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Ecological Modelling 221 (2010) 2243–2250

Contents lists available at ScienceDirect

Ecological Modelling

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

athematical problem definition for ecological restoration planning

arissa F. McBridea,∗, Kerrie A. Wilsonb, Jutta Burgerc, Yi-Chin Fangc, Megan Lulowc,avid Olsonc,d, Mike O’Connell c, Hugh P. Possinghamb

University of Melbourne, School of Botany, Melbourne, Victoria 3010, AustraliaUniversity of Queensland, School of Biological Sciences, St. Lucia, Queensland 4072, AustraliaIrvine Ranch Conservancy, 4727 Portola Parkway, Irvine, CA 92620-1914, USAConservation Earth Consulting, 4234 McFarlane Avenue, Burbank, CA 91505, USA

r t i c l e i n f o

rticle history:eceived 12 November 2009eceived in revised form 22 April 2010ccepted 23 April 2010vailable online 30 June 2010

eywords:cological restorationestoration priorities

a b s t r a c t

Ecological restoration is an increasingly important tool for managing and improving highly degraded oraltered environments. Faced with a large number of sites or ecosystems to restore, and a diverse arrayof restoration approaches, investments in ecological restoration must be prioritized. Nevertheless, thereare relatively few examples of the systematic prioritization of restoration actions. The development of ageneral theory for ecological restoration that is sufficiently sophisticated and robust to account for theinherent complexity of restoration planning, and yet is flexible and adaptable to ensure applicability toa diverse array of restoration problems is needed. In this paper we draw on principles from system-atic conservation planning to explicitly formulate the ‘restoration prioritization problem’. We develop

ecision theoryonservation planningcological thresholdscosystem management

a generalized theory for static and dynamic restoration planning problems, and illustrate how the basicproblem formulation can be expanded to allow for many factors characteristic of restoration problems,including spatial dependencies, the possibility of restoration failure, and the choice of multiple restora-tion techniques. We illustrate the applicability of our generic problem definition by applying it to a casestudy – restoration prioritization on The Irvine Ranch Natural Landmark in Southern California. Throughthis case study we illustrate how the definition of the general restoration problem can be extended to

onstr

account for the specific c

. Introduction

Ecological restoration, the process of “assisting the recovery ofn ecosystem that has been degraded, damaged, or destroyed” (e.g.obbs and Cramer, 2008), is an increasingly important tool forelivering a diverse range of conservation outcomes (Dobson etl., 1997; Young, 2000). Restoration is used extensively for: sta-ilising degraded soils, returning biodiversity and habitat valueso a landscape, reducing poverty and sequestering carbon diox-de from the atmosphere (Jordan et al., 1988; Dobson et al., 1997;amb et al., 2005; Hobbs and Cramer, 2008). Regardless of its util-ty, the assisted restoration of habitat is characteristically time- andesource-intensive (Noss et al., 2009). While ecological restorationay be achieved with limited financial outlay (through, for exam-

le, the implementation of prescribed burning or the removal of

ntroduced grazers), it can also entail costly reintroductions of rarepecies, replanting of diverse plant communities, and/or the cre-tion of specific ecological niches (Hobbs and Norton, 1996). Someandscapes may have the capacity to passively restore or naturally

∗ Corresponding author. Tel.: +61 3 8344 3305; fax: +61 3 9348 1620.E-mail address: m.mcbride@pgrad.unimelb.edu.au (M.F. McBride).

304-3800/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2010.04.012

aints and considerations of an on-the-ground restoration problem.© 2010 Elsevier B.V. All rights reserved.

regenerate, but others may fail to fully regain their natural biodi-versity and ecological integrity without active intervention (McIverand Starr, 2001). The likelihood of a successful outcome may alsobe linked to the cost and relative intensity of the restoration activ-ity selected (Dorrough et al., 2008). In such situations and whenresources are limited, decisions about which areas to restore, when,and what restoration techniques to use must be made (Hyman andLeibowitz, 2000; Beechie et al., 2008). Previous examples of prioritysetting for restoration have tended to be based on scoring or rank-ing methods, or on expert opinion (O’Neill et al., 1997; McAllisteret al., 2000; Cipollini et al., 2005; Petty and Thorne, 2005) althoughthere are an increasing number of examples of the use of system-atic techniques to prioritize areas for restoration to achieve a rangeof objectives (Crossman et al., 2007; Bryan and Crossman, 2008;Crossman and Bryan, 2009). Regardless of the solution method, theprocess of restoration planning needs to be underpinned by a well-defined problem (Possingham, 2001). This problem definition mustbe sufficiently sophisticated and robust to account for the inherent

complexity of ecological restoration, but also flexible enough toensure applicability to a diverse array of restoration problems.

To some extent a framework for restoration planning alreadyexists in the considerable body of research focused on the design ofprotected area networks, commonly referred to as systematic con-

2 l Modelling 221 (2010) 2243–2250

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Box 1: Restoration planning on the Irvine Ranch NaturalLandmarkThe Irvine Ranch Natural Landmark is a collection of perma-nently protected wildlands and parks located near the SantaAna Mountains in Southern California. The Ranch containssome of the largest remaining stands of coastal sage scrub,oak-sycamore woodland, native grassland, and chaparralvegetation types in southern California. It represents approxi-mately 44,000 acres of land, much of which has been degradedby agriculture, intensive grazing, woodland clearance, adja-cent development, invasive species, and too-frequent fire. TheIrvine Ranch Conservancy has been established as a non-profitorganization to provide coordinated, ecologically responsiblemanagement services to the landowners. Existing levels ofdegradation and vulnerabilities to further disturbances fromfire and invasive species mean that managers face the impor-tant problem of determining how best to target funds availablefor restoration to ensure the full recovery and the long termpreservation of the significant ecological values of the Land-mark. A total of 923 sites have been identified as potential

