Mathematical problem definition for ecological restoration planning

Ecological Modelling 221 (2010) 2243–2250

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Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel

Mathematical problem definition for ecological restoration planning Marissa F. McBride a,∗ , Kerrie A. Wilson b , Jutta Burger c , Yi-Chin Fang c , Megan Lulow c , David Olson c,d , Mike O’Connell c , Hugh P. Possingham b a

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

a r t i c l e

i n f o

Article history: Received 12 November 2009 Received in revised form 22 April 2010 Accepted 23 April 2010 Available online 30 June 2010 Keywords: Ecological restoration Restoration priorities Decision theory Conservation planning Ecological thresholds Ecosystem management

a b s t r a c t Ecological restoration is an increasingly important tool for managing and improving highly degraded or altered environments. Faced with a large number of sites or ecosystems to restore, and a diverse array of restoration approaches, investments in ecological restoration must be prioritized. Nevertheless, there are relatively few examples of the systematic prioritization of restoration actions. The development of a general theory for ecological restoration that is sufficiently sophisticated and robust to account for the inherent complexity of restoration planning, and yet is flexible and adaptable to ensure applicability to a diverse array of restoration problems is needed. In this paper we draw on principles from systematic conservation planning to explicitly formulate the ‘restoration prioritization problem’. We develop a generalized theory for static and dynamic restoration planning problems, and illustrate how the basic problem 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 restoration techniques. We illustrate the applicability of our generic problem definition by applying it to a case study – restoration prioritization on The Irvine Ranch Natural Landmark in Southern California. Through this case study we illustrate how the definition of the general restoration problem can be extended to account for the specific constraints and considerations of an on-the-ground restoration problem. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Ecological restoration, the process of “assisting the recovery of an ecosystem that has been degraded, damaged, or destroyed” (e.g. Hobbs and Cramer, 2008), is an increasingly important tool for delivering a diverse range of conservation outcomes (Dobson et al., 1997; Young, 2000). Restoration is used extensively for: stabilising degraded soils, returning biodiversity and habitat values to a landscape, reducing poverty and sequestering carbon dioxide from the atmosphere (Jordan et al., 1988; Dobson et al., 1997; Lamb et al., 2005; Hobbs and Cramer, 2008). Regardless of its utility, the assisted restoration of habitat is characteristically time- and resource-intensive (Noss et al., 2009). While ecological restoration may be achieved with limited financial outlay (through, for example, the implementation of prescribed burning or the removal of introduced grazers), it can also entail costly reintroductions of rare species, replanting of diverse plant communities, and/or the creation of specific ecological niches (Hobbs and Norton, 1996). Some landscapes may have the capacity to passively restore or naturally

∗ Corresponding author. Tel.: +61 3 8344 3305; fax: +61 3 9348 1620. E-mail address: [email protected] (M.F. McBride). 0304-3800/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2010.04.012

regenerate, but others may fail to fully regain their natural biodiversity and ecological integrity without active intervention (McIver and Starr, 2001). The likelihood of a successful outcome may also be linked to the cost and relative intensity of the restoration activity selected (Dorrough et al., 2008). In such situations and when resources are limited, decisions about which areas to restore, when, and what restoration techniques to use must be made (Hyman and Leibowitz, 2000; Beechie et al., 2008). Previous examples of priority setting for restoration have tended to be based on scoring or ranking methods, or on expert opinion (O’Neill et al., 1997; McAllister et al., 2000; Cipollini et al., 2005; Petty and Thorne, 2005) although there are an increasing number of examples of the use of systematic techniques to prioritize areas for restoration to achieve a range of objectives (Crossman et al., 2007; Bryan and Crossman, 2008; Crossman and Bryan, 2009). Regardless of the solution method, the process of restoration planning needs to be underpinned by a welldefined problem (Possingham, 2001). This problem definition must be sufficiently sophisticated and robust to account for the inherent complexity of ecological restoration, but also flexible enough to ensure applicability to a diverse array of restoration problems. To some extent a framework for restoration planning already exists in the considerable body of research focused on the design of protected area networks, commonly referred to as systematic con-

