Waiting can be an optimal conservation strategy, even in a crisis discipline

Significance

Every year, more species are driven to extinction by the combined pressures of habitat destruction, invasive species, and climate change. These ongoing losses have created a “crisis culture” in conservation, where project funds are spent as soon as they are received. We challenge this orthodoxy and demonstrate how strategic delays can improve efficiency. Waiting can allow agencies to leverage additional benefits from their funds, through investment, capacity building, or monitoring and research. With the right amount of delay, limited conservation resources can protect more species. Surprisingly, they can even do so in less time. Our results suggest that, in addition to their current focus on where to target resources, conservation managers should carefully choose when to spend these funds.

Abstract

Biodiversity conservation projects confront immediate and escalating threats with limited funding. Conservation theory suggests that the best response to the species extinction crisis is to spend money as soon as it becomes available, and this is often an explicit constraint placed on funding. We use a general dynamic model of a conservation landscape to show that this decision to “front-load” project spending can be suboptimal if a delay allows managers to use resources more strategically. Our model demonstrates the existence of temporal efficiencies in conservation management, which parallel the spatial efficiencies identified by systematic conservation planning. The optimal timing of decisions balances the rate of biodiversity decline (e.g., the relaxation of extinction debts, or the progress of climate change) against the rate at which spending appreciates in value (e.g., through interest, learning, or capacity building). We contrast the benefits of acting and waiting in two ecosystems where restoration can mitigate forest bird extinction debts: South Australia’s Mount Lofty Ranges and Paraguay’s Atlantic Forest. In both cases, conservation outcomes cannot be maximized by front-loading spending, and the optimal solution recommends substantial delays before managers undertake conservation actions. Surprisingly, these delays allow superior conservation benefits to be achieved, in less time than front-loading. Our analyses provide an intuitive and mechanistic rationale for strategic delay, which contrasts with the orthodoxy of front-loaded spending for conservation actions. Our results illustrate the conservation efficiencies that could be achieved if decision makers choose when to spend their limited resources, as opposed to just where to spend them.

Irreversible biodiversity loss makes conservation a race against time (1–3). In response to this state of “crisis” (4), conservation projects usually aim to maximize impact by “front-loading,” expending their resources on conservation activities as rapidly as possible (5–7). Many studies support this orthodoxy, with specific examples that recommend action as soon as resources become available (8–10). Delays incur opportunity costs, both because biodiversity losses are often irreversible [e.g., species, phylogenetic diversity, or pristine habitat (11, 12)] and because fewer options remain available to managers as time passes [e.g., properties become unavailable (13) or political will disappears (14)]. Although the front-loading of project spending is an understandable response to imminent threats, it may not be the most efficient decision. Finance theory stresses an optimal balance between immediate consumption and capital investment (15), even in the face of accelerating threats. Operations research similarly recognizes that the short-term costs incurred by delay can be offset by superior long-term outcomes (16–18).

Delayed actions improve long-term outcomes if deferment is used to build future capacity—the ability to pursue a desired outcome. Optimal growth theory offers a close ecological analogue to this phenomenon: plants aim to maximize their lifetime reproductive success, but they often achieve this by completely deferring any investment in reproductive activities [e.g., flowers or seeds (19)]. Fitness is instead maximized by early investments that increase an organism’s capacity to act (e.g., its photosynthetic system) or that allow it to store resources (e.g., root stock) while waiting for a period of lower competition (20). Delays can confer similar benefits on conservation actions via comparable mechanisms, including earning interest, learning, or building capital. Economic interest offers the most conceptually straightforward example: an increase in principal can be obtained by temporarily lending funds to other sectors of the economy. Learning through monitoring (21–23) and research (24, 25) delays actions but offers an opportunity to improve the efficiency of those actions when they are eventually taken. Finally, investments in technology, human capital, or infrastructure can also increase the efficiency of future spending (26).