244 M.F. McBride et al. / Ecologica

ervation planning (Westphal et al., 2003; Crossman and Bryan,006; Westphal et al., 2007). Systematic conservation planningims to identify representative, adequate, and efficient networksf protected areas (Margules and Pressey, 2000; Possingham et al.,000; Wilson et al., 2009). The allocation of funds for protected areastablishment can also be considered a special case of the more gen-ral conservation resource allocation problem, which entails therioritization of funds across a diverse array of conservation actions

n time and space (Wilson et al., 2007). Two general forms of theonservation resource allocation problem exist. In the maximal-overage problem the objective is to maximize the amount ofiodiversity protected (through the delivery of the required actionso abate the key threats) given a pre-specified resource constraintChurch and ReVelle, 1974). This form is appropriate where fundsre limited, which is commonly the case. In the alternative version,he minimum-set problem, the objective is to meet pre-specifiedoals while minimizing the resources expended (Pressey et al.,002). Within these two forms there are myriad possible formula-ions of the conservation resource allocation problem, dependingn the assumptions and simplifications that are made (Murdoch etl., 2007; Underwood et al., 2008; Joseph et al., 2009). An impor-ant distinction is between once-off allocations that preclude theossibility of strategic changes through time, and allocations overultiple time steps that allow for different allocations at different

imes.The static formulation of the conservation resource allocation

roblem assumes a single allocation of resources. A more gen-ral class is where decisions are made sequentially through timePossingham et al., 1993; Davis et al., 2006; Wilson et al., 2006).and protection is typically a resource-intensive process that is fea-ible only in stages. Under a protracted process of land protection,gnoring system dynamics and stochastic events when planningngoing investments may severely compromise the biodiversityutcomes: some areas may be degraded and lose their ecologi-al values, while other more important areas may come availableor protection (Meir et al., 2004). The same applies for restoration,hich is typically carried out over an extended time due to funding,

easonal, or logistical limitations. Disturbance processes such as firer drought, can dramatically alter the condition of sites in a land-cape, including those in which restoration is already underway.he effects of potential future changes and feedbacks in the systemre therefore important in the context of restoration prioritizationHobbs and Norton, 1996; Folke et al., 2004). Other potentiallymportant considerations are whether spatial dependencies areccounted for, whether desired outcomes are assumed to be real-zed with certainty, and whether multiple restoration techniquesre considered. The outcomes of ecological restoration are alsoighly influenced by the willingness of landholders to engage andupport restoration activities on their properties. Some landhold-rs will be more willing to engage than others and may thereforeepresent cost-effective opportunities for restoration.

The benefits of habitat protection and restoration are not con-trained to the protected or restored site, as there are both on-nd off-site impacts associated with such investments (Armswortht al., 2006). Spatial dependencies are of particular significancerom the perspective of restoration, since the speed and likeli-ood with which the benefits are realized is determined by the

nteractions between neighboring sites (Lindenmayer et al., 2002;uding et al., 2004). The allocation of resources, whether for pro-ection or restoration, must therefore be evaluated in the contextf the broader landscape. Whether planning for restoration at a

ite or landscape scale, a large proportion of restoration attemptsill fail and the possibility of failure is likely to vary between

ites and between restoration approaches and techniques (Zedlernd Callaway, 1999; Wilkins et al., 2003; Choi, 2004). The possi-ility that desired outcomes will not be realized is seldom dealt

candidates for restoration action and an annual budget of$700,000 is targeted each year over a 20-year time period forrestoration.

with in conservation (McBride et al., 2007; Hobbs, 2009), althoughaccounting for the possibility of failure will likely influence whichsites are prioritized for restoration and when the restoration shouldoccur.

Conservation practitioners implement a variety of strategiesto conserve biodiversity. There are similarly a diverse array ofapproaches and techniques used in restoration. Each approach hasdifferent costs associated with its implementation, delivers differ-ent ecological outcomes, and has varying likelihoods of success(McIver and Starr, 2001; Rayfield et al., 2005). Comprehensiveand informative restoration plans will be delivered by explicitlyaccounting for the benefits (both on- and off-site), costs, and risksof different restoration techniques, as opposed to considering thema single combined action (Hobbs and Cramer, 2008).

1.1. Towards a general theory for restoration planning

Despite the generality of the conservation resource allocationproblem, application of the theory and decision support tools toconservation actions other than protected area design is in itsinfancy (Possingham et al., 2001; Wilson et al., 2007). The first aimof this paper is to define the generic restoration problem. We out-line first the static formulation and illustrate how the basic problemformulation might be expanded to account for spatial dependen-cies, the possibility of restoration failure, and the choice betweendifferent restoration techniques. We then generalize this formu-lation to include temporal dynamics and to account for feedbacksand the likelihood that conditions will vary as restoration proceeds.The second aim of this paper is to demonstrate how the theory canbe applied and adapted to incorporate the specific constraints andconsiderations of an on-the-ground restoration problem. In orderto illustrate key concepts we refer to the restoration planning pro-cess on the Irvine Ranch Natural Landmark in Southern California(Box 1 ). We use this case study to test the utility of the theory wepresent.

2. The formulation of the one-step ecological restoration

prioritization problem

In the one-step formulation of the restoration prioritizationproblem, we assume a single once-off allocation of resources(Possingham et al., 2009). In this problem we ignore the poten-

l Modelling 221 (2010) 2243–2250 2245

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Box 2: An expanded set of budgetary constraints forrestorationAt the Irvine Ranch Natural Landmark, the cost of restoration ata site i is dependent on the desired habitat type, the restorationtechnique to be employed, and the area, slope, and acces-sibility of each site. Each site i = 1, . . ., Ns belongs to one of101 sub-watershed clusters – hereafter restoration clusters –k = 1, . . ., Nh, and a start-up cost CSC of $10,000 was includedfor initiating restoration at sites in each restoration cluster toaccount for the costs associated with moving equipment andpersonnel between sites. In some cases the start-up cost mayrepresent the cost of land purchase or easement establish-ment. We define Sk as the set of sites in restoration clusterk and ISk

(i) as an indicator function that equals 1 if site i is incluster k, and zero otherwise. The total cost for restoration eachyear is:

Ns∑i

ci(xi1, ˛i) +Nk∑k

[1 −(

Ns∏i

(1 − ISk(i))

)]CSC ≤ B, ˛i =⎡

⎢⎢⎣area

habitat typeslope

accessibility. . .