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servation planning (Westphal et al., 2003; Crossman and Bryan, 2006; Westphal et al., 2007). Systematic conservation planning aims to identify representative, adequate, and efficient networks of protected areas (Margules and Pressey, 2000; Possingham et al., 2000; Wilson et al., 2009). The allocation of funds for protected area establishment can also be considered a special case of the more general conservation resource allocation problem, which entails the prioritization of funds across a diverse array of conservation actions in time and space (Wilson et al., 2007). Two general forms of the conservation resource allocation problem exist. In the maximalcoverage problem the objective is to maximize the amount of biodiversity protected (through the delivery of the required actions to abate the key threats) given a pre-specified resource constraint (Church and ReVelle, 1974). This form is appropriate where funds are limited, which is commonly the case. In the alternative version, the minimum-set problem, the objective is to meet pre-specified goals while minimizing the resources expended (Pressey et al., 2002). Within these two forms there are myriad possible formulations of the conservation resource allocation problem, depending on the assumptions and simplifications that are made (Murdoch et al., 2007; Underwood et al., 2008; Joseph et al., 2009). An important distinction is between once-off allocations that preclude the possibility of strategic changes through time, and allocations over multiple time steps that allow for different allocations at different times. The static formulation of the conservation resource allocation problem assumes a single allocation of resources. A more general class is where decisions are made sequentially through time (Possingham et al., 1993; Davis et al., 2006; Wilson et al., 2006). Land protection is typically a resource-intensive process that is feasible only in stages. Under a protracted process of land protection, ignoring system dynamics and stochastic events when planning ongoing investments may severely compromise the biodiversity outcomes: some areas may be degraded and lose their ecological values, while other more important areas may come available for protection (Meir et al., 2004). The same applies for restoration, which is typically carried out over an extended time due to funding, seasonal, or logistical limitations. Disturbance processes such as fire or drought, can dramatically alter the condition of sites in a landscape, including those in which restoration is already underway. The effects of potential future changes and feedbacks in the system are therefore important in the context of restoration prioritization (Hobbs and Norton, 1996; Folke et al., 2004). Other potentially important considerations are whether spatial dependencies are accounted for, whether desired outcomes are assumed to be realized with certainty, and whether multiple restoration techniques are considered. The outcomes of ecological restoration are also highly influenced by the willingness of landholders to engage and support restoration activities on their properties. Some landholders will be more willing to engage than others and may therefore represent cost-effective opportunities for restoration. The benefits of habitat protection and restoration are not constrained to the protected or restored site, as there are both onand off-site impacts associated with such investments (Armsworth et al., 2006). Spatial dependencies are of particular significance from the perspective of restoration, since the speed and likelihood with which the benefits are realized is determined by the interactions between neighboring sites (Lindenmayer et al., 2002; Suding et al., 2004). The allocation of resources, whether for protection or restoration, must therefore be evaluated in the context of the broader landscape. Whether planning for restoration at a site or landscape scale, a large proportion of restoration attempts will fail and the possibility of failure is likely to vary between sites and between restoration approaches and techniques (Zedler and Callaway, 1999; Wilkins et al., 2003; Choi, 2004). The possibility that desired outcomes will not be realized is seldom dealt

Box 1: Restoration planning on the Irvine Ranch Natural Landmark The Irvine Ranch Natural Landmark is a collection of permanently protected wildlands and parks located near the Santa Ana Mountains in Southern California. The Ranch contains some of the largest remaining stands of coastal sage scrub, oak-sycamore woodland, native grassland, and chaparral vegetation types in southern California. It represents approximately 44,000 acres of land, much of which has been degraded by agriculture, intensive grazing, woodland clearance, adjacent development, invasive species, and too-frequent fire. The Irvine Ranch Conservancy has been established as a non-profit organization to provide coordinated, ecologically responsible management services to the landowners. Existing levels of degradation and vulnerabilities to further disturbances from fire and invasive species mean that managers face the important problem of determining how best to target funds available for restoration to ensure the full recovery and the long term preservation of the significant ecological values of the Landmark. A total of 923 sites have been identified as potential candidates for restoration action and an annual budget of $700,000 is targeted each year over a 20-year time period for restoration.

