached with Lm at one of the breakpoints.

Please cite this article in press as: Stralberg, D. et al., OptimiziConserv. (2008), doi:10.1016/j.biocon.2008.10.013

For the species abundances, the species density is expo-

nential in the model parameters. Since the exponential func-

tion is convex instead of concave (and the objective implies

that the optimization will maximize species density), we

use a piece-wise linear interpolation instead of a piece-wise

linear tangent approximation. To approximate exp(x) by ex

using the breakpoints b0 < b1 < � � � < bk, we introduce addi-

tional continuous variables y1, y2, . . ., yk and additional binary

variables z1, z2, . . ., zk and the constraints:

x ¼ b0 þ ðb1 � b0Þy1 þ ðb2 � b1Þy2 þ . . .þ ðbk � bk�1Þyk

ex ¼ expðb0Þ þ ðexpðb1Þ � expðb0ÞÞy1 þ ðexpðb2Þ � expðb1ÞÞy2 þ � �þ ðexpðbkÞ � expðbk�1ÞÞyk

yi 6 zi for i ¼ 1; 2; . . . ; k

ziþ1 6 yi for i ¼ 1; 2; . . . ; k� 1

These constraints force ex P exp(x), with equality if the zi are

integer and x is at one of the breakpoints.

Note that the equation defining Nh,m appears to contain an

additional nonlinearity, namely that Th,i (a variable) is multi-

plied by the exponential term, which is approximated by

the variable ex. The equation could be represented by a qua-

dratic constraint, but we can avoid the nonlinearity by using

the following linear constraints instead:

Nh;m ¼X

i

Nh;m;i

Nh;m;i 6 Aiex

Nh;m;i 6 Th;iAiMaxh;m;i;

where ex is the linear approximation to the exponential de-

scribed above, and Maxh,m,i is a constant upper bound on

the value of ex.

The second constraint bounds Nh,m correctly if Th,i = 1,

while the third constraint covers the other case, making

Nh,m,i = 0 if Th,i = 0.

For both the logarithmic objective and for species abun-

dances, the objective is maximized by maximizing the output

of the non-linear function. This is not the case for the log-

transformed wetland areas, since the influence of the wet-

land area on the objective depends on the sign of ch,l,m. That

is, if ch,l,m > 0, then increasing the wetland area increases the

objective, while if ch,l,m < 0, then increasing the wetland area

decreases the objective. Therefore, for the log-transformed

wetland areas we generated both the tangent and interpola-

tion piece-wise linear approximations and used the value

from the tangent approximation when ch,l,m > 0, and the value

from the interpolation when ch,l,m < 0.

For all three non-linearities in our model, we used an

approximation in which any approximation error overesti-

mates the resulting abundance. As a result, even though our

optimization problem is a mixed-integer, non-linear, non-

convex optimization problem, the bounds generated by our

mixed-integer linear approximation are still valid. However,

the bounds will be weak if the approximations are weak. To

obtain tight approximations without using too many break-

points (which significantly increase the size of the model)

we first solved using a small, coarse set of breakpoints and

limited the mixed-integer programming solver to 1000

branch-and-bound nodes. Then, if the optimum had variables

at values that are not at or very near breakpoints, we added

ng wetland restoration and management for avian ..., Biol.

14 B I O L O G I C A L C O N S E R V A T I O N x x x ( 2 0 0 8 ) x x x – x x x

ARTICLE IN PRESS

those values to the sequence of breakpoints and solved again.

This process continued until the optimum (or best solution

found before the branch-and-bound limit is reached) was at

or very near breakpoints. We then increased the branch-

and-bound limit to 10,000 nodes and continued adding values

and re-solving until the optimum or best solution was again

at or very near breakpoints. This process took an average of

230 min across the 20 sets of optimization parameters.

Appendix B

Detailed results from all 20 optimization runs, including per-

formance statistics and maps of species-specific densities by

pond, can be found on-line at http://www.prbo.org/wetlands/

SFBayOptimize/.

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