⎤⎥⎥⎦ ∀i,

where ˛i is matrix of site characteristics used in determiningcost. We assume the costs of restoration remain constant overtime and are independent of restoration undertaken elsewhere.Operational constraints also limit the total area that can be fea-sibly restored in any given year to approximately 80 hectares.This area constraint, AC, is represented by

Ns∑i=1

aixi1 ≤ AC,

M.F. McBride et al. / Ecologica

ial spatial dependencies and possible future events during projectmplementation. Our objective in the context of restoration mighte to select sites to restore in order to maximize the total utilitybenefit) gained for biodiversity from restoration, which is mea-ured in terms of a set of biodiversity assets j = 1, 2, . . ., Na, whichould include species, habitat types, or ecosystem processes. Weight aim to achieve this goal for a fixed financial budget B1.Initially we assume that each site, i = 1, 2, . . ., Ns, is either intact

r degraded. We represent the state of the landscape at time t withhe vector yt, where yit ∈ {0, 1} is the state of each individual site. If a site i is intact, yit = 1, if it is degraded yit = 0. We representhe control variable, the set of sites that can be restored, with theector xt, where xit ∈ {0, 1} and is equal to 1 if site i is selected forestoration, otherwise it is equal to 0. To begin with the landscapes in state y0 and following restoration, x1, it is in state y1. The stateynamics can be described by the equation:

i1 = 1 − (1 − yi0)(1 − xi1) ∀i, (1)

hich means that site i is intact at the next time step if it waslready intact, or it has been restored. Formally, our objective is toaximize Rs(y1), the utility from restoration across all sites in state

1:

s(y1) =Na∑j

wjfj(y1), (2)

here fj(y1) are asset-specific functions transforming representa-ion into a reward relevant to asset j. The weighting wj reflectshe degree to which biodiversity asset j contributes to the overallestoration utility. Where different assets are deemed more-or-lessritical to the overall outcomes of a restoration project, the weight-ng wj can be used to specify the importance of each. This utility iso be maximized subject to the budget constraint:

Ns

i=1

ci(x1) ≤ B, (3)

here ci(x1) is the cost of restoration at site i given other restora-ion actions, x1, in the area and B is the total budget constraint. Theonstraint can therefore be specified as a function of a matrix ofhe set of actions taken, x1, in addition to a set of site-specific vari-bles that modify the cost of restoring a particular site, ˛i (Box 2).dditional constraints can also be employed to account for non-nancial limitations, such as the amount of area able to be restored,he availability of seed, or a lack of sufficient field support (Box 2).

ithin this basic problem formulation we can also incorporateore specific details into the utility function.

.1. The objective of ecological restoration and the utility function

In Eq. (2), fj(y1) are biodiversity asset-specific functions, whichransform the state of the landscape into a utility. This can be aimple linear function or a more complex function which includesependencies between sites. In the simplest case, we might assumehat the objective is to maximize the representation of each biodi-ersity asset, given a linear utility function and knowledge of themount, rij, of each biodiversity asset that will be present at eachite when it is restored. We therefore state the utility of restoration

s:

j(y1) =Ns∑i

rijyi1, (4)

where ai is the area of site i which would receive restorationinvestment if xi1 = 1.

or more generally,

fj(y1) =Ns∑i

rijyi1 + (1 − yi1)sij, (5)

where sij is the amount of the biodiversity asset j in each site iwhen it is degraded. The summation over all assets in Eqs. (4) and(5) assumes the utility obtained from each is independent.

In more complicated situations, representation levels and costswill be a function of the investment made. Depending on howrepresentation of a particular biodiversity asset increases withincreasing investment, the utility function can take a wide varietyof different forms (for example, sigmoidal, concave, or threshold).A concave function can represent cases where there are diminish-ing returns with increasing investment (Wilson et al., 2007; Hobbsand Cramer, 2008). Similarly, threshold effects can be accountedfor, such as when a certain level of investment is necessary beforeany benefits are realized (Fahrig, 2002; Huggett, 2005; Rhodes etal., 2008).

There are many variations on the generalized version of the util-ity function presented in Eq. (2). We can, for example, account forspatial dependencies by incorporating a weighting term for the con-nectedness of restored sites. This would add a further term to the

utility function Rs(y1):

−bLf (L(y1), A(y1)), (6)

where L(y1) is the boundary length of the restored system, A(y1) isthe area of the restored system, and f is a function of L(y1) and A(y1).

2 l Modelling 221 (2010) 2243–2250

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Box 3: The utility of restorationThe objective of the restoration program on the Irvine RanchNatural Landmark is to maximize both the area restored andthe habitat restored for species of special concern, whileenhancing the resilience at both site and landscape scales. Wedefine resilience as the ability of the system to recover coreecological functions after disturbance (i.e. after fire or mechan-ical disturbance). The contribution of restoration to enhancingresilience can be measured in terms of the achievement of a setof additional criteria that target particular ecological processes,specifically climate change and fire. For logistical reasons thereis also a preference towards the selection of sites that are clus-tered. The utility of restoration on the Irvine Ranch NaturalLandmark can therefore be measured as:

Rs(y1) =

⎡⎢⎢⎣w1

Ns∑i

f1(ai, yi1) w2

Ns∑i

f2(SSCi1, yi1)

+ w3

⎛⎜⎜⎝∑

m

∑n

blmn(ym1(1 − ym1yn1))

∑m

amym1

⎞⎟⎟⎠+ G

⎤⎥⎥⎦ ,

where {w1, w2, w3, ...} are constants used to weight the rela-tive importance of each of the terms, blmn is the edge perimeter(the boundary length) between sites m and n and f1, f2 arefunctions converting the state of the habitat (highly degradedor partially degraded) into a reward. Here, w1 is the weightgiven to the amount of area restored, w2 is the weight givento the predicted number of species of special concern (SSC) inrestored areas, and w3 is the weight given for enhancing theconnectivity of restored sites in the system. The level of con-nectivity is measured in terms of the ratio of boundary to thearea for the restored portion of the landscape, with lower val-ues desired and indicative of a more connected system (Fig. 3).The weight w3 is used to adjust the relative importance ofachieving connectivity versus enhancing other components ofthe utility function. G is the total utility derived from restor-ing habitat in preferred locations (in riparian corridors, climatechange corridors and high fire risk zones):

G = w4

Ns∑i

yiISRC(i) + w5

Ns∑i

yiISCC(i) + w6

Ns∑i

yiISFZ(i), (3.1)

where IS(i) is an indicator function that equals one if site i is ina set S, and zero otherwise, and

SRC: is the set of all sites that are located within riparian cor-ridors;SCC: is the set of all sites located within climate change corri-dors;SFZ: is the set of all sites in high fire risk zones.