with in conservation (McBride et al., 2007; Hobbs, 2009), although accounting for the possibility of failure will likely influence which sites are prioritized for restoration and when the restoration should occur. Conservation practitioners implement a variety of strategies to conserve biodiversity. There are similarly a diverse array of approaches and techniques used in restoration. Each approach has different costs associated with its implementation, delivers different ecological outcomes, and has varying likelihoods of success (McIver and Starr, 2001; Rayfield et al., 2005). Comprehensive and informative restoration plans will be delivered by explicitly accounting for the benefits (both on- and off-site), costs, and risks of different restoration techniques, as opposed to considering them a single combined action (Hobbs and Cramer, 2008). 1.1. Towards a general theory for restoration planning Despite the generality of the conservation resource allocation problem, application of the theory and decision support tools to conservation actions other than protected area design is in its infancy (Possingham et al., 2001; Wilson et al., 2007). The first aim of this paper is to define the generic restoration problem. We outline first the static formulation and illustrate how the basic problem formulation might be expanded to account for spatial dependencies, the possibility of restoration failure, and the choice between different restoration techniques. We then generalize this formulation to include temporal dynamics and to account for feedbacks and the likelihood that conditions will vary as restoration proceeds. The second aim of this paper is to demonstrate how the theory can be applied and adapted to incorporate the specific constraints and considerations of an on-the-ground restoration problem. In order to illustrate key concepts we refer to the restoration planning process on the Irvine Ranch Natural Landmark in Southern California (Box 1 ). We use this case study to test the utility of the theory we present. 2. The formulation of the one-step ecological restoration prioritization problem In the one-step formulation of the restoration prioritization problem, we assume a single once-off allocation of resources (Possingham et al., 2009). In this problem we ignore the poten-

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tial spatial dependencies and possible future events during project implementation. Our objective in the context of restoration might be to select sites to restore in order to maximize the total utility (benefit) gained for biodiversity from restoration, which is measured in terms of a set of biodiversity assets j = 1, 2, . . ., Na , which could include species, habitat types, or ecosystem processes. We might aim to achieve this goal for a fixed financial budget B1 . Initially we assume that each site, i = 1, 2, . . ., Ns , is either intact or degraded. We represent the state of the landscape at time t with the vector yt , where yit ∈ {0, 1} is the state of each individual site i. If a site i is intact, yit = 1, if it is degraded yit = 0. We represent the control variable, the set of sites that can be restored, with the vector xt , where xit ∈ {0, 1} and is equal to 1 if site i is selected for restoration, otherwise it is equal to 0. To begin with the landscape is in state y0 and following restoration, x1 , it is in state y1 . The state dynamics can be described by the equation: yi1 = 1 − (1 − yi0 )(1 − xi1 ) ∀i,

(1)

which means that site i is intact at the next time step if it was already intact, or it has been restored. Formally, our objective is to maximize Rs (y1 ), the utility from restoration across all sites in state y1 : Rs (y1 ) =

Na 

wj fj (y1 ),

(2)

j

where fj (y1 ) are asset-specific functions transforming representation into a reward relevant to asset j. The weighting wj reflects the degree to which biodiversity asset j contributes to the overall restoration utility. Where different assets are deemed more-or-less critical to the overall outcomes of a restoration project, the weighting wj can be used to specify the importance of each. This utility is to be maximized subject to the budget constraint: Ns 

ci (x1 ) ≤ B,

Box 2: An expanded set of budgetary constraints for restoration At the Irvine Ranch Natural Landmark, the cost of restoration at a site i is dependent on the desired habitat type, the restoration technique to be employed, and the area, slope, and accessibility of each site. Each site i = 1, . . ., Ns belongs to one of 101 sub-watershed clusters – hereafter restoration clusters – k = 1, . . ., Nh , and a start-up cost CSC of $10,000 was included for initiating restoration at sites in each restoration cluster to account for the costs associated with moving equipment and personnel between sites. In some cases the start-up cost may represent the cost of land purchase or easement establishment. We define Sk as the set of sites in restoration cluster k and ISk (i) as an indicator function that equals 1 if site i is in cluster k, and zero otherwise. The total cost for restoration each year is: Ns 

ci (xi1 , ˛i ) +

⎡i

area

Nk 



1−

N s 

⎤k



(1 − ISk (i))

CSC ≤ B,

˛i =

i

⎢ habitat type ⎥ ⎢ slope ⎥ ∀i, ⎣ ⎦ accessibility ...