Regardless of the mechanism, delays are only optimal when their benefits outweigh the concurrent increase in threats. In conservation, the optimal timing is determined by a competition between two rates: the rate of decline of conservation assets and the rate at which capacity increases. These rates operate in different directions and also have different characteristics. The decline of conservation assets can follow complex trajectories: the simultaneous accumulation and relaxation of extinction debts (27); the slow–fast–slow dynamics of regional habitat loss (28); or critical threshold dynamics found in nonlinear ecological and climate systems (29, 30). Conservation capacity in the simplest sense could be money in a bank, which can exponentially increase over time due to compound interest but is subject to economic shocks and reversals (14, 31). Learning generally improves conservation outcomes, but at a diminishing marginal rate (21). Conservation dynamics are therefore temporally heterogeneous, and this creates periods of time during which actions will be more effective. Systematic conservation planning improves outcomes by identifying the most efficient locations to act in space. In a direct analogy, a systematic approach is needed to identify the most efficient points in time to undertake actions.

In this paper, we formally model the general trade-off between the benefits of waiting and the costs of delay. We consider this trade-off in the context of an extensively studied conservation management problem, species extinction driven by habitat loss (32). We parameterize the model for two case studies—the Mount Lofty Ranges in Australia and the Atlantic Forests in Paraguay—where widespread clearing has created an extinction debt that can only be addressed through habitat restoration. In these examples, managers can defer spending to accrue financial interest, but species will continue to be lost during this delay as the extinction debt relaxes (33, 34).

Dynamic Habitat Model

We model the impacts of conservation projects in a landscape using a system of deterministic differential equations that link conservation actions through time with a biodiversity conservation goal. The model captures the essential dynamics of restoration and extinction debts in degraded landscapes, but does not include factors (e.g., stochasticity, spatial heterogeneity) which will have limited influence on the optimal timing but which preclude closed-form solutions. Its structural simplicity therefore emphasizes the contrast between waiting and spending.

At any given time, habitat is either not supporting species because it is cleared $C\left(t\right)$, or is intact $R\left(t\right)=1-C\left(t\right)$ and can support species [$0\le C\left(t\right),R\left(t\right)\le 1$]. Managers are given a single endowment of funds, $B\left(t=0\right)=B_0$, which they can spend at any time to convert cleared habitat into intact habitat by restoration, at a cost $c_R$ per unit area. Unspent funds are invested and accrue interest at proportional rate r. In the analyses that follow, we use inflation-adjusted interest rates, which allow us to use a constant value for $c_R$. Managers define a proportional spending schedule $0\le u\left(t\right)\le 1$, with the objective of minimizing the number of extinctions. These dynamics are described mathematically using two habitat equations and a budget equation:

$$\frac{dR}{dt}=\frac{u\left(t\right)B\left(t\right)}{c_R},$$ [1a]

$$\frac{dC}{dt}=-\frac{u\left(t\right)B\left(t\right)}{c_R},$$ [1b]

$$\frac{dB}{dt}=\left(r-u\left(t\right)\right)B\left(t\right).$$ [1c]

We assume that the species–area relationship (SAR) allows equilibrium species richness to be predicted based on the area of intact habitat:

$$S^{\ast }=\alpha R{\left(t\right)}^z,$$ [2]

where $\alpha$ represents regional species richness and $z$ denotes the species accumulation rate in the region. Historic species richness is represented by $S^{\ast }=\alpha$ when R = 1 and all of the habitat is intact. If the amount of current intact habitat is insufficient to support the current number of extant species [i.e., if $S\left(t\right)>S^{\ast }$], an extinction debt exists and species will be lost as this debt relaxes at proportional rate $\theta$:

$$\frac{dS\left(t\right)}{dt}=-\theta \left[S\left(t\right)-S^{\ast }\right]=-\theta \left[S\left(t\right)-\alpha R{\left(t\right)}^z\right].$$ [3]

The managers’ objective is to identify the spending schedule that maximizes the number of extant species at some future time $T$:

$$\underset{0\le u\left(t\right)\le 1}{\max }S\left(T\right).$$ [4]

Analytic Solution for Optimal Single Disbursement.