The general form of the utility function in Eq. (3.1) can thenbe modified from a linear representation to a sigmoidal rela-tionship to reflect the assumption that low levels of investmentwill deliver low returns, followed by high returns, which thentaper off towards the end of restoration program (Goldstein etal., 2008; Hartig and Drechsler, 2008). The final utility functionR̃s(y1) is a sigmoidal transformation of the previous versionRs(y1):

R̃ (y ) =[

1]

R (y ) (3.2)

246 M.F. McBride et al. / Ecologica

he parameter bL is used to trade-off the degree of connectednessn the system against the utility derived (Ball et al., 2009). Moreomplicated functions for L(y1) and A(y1) can be substituted whereonnectivity between some sites or specific spatial configurationsre considered particularly important (Calabrese and Fagan, 2004).

Restoration goals are usually stated in terms of restoring mea-urable attributes of a system (e.g. restoring the area of habitat forarticular species), while also restoring the overall resilience of the

andscape (see Box 3 ). In the simplest case there is a single restora-ion goal, but restoration goals are frequently specified in terms of

ultiple objectives – both ecological and non-ecological – such asnhancing recreational opportunities, and the delivery of ecosys-em services (Hajkowicz et al., 2000). Where multiple objectives areo be included, they should be investigated independently and thenresented to decision-makers as trade-off curves. This is becausebjectives can be contradictory, and maximising one objective gen-rally compromises the achievement of another (Lawrence et al.,997; Maron and Cockfield, 2008), so that the utility function Rs(y1)ill no longer be a simple summation of the utility derived for each

sset. However, where appropriate weights for contrasting objec-ives can be defined, outcomes can be presented as a weighted sumf disparate objectives.

.2. Uncertain outcomes

In many cases restoration fails, with certain sites or techniquesore prone to failure than others and environmental factors such

s fire and drought influencing success (Box 4 ; Lindenmayer et al.,002; Vallauri et al., 2002). The inclusion of uncertainty about theutcomes of restoring a particular site means that the transitionrom system state y0 to new system state y1 is no longer determin-stic and our objective is to maximize the expected utility over allhe possible outcomes.

We define the dynamics of a system being restored with a setf conditional transition probabilities, P(y1|y0, x1), the probabil-ty that the system is in state y1 given the initial state y0, and ourestoration actions x1. The function P: y0 × x1 → P(y1) defines forach state-action pair a probability distribution over y1, based onhe probability of restoration success at each site. Rewriting ourtility function (Eq. (2)) to maximize the expected utility gives:

y1

P(y1|y0, x1)Na∑j

wjfj(y1), (7)

hich is the sum of the benefits achieved from being in state y1eighted by the probability of being in each state, and is subject to

he same cost constraint in Eq. (3).For example,

P(yi1 = 1) = 1 if yi0 = 1,= pi1 if yi0 = 0, and xi1 = 1,

(yi1 = 0) if yi0 = 0 and xi1 = 0. (8)

or pi1, the probability that restoration action i succeeds at site j.A similar framework can be used for maximizing expected

tility over other uncertainties, such as knowledge uncertainty,nvironmental stochasticity, and future conditions under climatehange (Halpern et al., 2006; McBride et al., 2007). Where theres not enough information to define a set of transition proba-ilities, non-probabilistic approaches such as interval bounds or

nformation-gap analysis may represent alternative methods toncorporating the uncertainty about state transitions. Under annformation-gap approach, for example, uncertainty could be mod-led as a non-probabilistic set of bounds of uncertainty around therue future state y1 under each possible set of actions x1. The utility

s 1 1 + e−ˇ(xa−0.5) s 1

where xa = [a(y1)/A], is the proportion of the total area Arestored in system y1, and ˇ is a constant defined for thesigmoidal relationship.

M.F. McBride et al. / Ecological Modelling 221 (2010) 2243–2250 2247

Box 4: Uncertainty in restoration outcomes–incorporating stochastic eventsIn our application to the Irvine Ranch Natural Landmark, thelikelihood of success, and therefore the expected utility ofrestoration, varies depending on the restoration action, degra-dation state, and desired habitat type. It also varies with theslope of the site, its aspect, and the condition of neighbouringsites. We can include a probability pi that restoration action ata site will succeed, with a corresponding probability (1 − pi)that it will fail, and the benefits of restoration not realized.We assume that for sites where restoration fails that the siteswill revert back to their original condition. Sites undergoingrestoration on the Irvine Ranch Natural Landmark are also vul-nerable to fire and drought. Fires occur on average, once every12 years, and we can therefore assume a once-off probabilitypf of 0.08 that a site undergoing restoration will be affectedby fire and revert to its original condition. Drought conditionsoccur on average, once every 4 years, and therefore there is a

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applied to the multiple time-step formulation. In addition, withinthe dynamic framework we can also define a variety of systemdynamics for how the site conditions and constraints vary through

probability pd of 0.25 that a drought will occur, during whichthe likelihood of success pi of all sites will be reduced.

unction in Eq. (7) could then be used to select for the set of actionsaximizing the minimum possible performance under a specified

hreshold level of uncertainty.

.3. Multiple states and actions

In the basic formulation of the restoration prioritization prob-em we allow for the implementation of only one restorationpproach at a site, and consider a site to be either in a degradedr intact state. More commonly, the degradation state of a site isikely to vary along the spectrum from degraded to intact, witharticular restoration approaches being necessary to deliver theesired outcomes depending on the degradation state. Redefininghe basic problem in Eq. (2), we can generalize our action and statepace to include multiple possible degradation states and restora-ion actions. The choice of multiple actions means that the decisionariable x1 is no longer a {0, 1} vector and we define a discrete set oftates yi1 ∈ {1, 2, . . ., Yi1}where Yi1 is the set of possible degradationtates at site i, and xi1 ∈ {1, 2, . . ., Xi1}, where Xi1 is the set of possiblections at site i. An example of a multiple restoration state-actionroblem with three degradation states and a single action for eachtate is shown in Fig. 1.