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

ai xi1 ≤ AC ,

i=1

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

(3)

i=1

or more generally,

where ci (x1 ) is the cost of restoration at site i given other restoration actions, x1 , in the area and B is the total budget constraint. The constraint can therefore be specified as a function of a matrix of the set of actions taken, x1 , in addition to a set of site-specific variables that modify the cost of restoring a particular site, ˛i (Box 2). Additional constraints can also be employed to account for nonfinancial limitations, such as the amount of area able to be restored, the availability of seed, or a lack of sufficient field support (Box 2). Within this basic problem formulation we can also incorporate more specific details into the utility function. 2.1. The objective of ecological restoration and the utility function In Eq. (2), fj (y1 ) are biodiversity asset-specific functions, which transform the state of the landscape into a utility. This can be a simple linear function or a more complex function which includes dependencies between sites. In the simplest case, we might assume that the objective is to maximize the representation of each biodiversity asset, given a linear utility function and knowledge of the amount, rij , of each biodiversity asset that will be present at each site when it is restored. We therefore state the utility of restoration as: fj (y1 ) =

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Ns  i

fj (y1 ) =

Ns 

rij yi1 + (1 − yi1 )sij ,

where sij is the amount of the biodiversity asset j in each site i when 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 costs will be a function of the investment made. Depending on how representation of a particular biodiversity asset increases with increasing investment, the utility function can take a wide variety of different forms (for example, sigmoidal, concave, or threshold). A concave function can represent cases where there are diminishing returns with increasing investment (Wilson et al., 2007; Hobbs and Cramer, 2008). Similarly, threshold effects can be accounted for, such as when a certain level of investment is necessary before any benefits are realized (Fahrig, 2002; Huggett, 2005; Rhodes et al., 2008). There are many variations on the generalized version of the utility function presented in Eq. (2). We can, for example, account for spatial dependencies by incorporating a weighting term for the connectedness of restored sites. This would add a further term to the utility function Rs (y1 ): −bL f (L(y1 ), A(y1 )),

rij yi1 ,

(4)

(5)

i

(6)

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

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The parameter bL is used to trade-off the degree of connectedness in the system against the utility derived (Ball et al., 2009). More complicated functions for L(y1 ) and A(y1 ) can be substituted where connectivity between some sites or specific spatial configurations are considered particularly important (Calabrese and Fagan, 2004). Restoration goals are usually stated in terms of restoring measurable attributes of a system (e.g. restoring the area of habitat for particular species), while also restoring the overall resilience of the landscape (see Box 3 ). In the simplest case there is a single restoration goal, but restoration goals are frequently specified in terms of multiple objectives – both ecological and non-ecological – such as enhancing recreational opportunities, and the delivery of ecosystem services (Hajkowicz et al., 2000). Where multiple objectives are to be included, they should be investigated independently and then presented to decision-makers as trade-off curves. This is because objectives can be contradictory, and maximising one objective generally compromises the achievement of another (Lawrence et al., 1997; Maron and Cockfield, 2008), so that the utility function Rs (y1 ) will no longer be a simple summation of the utility derived for each asset. However, where appropriate weights for contrasting objectives can be defined, outcomes can be presented as a weighted sum of disparate objectives.

Box 3: The utility of restoration The objective of the restoration program on the Irvine Ranch Natural Landmark is to maximize both the area restored and the habitat restored for species of special concern, while enhancing the resilience at both site and landscape scales. We define resilience as the ability of the system to recover core ecological functions after disturbance (i.e. after fire or mechanical disturbance). The contribution of restoration to enhancing resilience can be measured in terms of the achievement of a set of additional criteria that target particular ecological processes, specifically climate change and fire. For logistical reasons there is also a preference towards the selection of sites that are clustered. The utility of restoration on the Irvine Ranch Natural Landmark can therefore be measured as:

⎡

Ns Ns  ⎢  ⎢w1 f1 (ai , yi1 ) w2 f2 (SSCi1 , yi1 ) ⎣

Rs (y1 ) =

i

i

⎛  ⎜ ⎝

+ w3 ⎜

m

blmn (ym1 (1 − ym1 yn1 ))

n



2.2. Uncertain outcomes

Na 

P(y1 |y0 , x1 )

y1

wj fj (y1 ),

(7)

G = w4

Ns 

P(yi1 = 0)

if yi0 = 0 and xi1 = 0.

yi ISRC (i) + w5

Ns 

i

which is the sum of the benefits achieved from being in state y1 weighted by the probability of being in each state, and is subject to the same cost constraint in Eq. (3). For example, = 1 if yi0 = 1, = pi1 if yi0 = 0, and xi1 = 1,