The simplest formulation of our problem begins with no intact habitat [$R\left(0\right)=0$], but with a large extinction debt (i.e., all species remain extant: $S_0=\alpha$). A manager delays spending for $t_S$ years, and then spends all of the accumulated funds in a “single disbursement,” allowing the restoration of area $R\left(t_s\right)=B\left(t_s\right)/c_R$. The optimal disbursement time maximizes the number of protected species, defined as follows:

$$S_p=\underset{0\le t_s\le T}{\max }\left[\min \left\{\alpha R{\left(t_s\right)}^z;S\left(t_s\right)\right\}\right],$$ [5]

The first term in the set applies when the number of species extant at $t_s$ equals or exceeds the number that can be supported by the restored land. The second term applies when the extinction debt has relaxed to below the level that can be supported by the restored land (i.e., the managers waited too long). Since the increase in the first term and the decrease in the second term are both monotonic, the optimal time to disburse the funds occurs when the functions intersect (Fig. 1). For r > 0, this occurs at time:

$$t_s^{\ast }=z\frac{\ln \left(\frac{c_R}{B_0}\right)}{rz+\theta }.$$ [6]

This solution shows that, for a single disbursement, some length of delay is always optimal, given two common conservation conditions. First, the initial budget must be insufficient to immediately eliminate the extinction debt. This condition is ubiquitous in conservation (35, 36). Second, inflation-adjusted real interest rates must be positive. This condition is almost always true, even for conservative instruments such as Treasury bonds (Fig. S1). The optimal length of the delay is determined by the initial restoration budget, and the rates of change within the system (SI Model of the System Dynamics for details). The numerator shows that expensive restoration costs (relative to the budget: $c_R/B_0$) will encourage longer delays, while the denominator indicates that faster extinction debt relaxation ($\theta$) and higher interest rates (r) incentivize front-loading. Higher interest rates make investment more attractive because capacity to act increases more rapidly. Higher interest rates therefore result in shorter delays because it takes less time to amass the resources required to fund all necessary restoration activities (Fig. 1).

Fig. 1.

The optimal delay is determined by the intersection of two curves, each defined by a critical system rate. The red line shows the number of species remaining extant if funds are not spent. The extinction debt relaxation rate $\theta$ defines the downward slope of this curve. The blue lines show the number of species that could be protected if accumulated funding were disbursed at a given point in time. The upward slope of these two curves reflects an increase in resources via the financial interest rate r, shown for two different values. To generate this figure, $z=0.3$, $\alpha =1$, $B_0=1$, $c_R=20$, and $0\le R\left(t\right)\le 1$.

Fig. S1.

Inflation-adjusted interest rate for US Treasury securities. Our analytic solution suggests that, when interest rates are positive, some delay is optimal. This figure shows that this constraint is reasonable because inflation-adjusted interest rates offered on 10-y US Treasury inflation-indexed securities are almost always positive. These highly conservative investments represent the lower end of possible financial returns. Over the past 15 y—during which time interest rates have been historically low—these securities have had positive returns for 90% of the time. Data were replotted from Federal Reserve Economic Data, accessed at https://fred.stlouisfed.org on August 5, 2017.

Optimal Disbursement Schedule.

The single disbursement solution reveals the essential factors that govern the duration of the optimal delay, but conservation resources are often obtained and expended over a longer period. In the following two case studies, we identify a disbursement schedule that maximizes bird species richness at the end of the planning horizon. We use each case study to illustrate a different project funding scenario: the Mount Lofty Ranges case study receives a recurrent annual budget from a central funding organization; the Atlantic Forests case study is structured as a “sinking fund,” where an initial lump sum is raised to fund a single initiative, without an expectation of ongoing funds (37). In both cases, the managers can choose to spend a proportion of their resources every year and accrue interest on the remainder. Operationally, we used numerical optimization (Matlab R2016a; MathWorks) to identify the disbursement schedule $u\left(t\right)$ that maximizes Eq. 4 under the constraints of Eqs. 1–3. See SI Model of the System Dynamics for further details.

Case Study 1: Mount Lofty Ranges (Australia).

The Mount Lofty Ranges woodlands ecoregion (MLR) in Southern Australia acts as a habitat island for woodland bird species because it experiences substantially higher rainfall (400–1,100 mm/y) than surrounding ecosystems. Historically, 115 species of birds were found in this ≈500,000-ha region. Extensive land clearing has reduced the forest cover by 87% (38) (Table S1), and the SAR predicts that this amount of habitat loss will eventually cause 35 bird extinctions (39). At present, only eight extinctions have occurred, with a further eight species considered near extinction (40) (Table S2); this indicates the presence of a sizeable extinction debt (39). Biogeographic analyses of southern Australian forest bird communities estimate the relevant SAR parameters to be z = 0.17 and $\alpha$ = 13.6 (39), and historical survey data and forest cover records indicate an extinction debt relaxation rate of $\theta =2.5\times {10}^{-3}$ y⁻¹. Restoration costs $\left(c_R\right)$ are estimated at $1,132 (2016 AUD per hectare). We assume that decision makers can secure a constant 5.8% interest rate, based on Australia’s inflation-adjusted lending interest rate over the past 25 years (41). Revegetation has been shown to counteract further bird decline in the MLR (42, 43), and at least $500,000 (2016 AUD) per year is spent protecting and restoring priority areas within the ecoregion. See SI Model of the System Dynamics and Fig. S2 for an explanation of these parameters.