More complicated scenarios allow for multiple possible restora-ion approaches for each state with likelihood transition matricesepresenting the probability of moving between each state give the

pproach implemented (Fig. 2). We can also allow for dependenciesetween sites, by specifying a likelihood that a site is in a particulartate given the restoration approach employed at the site and atdjoining sites. Such likelihoods would be dependent on the initial

ig. 1. An example state transition diagram for multiple degradation states andultiple actions, and corresponding transition probabilities for restoration at the

rvine Ranch Natural Landmark. Restoration is successful with probability pi , andre occurs with probability pf .

Fig. 2. An example state transition diagram for multiple degradation states andmultiple actions.

condition of the entire landscape y0, not just the starting state ofthe individual site. The states and actions yi1 and xi1 could also bedefined as continuous variables.

3. The dynamic formulation of the ecological restorationprioritization problem

The formulation described in the previous section details astatic, single time-step version of the restoration prioritizationproblem. Conditions relevant to the restoration planning problemare assumed to remain constant while sites are restored. Changesin site characteristics and potential feedbacks are ignored. In adynamic formulation, resources are no longer allocated statically,but sequentially through time, usually in accordance with an annualbudget or other constraint.

The dynamic version follows naturally as the multiple time-stepversion of Eq. (2). One possible objective is to maximize the expectedutility achieved at the end of some timeframe of interest T:

Na∑j

T∑t

wjfj(yt). (9)

Each year the costs of all the actions across all sites must be lessthan the overall constraint:

Ns∑i=1

cixit ≤ Bt, for every year t, (10)

where Bt = f (Bt−1, yt) is a function describing the budget (time,money or other resources) available each year based on investmentin the previous year and the current state of the landscape.

In the simplest case, sites can be either restored or intact,yit ∈ {0, 1} and restoration is the only possible action. The exten-sions considered above for the single time-step problem can also be

time. For example, potential restoration sites might have a prob-ability dit that they will be lost to development and irreversibly

Fig. 3. Example of the effect of incorporating the connectivity penalty in the IrvineRanch Natural Landmark prioritization. The shaded area represents intact habitatand the unshaded area represents degraded habitat. The dashed line indicates thecurrent penalty each cluster contributes to the overall connectivity penalty for thesystem. This equates to a penalty of 8 for cluster X and 14 for cluster Y. All else beingequal between clusters X and Y, restoration is preferred at cluster Y over cluster X,as restoration of sites within cluster Y will reduce the boundary to area ratio by agreater amount. Depending on the budget available, restoration will be applied toall sites within the cluster.

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248 M.F. McBride et al. / Ecologica

egraded. Alternatively, there might be varying availability of sitesor restoration or the suitability of sites for restoration may change,ue to, for example, invasion by a particularly noxious invasivepecies or identification of a species of concern.

Within the static formulation, a single set of sites can be pri-ritized for restoration, whereas within a dynamic formulation, achedule for restoration over a specified time period can be iden-ified. The sites identified for restoration in any one time-step canhen be updated based on the success or failure of restoration inrevious years. Interactions between sites as restoration proceedsan also be accounted for. For example, sites surrounded by intactabitat might be considered to have a greater likelihood of successhan those surrounded by degraded habitat (e.g. Field, 1998; Mullert al., 1998). To incorporate such interactions, the likelihood of suc-ess can be modeled as a function of the condition of neighboringites. For example, sites with an edge perimeter adjacent to intact orestored sites that is greater than a specified threshold (e.g. greaterhan 50%) might have a greater likelihood of success, which woulde calculated via:

i(t) ={

pit + 0.05 if Edi(x) ≥ 0.5pit if Edi(x) < 0.5

, (11)

here Edi(x) is the proportion of the edge perimeter of site i that isntact.

Within the dynamic formulation there is also the possibility todjust the priority of restoring each site based on the outcomes oftochastic events, including natural catastrophes such as fire androught, and changes in socioeconomic factors (Hobbs and Harris,001; McBride et al., 2007). Similarly, changes in site value fromrocesses such as passive restoration can be accounted for.

.1. Solution methods

The overall aim of a restoration prioritization analysis is to findsolution through manipulation of the control variable (whetherr not to restore a site) that has the highest possible score for thetility function subject to the constraints. Solution methods cane classified as ‘optimal’ or ‘heuristic’. Optimal methods are math-matically proven to result in the optimal solution, and identifyhe set of sites that uniquely maximizes (or minimizes) the utilityunction. The need to search through large portions of the solutionpace means that solution times increase exponentially with theroblem size, and identifying optimal solutions is precluded for

arge and complex combinatorial optimization problems (knowns computationally hard or ‘NP-hard’). Explicitly accounting forhe spatial relationship between sites, for example, will consider-bly increase problem complexity and solution time. A number ofeviews exist on the use of optimal methods in reserve selectionroblems (e.g. Rodrigues and Gaston, 2002; Williams et al., 2004;aight and Snyder, 2009).

Heuristic methods are algorithms used to approximate optimalethods, and provide no guarantee about the quality of solution

enerated. Global heuristic search algorithms, such as genetic algo-ithms and simulated annealing, are commonly used to generateear-optimal solutions for large, complex problems that are tooomputationally intensive to be solved exactly. Such algorithmsvaluate sites jointly as a set, and have generally been found toerform well relative to an optimal solution (Pressey et al., 1996;ossingham et al., 2000; Costello and Polasky, 2004; Westphal etl., 2007; Thomson et al., 2009). Alternatively sites can be ranked

ccording to various performance metrics (Petty and Thorne, 2005),ut such ranking approaches evaluate sites on an individual basisnd therefore do not account for spatial dependencies and feed-acks and cannot dynamically update. Search algorithms suchs simulated annealing generally out-perform scoring and rank-

elling 221 (2010) 2243–2250

ing methods, and deliver comparatively more efficient solutions(Pressey and Nicholls, 1989).