⎟ ⎥ ⎟ + G⎥ , ⎠ ⎦

where {w1 , w2 , w3 , ...} are constants used to weight the relative importance of each of the terms, blmn is the edge perimeter (the boundary length) between sites m and n and f1 , f2 are functions converting the state of the habitat (highly degraded or partially degraded) into a reward. Here, w1 is the weight given to the amount of area restored, w2 is the weight given to the predicted number of species of special concern (SSC) in restored areas, and w3 is the weight given for enhancing the connectivity of restored sites in the system. The level of connectivity is measured in terms of the ratio of boundary to the area for the restored portion of the landscape, with lower values desired and indicative of a more connected system (Fig. 3). The weight w3 is used to adjust the relative importance of achieving connectivity versus enhancing other components of the utility function. G is the total utility derived from restoring habitat in preferred locations (in riparian corridors, climate change corridors and high fire risk zones):

j

P(yi1 = 1)

⎤

m

In many cases restoration fails, with certain sites or techniques more prone to failure than others and environmental factors such as fire and drought influencing success (Box 4 ; Lindenmayer et al., 2002; Vallauri et al., 2002). The inclusion of uncertainty about the outcomes of restoring a particular site means that the transition from system state y0 to new system state y1 is no longer deterministic and our objective is to maximize the expected utility over all the possible outcomes. We define the dynamics of a system being restored with a set of conditional transition probabilities, P(y1 |y0 , x1 ), the probability that the system is in state y1 given the initial state y0 , and our restoration actions x1 . The function P: y0 × x1 → P(y1 ) defines for each state-action pair a probability distribution over y1 , based on the probability of restoration success at each site. Rewriting our utility function (Eq. (2)) to maximize the expected utility gives:



am ym1

⎞

(8)

For pi1 , the probability that restoration action i succeeds at site j. A similar framework can be used for maximizing expected utility over other uncertainties, such as knowledge uncertainty, environmental stochasticity, and future conditions under climate change (Halpern et al., 2006; McBride et al., 2007). Where there is not enough information to define a set of transition probabilities, non-probabilistic approaches such as interval bounds or information-gap analysis may represent alternative methods to incorporating the uncertainty about state transitions. Under an information-gap approach, for example, uncertainty could be modeled as a non-probabilistic set of bounds of uncertainty around the true future state y1 under each possible set of actions x1 . The utility

yi ISCC (i) + w6

i

Ns 

yi ISFZ (i),

(3.1)

i

where IS (i) is an indicator function that equals one if site i is in a set S, and zero otherwise, and SRC : is the set of all sites that are located within riparian corridors; SCC : is the set of all sites located within climate change corridors; SFZ : is the set of all sites in high fire risk zones. The general form of the utility function in Eq. (3.1) can then be modified from a linear representation to a sigmoidal relationship to reflect the assumption that low levels of investment will deliver low returns, followed by high returns, which then taper off towards the end of restoration program (Goldstein et al., 2008; Hartig and Drechsler, 2008). The final utility function ˜ s (y1 ) is a sigmoidal transformation of the previous version R Rs (y1 ): ˜ s (y1 ) = R



1 1 + e−ˇ(xa −0.5)



Rs (y1 )

(3.2)

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

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Box 4: Uncertainty in restoration outcomes– incorporating stochastic events In our application to the Irvine Ranch Natural Landmark, the likelihood of success, and therefore the expected utility of restoration, varies depending on the restoration action, degradation state, and desired habitat type. It also varies with the slope of the site, its aspect, and the condition of neighbouring sites. We can include a probability pi that restoration action at a 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 sites will revert back to their original condition. Sites undergoing restoration on the Irvine Ranch Natural Landmark are also vulnerable to fire and drought. Fires occur on average, once every 12 years, and we can therefore assume a once-off probability pf of 0.08 that a site undergoing restoration will be affected by fire and revert to its original condition. Drought conditions occur on average, once every 4 years, and therefore there is a probability pd of 0.25 that a drought will occur, during which the likelihood of success pi of all sites will be reduced.