Table S1.

Forest cover in Mount Lofty Ranges

Date	Forest remaining, ha	Ref(s).
1880	500,000	(38)
1945	240,000	(40)
1980	90,000	(40)
2015	65,000	(38)

Table S2.

Species lost in Mount Lofty Ranges [from Possingham and Field (40)]

Species	Scientific name	Year extinct in region
Glossy black cockatoo	Calyphorhynchus lathami	1980
Barking owl	Ninox connivens	?
Eastern ground parrot (swamp parrot)	Pezoporus wallicus	1900
King quail	Coturnix chinensis	1900
Rufous fieldwren	Calamanthus (Calamanthus) campestris	1954
Regent honeyeater	Xanthomyza phrygia	1949
Swift parrot	Lathmus discolor	Irregular visitor?
Azure kingfisher	Ceyx azureus	1950s
Square-tailed kite	Lophoictinia isura	Near extinction
Bush stone-curlew	Burhinus (Burhinus) grallarius	Near extinction
Little lorikeet	Glossopsitta pusilla	Near extinction
White-throated gerygone	Gerygone olivacea	Near extinction
Spotted quail-thrush	Cinclosoma (Cinclosoma) punctatum anachoreta	Near extinction
Olive-backed oriole	Oriolus (Mimeta) sagittatus	Near extinction
Brown quail	Coturnix (Synoicus) ypsilophora	Near extinction
Flame robin	Petroica (Littlera) phoenicea	Near extinction

Fig. S2.

Estimate of Mount Lofty Ranges extinction debt dynamics. Estimate of the extinction debt for bird species in the Mount Lofty Ranges, South Australia. Red markers indicate the observed number of extant species (with linear fit), and blue markers indicate the number of species expected at equilibrium according to available habitat and the fitted species area relationship (with linear fit). The distance between the two lines measures the extinction debt.

Fig. 2 shows that, if this level of annual funding is continually and immediately invested into direct restoration of habitats, the existing extinction debt will be negated in 250 years. Such front-loading of the disbursement of funds will conserve 102 species by restoring 116,400 ha of habitat. In contrast, if managers are allowed to delay spending, this extinction debt could be negated within 78 years, with 147,300 ha of habitat restored. The optimal solution invests the annual budget for more than 40 years before undertaking direct conservation actions, with the accumulated funds gradually liquidated over the subsequent three decades. During the initial phase of delay, more extinctions occur than in the front-loaded schedule (Fig. 2). However, the optimal spending schedule eventually reduces the number of extinctions by 51% over business as usual and allows the extinction debt to be eliminated within 78 years. A delay therefore allows managers to achieve their aims in less than half the time.

Fig. 2.

Optimal schedule for the Mount Lofty Ranges, Australia, case study. Top shows extant (red lines) and protected (blue lines) species under the optimal (dashed lines) and front-loading strategy (solid lines). The distance between the blue and red lines reflects the size of the extinction debt. Note the different timescales of the plots. Bottom shows the amount of funding available to the manager over time (green), and the amount spent (bars) in yearly increments under the optimal strategy.

Our optimization approach allows the optimal solution to take the form of a single disbursement, but the best decision is to gradually spend the invested funds over a series of years (Fig. 2). A small amount of early spending slows down the species loss rate (Eq. 3), allowing managers more time to invest the remaining resources, and therefore to protect more species (SI Model of the System Dynamics and Fig. S3). A sensitivity test with a more conservative interest rate estimate, based on Australian Treasury bond returns, resulted in a longer optimal delay (SI Model of the System Dynamics and Fig. S4). However, we note that our results do not represent a strict recommendation to wait decades before acting, since a delay of this length would be untenable for most conservation organizations. Instead, the optimal solution emphasizes the potential efficiencies that are foregone by a decision to front-loading spending.