4. Discussion

We provide a generalized theory for ecological restorationplanning and prioritization. We detail the static and dynamic for-mulations and outline the key factors requiring consideration in arestoration context: spatial dependencies, the likelihood of restora-tion success, and multiple restoration actions and degradationstates. Using the Irvine Ranch Natural Landmark as a case study, weprovide examples of how the general framework can be adapted tospecific situations and reveal the ease of incorporation of real-worldcomplexities. While there are many examples and applications ofsystematic conservation planning principles and tools to protectedarea design, there are few examples of their application to restora-tion. Explicit formulation of the restoration prioritization problemallows for methods from mathematics and operations research tobe applied. Through the delivery of a generalized theory we envis-age further development of decision support tools (and associatedalgorithms) and their application to real-world restoration plan-ning problems. Important limitations in the existing theory willbe the availability of appropriate and accurate data, and computa-tional power for solving complex applications.

While we deliver a general theory for restoration planning andprioritization, this theory need not be considered separate frommore traditional conservation resource allocation problems, suchas the prioritization of sites for habitat protection. We are nowpresented with an opportunity to jointly prioritize the restora-tion and protection of habitat and relax the traditional assumptionin conservation planning that degraded sites are unavailable forinvestment (Costello and Polasky, 2004). Such integration wouldmove us closer to a comprehensive conservation investment frame-work, where habitat destruction and reconstruction through timeare acknowledged and accounted for. The costs and benefits ofrestoring degraded sites could be evaluated and the benefits ofacquiring new sites explicitly weighted against the managementand restoration that they may require.

Such a multiple-action dynamic framework allows us to takea much wider ranging view in planning for conservation invest-ments across a region. With increasing competition for land, newprotected areas are likely to contain habitat that presents the small-est foregone opportunity to other uses and are the most readilyavailable for acquisition. Protected area networks may be inad-vertently deficient in habitat types that have been subjected toextensive degradation. Within a comprehensive resource alloca-tion framework, habitat protection targets could be met throughthe allocation of resources to both habitat protection and habi-tat restoration, with the aim to achieve equitable protection of allhabitats. Similarly, the full suite of conservation actions, such asprescribed burning or invasive species control, could be consid-ered and the relative cost-effectiveness of all conservation actionsexplicitly evaluated. Cost-effectiveness is however only one aspectto be considered when prioritizing restoration activities with pri-ority areas, and activities are also determined by the willingness oflandholders to engage in restoration activities.

A focus of future research will be on the prioritization of restora-tion at multiple ecological scales. Preliminary attention has alsobeen given to accounting for the time lags associated with restora-tion action (Thomson et al., 2009). A simple way of accounting fortime gaps in the dynamic formulation is to account for the utility

of restoration only after a pre-specified period of time has elapsed.However, the time difference between restoring a site and when thefull extent of ecological benefits are realized also varies depend-ing on how difficult a site or system is to restore and what theecological target is, and there can be substantial uncertainty in its

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stimation. Methods and data to account for these components ofhe restoration prioritization problem require further developmentnd exploration. Better methods for handling uncertainty will alsoe important for making decisions in data-scarce or poorly under-tood systems.

Restoration is characterized by multiple objectives, a diversity ofpproaches and techniques, complex system dynamics, and uncer-ain outcomes. There is recognition in the restoration communityor the need for broad scale planning and a clear approach to set-ing restoration priorities that accounts for the characteristics of theystem restored (Hobbs and Norton, 1996; Hobbs and Kristjanson,003). The unified theory we present provides a method for restora-ion planning and prioritization that accounts for these factorsithin a properly formulated decision-theoretic framework. Ecol-

gists and restoration scientists can build upon this framework andncapsulate the increasing body of knowledge on the ecology andynamics of ecological systems undergoing restoration.

cknowledgements

We thank The Resources Legacy Foundation Fund, The Irvineompany Environmental Enhancement Fund, The Australianesearch Council, and The Australian Centre of Excellence for Risknalysis for providing funding.

eferences

rmsworth, P.R., Daily, G.C., Kareiva, P., Sanchirico, J.N., 2006. Land market feed-backs can undermine biodiversity conservation. Proc. Natl. Acad. Sci. U.S.A. 103,5403–5408.

all, I.R., Possingham, H.P., Watts, M.E., 2009. Marxan and relatives: software forspatial conservation prioritization. In: Moilanen, A., Wilson, K.A., Possingham,H.P. (Eds.), Spatial Conservation Prioritisation: Quantitative Methods and Com-putational Tools. Oxford University Press, Oxford, pp. 185–195.

eechie, T., Pess, G., Roni, P., 2008. Setting river restoration priorities: a review ofapproaches and a general protocol for identifying and prioritising actions. N.Am. J. Fish. Manage. 28, 891–905.

ryan, B., Crossman, N.D., 2008. Systematic regional planning for multiple objectivenatural resource management. J. Environ. Manage. 88, 1175–1189.

alabrese, J.M., Fagan, W.F., 2004. A comparison-shopper’s guide to connectivitymetrics. Front. Ecol. Environ. 2, 529–536.

hoi, Y.D., 2004. Theories for ecological restoration in changing environment:toward futuristic restoration. Ecol. Res. 19, 75–81.

hurch, R., ReVelle, C., 1974. The maximal coverage location problem. Pap. Reg. Sci.32, 101–118.

ipollini, K.A., Maruyama, A.L., Zimmerman, C.L., 2005. Planning for restoration: adecision analysis approach to prioritization. Restor. Ecol. 13, 460–470.

ostello, C., Polasky, S., 2004. Dynamic reserve site selection. Resource Energy Econ.26, 157–174.

rossman, N.D., Bryan, B.A., 2006. Systematic landscape restoration using integerprogramming. Biol. Conserv. 28, 369–383.

rossman, N.D., Bryan, B.A., Ostendorf, B., Collins, S., 2007. Systematic landscaperestoration in the rural–urban fringe: meeting conservation planning and policygoals. Biodiv. Conserv. 16, 3781–3802.

rossman, N.D., Bryan, B.A., 2009. Identifying cost-effective hotspots for restor-ing natural capital and enhancing landscape multifunctionality. Ecol. Econ. 68,654–668.