function in Eq. (7) could then be used to select for the set of actions maximizing the minimum possible performance under a specified threshold level of uncertainty. 2.3. Multiple states and actions In the basic formulation of the restoration prioritization problem we allow for the implementation of only one restoration approach at a site, and consider a site to be either in a degraded or intact state. More commonly, the degradation state of a site is likely to vary along the spectrum from degraded to intact, with particular restoration approaches being necessary to deliver the desired outcomes depending on the degradation state. Redefining the basic problem in Eq. (2), we can generalize our action and state space to include multiple possible degradation states and restoration actions. The choice of multiple actions means that the decision variable x1 is no longer a {0, 1} vector and we define a discrete set of states yi1 ∈ {1, 2, . . ., Yi1 } where Yi1 is the set of possible degradation states at site i, and xi1 ∈ {1, 2, . . ., Xi1 }, where Xi1 is the set of possible actions at site i. An example of a multiple restoration state-action problem with three degradation states and a single action for each state is shown in Fig. 1. More complicated scenarios allow for multiple possible restoration approaches for each state with likelihood transition matrices representing the probability of moving between each state give the approach implemented (Fig. 2). We can also allow for dependencies between sites, by specifying a likelihood that a site is in a particular state given the restoration approach employed at the site and at adjoining sites. Such likelihoods would be dependent on the initial

Fig. 1. An example state transition diagram for multiple degradation states and multiple actions, and corresponding transition probabilities for restoration at the Irvine Ranch Natural Landmark. Restoration is successful with probability pi , and fire occurs with probability pf .

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

condition of the entire landscape y0 , not just the starting state of the individual site. The states and actions yi1 and xi1 could also be defined as continuous variables. 3. The dynamic formulation of the ecological restoration prioritization problem The formulation described in the previous section details a static, single time-step version of the restoration prioritization problem. Conditions relevant to the restoration planning problem are assumed to remain constant while sites are restored. Changes in site characteristics and potential feedbacks are ignored. In a dynamic formulation, resources are no longer allocated statically, but sequentially through time, usually in accordance with an annual budget or other constraint. The dynamic version follows naturally as the multiple time-step version of Eq. (2). One possible objective is to maximize the expected utility achieved at the end of some timeframe of interest T: Na T   j

wj fj (yt ).

(9)

t

Each year the costs of all the actions across all sites must be less than the overall constraint: Ns 

ci xit ≤ Bt ,

for every year t,

(10)

i=1

where Bt = f (Bt−1 , yt ) is a function describing the budget (time, money or other resources) available each year based on investment in 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 extensions considered above for the single time-step problem can also be applied to the multiple time-step formulation. In addition, within the dynamic framework we can also define a variety of system dynamics for how the site conditions and constraints vary through time. For example, potential restoration sites might have a probability dit that they will be lost to development and irreversibly

Fig. 3. Example of the effect of incorporating the connectivity penalty in the Irvine Ranch Natural Landmark prioritization. The shaded area represents intact habitat and the unshaded area represents degraded habitat. The dashed line indicates the current penalty each cluster contributes to the overall connectivity penalty for the system. This equates to a penalty of 8 for cluster X and 14 for cluster Y. All else being equal 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 a greater amount. Depending on the budget available, restoration will be applied to all sites within the cluster.

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degraded. Alternatively, there might be varying availability of sites for restoration or the suitability of sites for restoration may change, due to, for example, invasion by a particularly noxious invasive species or identification of a species of concern. Within the static formulation, a single set of sites can be prioritized for restoration, whereas within a dynamic formulation, a schedule for restoration over a specified time period can be identified. The sites identified for restoration in any one time-step can then be updated based on the success or failure of restoration in previous years. Interactions between sites as restoration proceeds can also be accounted for. For example, sites surrounded by intact habitat might be considered to have a greater likelihood of success than those surrounded by degraded habitat (e.g. Field, 1998; Muller et al., 1998). To incorporate such interactions, the likelihood of success can be modeled as a function of the condition of neighboring sites. For example, sites with an edge perimeter adjacent to intact or restored sites that is greater than a specified threshold (e.g. greater than 50%) might have a greater likelihood of success, which would be calculated via:



pi (t) =

pit + 0.05 pit

if Edi (x) ≥ 0.5 , if Edi (x) < 0.5

(11)

where Edi (x) is the proportion of the edge perimeter of site i that is intact. Within the dynamic formulation there is also the possibility to adjust the priority of restoring each site based on the outcomes of stochastic events, including natural catastrophes such as fire and drought, and changes in socioeconomic factors (Hobbs and Harris, 2001; McBride et al., 2007). Similarly, changes in site value from processes such as passive restoration can be accounted for. 3.1. Solution methods The overall aim of a restoration prioritization analysis is to find a solution through manipulation of the control variable (whether or not to restore a site) that has the highest possible score for the utility function subject to the constraints. Solution methods can be classified as ‘optimal’ or ‘heuristic’. Optimal methods are mathematically proven to result in the optimal solution, and identify the set of sites that uniquely maximizes (or minimizes) the utility function. The need to search through large portions of the solution space means that solution times increase exponentially with the problem size, and identifying optimal solutions is precluded for large and complex combinatorial optimization problems (known as computationally hard or ‘NP-hard’). Explicitly accounting for the spatial relationship between sites, for example, will considerably increase problem complexity and solution time. A number of reviews exist on the use of optimal methods in reserve selection problems (e.g. Rodrigues and Gaston, 2002; Williams et al., 2004; Haight and Snyder, 2009). Heuristic methods are algorithms used to approximate optimal methods, and provide no guarantee about the quality of solution generated. Global heuristic search algorithms, such as genetic algorithms and simulated annealing, are commonly used to generate near-optimal solutions for large, complex problems that are too computationally intensive to be solved exactly. Such algorithms evaluate sites jointly as a set, and have generally been found to perform well relative to an optimal solution (Pressey et al., 1996; Possingham et al., 2000; Costello and Polasky, 2004; Westphal et al., 2007; Thomson et al., 2009). Alternatively sites can be ranked according to various performance metrics (Petty and Thorne, 2005), but such ranking approaches evaluate sites on an individual basis and therefore do not account for spatial dependencies and feedbacks and cannot dynamically update. Search algorithms such as simulated annealing generally out-perform scoring and rank-

ing methods, and deliver comparatively more efficient solutions (Pressey and Nicholls, 1989). 4. Discussion We provide a generalized theory for ecological restoration planning and prioritization. We detail the static and dynamic formulations and outline the key factors requiring consideration in a restoration context: spatial dependencies, the likelihood of restoration success, and multiple restoration actions and degradation states. Using the Irvine Ranch Natural Landmark as a case study, we provide examples of how the general framework can be adapted to specific situations and reveal the ease of incorporation of real-world complexities. While there are many examples and applications of systematic conservation planning principles and tools to protected area design, there are few examples of their application to restoration. Explicit formulation of the restoration prioritization problem allows for methods from mathematics and operations research to be applied. Through the delivery of a generalized theory we envisage further development of decision support tools (and associated algorithms) and their application to real-world restoration planning problems. Important limitations in the existing theory will be the availability of appropriate and accurate data, and computational power for solving complex applications. While we deliver a general theory for restoration planning and prioritization, this theory need not be considered separate from more traditional conservation resource allocation problems, such as the prioritization of sites for habitat protection. We are now presented with an opportunity to jointly prioritize the restoration and protection of habitat and relax the traditional assumption in conservation planning that degraded sites are unavailable for investment (Costello and Polasky, 2004). Such integration would move us closer to a comprehensive conservation investment framework, where habitat destruction and reconstruction through time are acknowledged and accounted for. The costs and benefits of restoring degraded sites could be evaluated and the benefits of acquiring new sites explicitly weighted against the management and restoration that they may require. Such a multiple-action dynamic framework allows us to take a much wider ranging view in planning for conservation investments across a region. With increasing competition for land, new protected areas are likely to contain habitat that presents the smallest foregone opportunity to other uses and are the most readily available for acquisition. Protected area networks may be inadvertently deficient in habitat types that have been subjected to extensive degradation. Within a comprehensive resource allocation framework, habitat protection targets could be met through the allocation of resources to both habitat protection and habitat restoration, with the aim to achieve equitable protection of all habitats. Similarly, the full suite of conservation actions, such as prescribed burning or invasive species control, could be considered and the relative cost-effectiveness of all conservation actions explicitly evaluated. Cost-effectiveness is however only one aspect to be considered when prioritizing restoration activities with priority areas, and activities are also determined by the willingness of landholders to engage in restoration activities. A focus of future research will be on the prioritization of restoration at multiple ecological scales. Preliminary attention has also been given to accounting for the time lags associated with restoration action (Thomson et al., 2009). A simple way of accounting for time 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 the full extent of ecological benefits are realized also varies depending on how difficult a site or system is to restore and what the ecological target is, and there can be substantial uncertainty in its

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