avis, F.W., Costello, C., Stoms, D., 2006. Efficient conservation in a utility max-imization framework. Ecol. Soc. 11, 33, http://www.ecologyandsociety.org/vol11/iss31/art33/.

obson, A.P., Bradshaw, A.D., Baker, J.M., 1997. Hopes for the future: restorationecology and conservation biology. Science 277, 515–522.

orrough, J., Vesk, P.A., Moll, J., 2008. Integrating ecological uncertainty and farm-scale economics when planning restoration. J. Appl. Ecol. 45, 288–295.

ahrig, L., 2002. Effect of habitat fragmentation on the extinction threshold: a syn-thesis. Ecol. Appl. 12, 346–353.

ield, C.D., 1998. Rehabilitation of mangrove ecosystems: an overview. Mar. Poll.Bull. 37, 383–392.

olke, C., Carpenter, S., Walker, B., Scheffer, M., Elmqvist, T., Gunderson, L., Holling,C.S., 2004. Regime shifts, resilience, and biodiversity in ecosystem management.Annu. Rev. Ecol. Evol. Syst. 35, 557–581.

oldstein, J.H., Pejchar, L., Daily, G.C., 2008. Using return-on-investment to guiderestoration: a case study from Hawaii. Conserv. Lett. 1, 236–243.

aight, R.G., Snyder, S.A., 2009. Integer programming methods for reserve selectionand design. In: Moilanen, A., Wilson, K.A., Possingham, H.P. (Eds.), Spatial Con-servation Prioritization: Quantitative Methods and Computational Tools. OxfordUniversity Press, Oxford.

elling 221 (2010) 2243–2250 2249

Hajkowicz, S.A., McDonald, G.T., Smith, P.N., 2000. An evaluation of multiple objec-tive decision support weighting techniques in natural resource management. J.Environ. Econ. Manage. 43, 505–518.

Halpern, B.S., Regan, H.M., Possingham, H.P., McCarthy, M.A., 2006. Accounting foruncertainty in marine reserve design. Ecol. Lett. 9, 2–11.

Hartig, F., Drechsler, M., 2008. The time horizon and its role in multiple speciesconservation planning. Biol. Conserv. 141, 2625–2631.

Hobbs, R., 2009. Looking for the silver lining: making the most of failure. Restor.Ecol. 17, 1–3.

Hobbs, R.J., Cramer, V.A., 2008. Restoration ecology: interventionist approaches forrestoring and maintaining ecosystem function in the face of rapid environmentalchange. Annu. Rev. Energ. Environ. 33, 39–61.

Hobbs, R.J., Harris, J.A., 2001. Restoration ecology: repairing the Earth’s ecosystemsin the new millennium. Restor. Ecol. 2, 239–246.

Hobbs, R.J., Kristjanson, L.J., 2003. Triage: how do we prioritize health care for land-scapes? Ecol. Manage. Restor. 4, 39–45.

Hobbs, R.J., Norton, D.A., 1996. Towards a conceptual framework for restorationecology. Restor. Ecol. 4, 93–110.

Huggett, A.J., 2005. The concept and utility of ecological thresholds in biodiversityconservation. Biol. Conserv. 124, 301–310.

Hyman, J.B., Leibowitz, S.G., 2000. A general framework for prioritizing landunits forecological restoration and protection. Environ. Manage. 25, 23–35.

Jordan, W.R., Peters, R.L., Allen, E.B., 1988. Ecological restoration as a strategy forconserving biological diversity. Environ. Manage. 12, 55–72.

Joseph, L.N., Maloney, R.F., Possingham, H.P., 2009. Evaluating costs, benefits andprobability of success for threatened species management: a project prioritisa-tion protocol. Conserv. Biol. 23, 328–338.

Lamb, D., Erskine, P.D., Parrotta, J.A., 2005. Restoration of degraded tropical forestlandscapes. Science 310, 1628–1632.

Lawrence, P.A., Stone, J.J., Heilman, P., Lane, L.J., 1997. Using measured data andexpert opinion in a multiple objective decision support system for semiaridrangelands. Trans. ASAE 40, 1589–1597.

Lindenmayer, D.B., Manning, A.D., Smith, P.L., Possingham, H.P., Fischer, J., Oliver, I.,McCarthy, M.A., 2002. The focal-species approach and landscape restoration: acritique. Conserv. Biol. 16, 338–345.

Margules, C.R., Pressey, R.L., 2000. Systematic conservation planning. Nature 405,243–253.

Maron, M., Cockfield, G., 2008. Managing trade-offs in landscape restoration andrevegetation projects. Ecol. Appl. 18, 2041–2049.

McAllister, L.S., Peniston, B.E., Leibowitz, S.G., Abbruzzese, B., Hyman, J.B., 2000.A synoptic assessment for prioritizing wetland restoration efforts to optimizeflood attenuation. Wetlands 20, 70–83.

McBride, M.F., Wilson, K.A., Bode, M., Possingham, H.P., 2007. Incorporating theeffects of socioeconomic uncertainty into priority setting for conservationinvestment. Conserv. Biol. 21, 1463–1474.

McIver, J., Starr, L., 2001. Restoration of degraded lands in the interior ColumbiaRiver basin: passive vs. active approaches. Forest Ecol. Manage. 153, 15–28.

Meir, E., Andelman, S., Possingham, H.P., 2004. Does conservation planning matterin a dynamic and uncertain world? Ecol. Lett. 7, 615–622.

Muller, S., Dutoit, T., Alard, D., Grevilliot, F., 1998. Restoration and rehabilitation ofspecies-rich grassland ecosystems in France: a review. Restor. Ecol. 6, 94–101.

Murdoch, W., Polasky, S., Wilson, K.A., Possingham, H.P., Kareiva, P., Shaw, R., 2007.Maximising return on investment in conservation. Biol. Conserv. 139, 375–388.

Noss, R., Nielsen, S., Vance-Borland, K., 2009. Prioritising ecosystems, species andsites for restoration. In: Moilanen, A., Wilson, K.A., Possingham, H. (Eds.), Spa-tial Conservation Prioritization: Quantitative Methods and Computational Tools.Oxford University Press.

O’Neill, M.P., Schmidt, J.C., Dobrowolski, J.P., Hawkins, C.P., Neale, C.M.U., 1997. Iden-tifying sites for riparian wetland restoration: application of a model to the UpperArkansas River basin. Restor. Ecol. 5, 85–102.

Petty, J.T., Thorne, D., 2005. An ecologically based approach to identifying restorationpriorities in an acid-impacted watershed. Restor. Ecol. 13, 348–357.

Possingham, H.P., 2001. Models, Problems and Algorithms: Perceptions About theirApplication to Conservation Biology. MODSIM.

Possingham, H.P., Andelman, S.J., Noon, B.R., Trombulak, S., Pulliam, H.R., 2001.Making smart conservation decisions. In: Orians, G., Soulé, M. (Eds.), ResearchPriorities for Conservation Biology. Island Press, California, pp. 225–244.

Possingham, H., Ball, I., Andelman, S., 2000. Mathematical methods for identifyingrepresentative reserve networks. In: Ferson, S., Burgman, M. (Eds.), Quantita-tive Methods for Conservation Biology. Springer-Verlag, New York, pp. 291–305.

Possingham, H., Day, J., Goldfinch, M., Salzborn, F., 1993. The mathematics of design-ing a network of protected areas for conservation. In: Sutton, D., Cousins, E.,Pearce, C. (Eds.), Decision Sciences: Tools for Today, 12th Australian OperationsResearch Conference. ASOR, University of Adelaide, Adelaide, pp. 536–545.

Possingham, H.P., Moilanen, A., Wilson, K.A., 2009. Accounting for habitat loss ratesin conservation planning. In: Moilanen, A., Wilson, K.A., Possingham, H.P. (Eds.),Spatial Conservation Prioritisation: Quantitative Methods and ComputationalTools. Oxford University Press, Oxford, pp. 135–144.

Pressey, R.L., Nicholls, A.O., 1989. Efficiency in conservation planning: scoring versus

iterative approaches. Biol. Conserv. 50, 199–218.

Pressey, R.L., Possingham, H.P., Margules, C.R., 1996. Optimality in reserve selectionalgorithms: when does it matter and how much? Biol. Conserv. 76, 259–267.

Pressey, R.L., Whish, G.L., Barrett, T.W., Watts, M.E., 2002. Effectiveness of protectedareas in north-eastern New South Wales: recent trends in six measures. Biol.Conserv. 106, 57–69.

2 l Mod

R

R

R

S

T

U

V

W

P., Shaw, M.R., Possingham, H.P., 2007. Conserving biodiversity efficiently: what

250 M.F. McBride et al. / Ecologica

ayfield, B., Anand, M., Laurence, S., 2005. Assessing simple versus complex restora-tion strategies for industrially disturbed forests. Restor. Ecol. 13, 639–650.

hodes, J.R., Callaghan, J.G., McAlpine, C.A., Jong, C.d., Bowen, M.E., Mitchell, D.L.,Lunney, D., Possingham, H.P., 2008. Regional variation in habitat—occupancythresholds: a warning for conservation planning. J. Appl. Ecol. 45, 549–557.

odrigues, A.S.L., Gaston, K.J., 2002. Optimisation in reserve selectionprocedures—why not? Biol. Conserv. 107, 123–129.

uding, K.N., Gross, K.L., Houseman, G.R., 2004. Alternative states and positive feed-backs in restoration ecology. Trends Ecol. Evol. 19, 46–53.

homson, J.R., Moilanen, A.J., Vesk, P.A., Bennett, A.F., MacNally, R., 2009. Where andwhen to revegetate: a quantitative model for scheduling landscape reconstruc-tion. Ecol. Appl. 19, 817–828.

nderwood, E.C., Shaw, M.R., Wilson, K.A., Kareiva, P., Klausmeyer, K.R., McBride,M.F., Bode, M., Morrison, S.A., Hoekstra, J.M., Possingham, H.P., 2008. Protectingbiodiversity when money matters: maximizing return on investment. PLoS One3, e1515.

allauri, D.R., Aronson, J., Barbero, M., 2002. An analysis of forest restoration 120years after reforestation on badlands in the Southwestern Alps. Restor. Ecol. 10,16–26.

estphal, M.I., Field, S.A., Possingham, H.P., 2007. Optimizing landscape configura-tion: a case study of woodland birds in the Mount Lofty Ranges, South Australia.Landsc. Urban Plan. 81, 56–66.

elling 221 (2010) 2243–2250

Westphal, M.I., Pickett, M., Getz, W.M., Possingham, H.P., 2003. The use of stochasticdynamic programming in optimal landscape reconstruction for metapopula-tions. Ecol. Appl. 13, 543–555.

Wilkins, S., Keith, D.A., Adam, P., 2003. Measuring success: evaluating the restora-tion of a Grassy Eucalypt Woodland on the Cumberland Plain, Sydney, Australia.Restor. Ecol. 11, 489–503.

Williams, J.C., Re Velle, C.S., Levin, S.A., 2004. Using mathematical optimization mod-els to design nature reserves. Front. Ecol. Environ. 2, 98–105.

Wilson, K.A., Carwardine, J., Possingham, H.P., 2009. Setting conservation priorities.Ann. NY Acad. Sci. 1162, 237–264.

Wilson, K.A., McBride, M.F., Bode, M., Possingham, H.P., 2006. Prioritising globalconservation efforts. Nature 440, 337–340.

Wilson, K.A., Underwood, E.C., Morrison, S.A., Klausmeyer, K.R., Murdoch, W.W., Rey-ers, B., Wardell-Johnson, G., Marquet, P.A., Rundel, P.W., McBride, M.F., Pressey,R.L., Bode, M., Hoekstra, J.M., Andelman, S.J., Looker, M., Rondinini, C., Kareiva,

to do, where and when. PLoS Biol. 5, e223.Young, T.P., 2000. Restoration ecology and conservation biology. Biol. Conserv. 92,

73–83.Zedler, J.B., Callaway, J.C., 1999. Tracking wetland restoration: do mitigation sites

follow desired trajectories. Restor. Ecol. 7, 69–73.