Fig. S3.

Comparison of single disbursement with gradual disbursement. The analytic model we analyze in the main text assumes that managers disburse all accumulated funds at a single point in time. However, the optimal solution to the case studies was not constrained by this assumption and found that a gradual disbursement generated superior outcomes. Note that the optimization algorithm is flexible enough to choose a single disbursement strategy but found that it was suboptimal. Fig. S3 offers an explanation of why the gradual option is superior. In all three panels, the black line describes the best single disbursement solution, found to be $t_s=13$ y using Eq. 5 in the main text. The red line indicates a variant on this solution, where 50% of accumulated funds are spent 5 y before the optimal single-disbursement solution. For this “dual-disbursement” solution, the first disbursement has three immediate consequences. First, with one-half of the funding expended, the accumulation rate of the budget is slowed considerably (A). Second, this expenditure creates an initial protected area (B). Third, as a consequence of this protection, the rate of species loss immediately slows (C). With this slower rate of species loss, the dual-disbursement manager is not forced to spend the remainder of their resources as soon as the single-disbursement manager, which allows them to accumulate additional funds until $t=23$ y (A). This process takes more time than the single-disbursement solution, but the slower relaxation of the extinction debt means that more species still remain extant when the dual-disbursement manager finally acts. (Note that this is not the optimal dual disbursement solution.) Fig. S3 was generated by the model described in the main text, with parameters $z=0.2$, $\theta =0.06$, $r=0.025$, $c_R=50$, $B_0=1$, $S_0=\alpha =1$, and $0\le R\left(t\right)\le 1$. The dynamics shown here are representative of many different parameter combinations.

Fig. S4.

Mount Lofty Ranges—Treasury bond interest rates. This figure shows the optimal schedule for the Mount Lofty Ranges, Australia, case study when the interest rate is set to 4% to represent the more conservative returns available from 25-y average Australian Treasury bond rates. Top panels show extant (red lines) and protected (blue lines) species under the optimal (dashed lines) and front-loading strategy (solid lines). The distance between the blue and red lines reflects the size of the extinction debt. Note the different timescales of the plots. Bottom shows the amount of funding available to the manager over time (green), and the amount spent (bars) in yearly increments under the optimal strategy.

Case Study 2: Atlantic Forest (Paraguay).

South America’s Atlantic Forest is a consensus global conservation priority (32). There is evidence for a substantial extinction debt in the ecoregion (33), with dramatic clearing in the 1980s leaving little intact habitat (≈25%) and many threatened species, but causing few immediate extinctions. International and national initiatives (e.g., Pacto Mata Atlantica, The Nature Conservancy’s One Billion Trees program, World Wildlife Fund, Paraguay Biodiversidad) aim to restore forest before any of the 124 forest-dependent endemic bird species go extinct (33).

Habitat dynamics for the region are parameterized using $\theta =0.03$ y⁻¹ (44), z = 0.18, and $\alpha =16$ (45), and restoration costs are estimated at $153,000 km⁻² (46) (2016 USD). All resources made available to the project are placed into a “sinking fund” in the first year of the project (37), and we assume an initial investment equivalent to Brazil’s FUNBIO fund of $500 million (2016 USD). While invested, the sinking fund returns a rate of 3.6% per year, based on the US lending interest rate adjusted for inflation over the past 25 years (41). See SI Model of the System Dynamics for an explanation of these parameters.

Fig. 3 shows that, if spending is heavily front-loaded (specifically, entirely within the first 5 years), species loss will halt in 143 years, protecting 87 species with 3,340 km² of additional restored habitat. Alternatively, in the optimal delayed schedule, managers accrue interest on their unspent funds for 40 years before they start to disburse them. When the accumulated funds are spent over the following nearly 20 years, the restored habitat will protect 96 species, restoring 12,000 km² of forest habitat in the process, and halting extinction within 58 years. Once again, an optimal delay accelerated conservation outcomes and achieved a 23% reduction in expected extinctions. A larger initial budget or higher interest rate will result in a shorter delay (including a small amount of spending in the first year) but front-loading remains suboptimal (SI Model of the System Dynamics and Fig. S5). More conservative interest rate estimates, based on Federal Reserve bond returns, result in a longer optimal delay (SI Model of the System Dynamics and Fig. S6).

Fig. 3.

Optimal schedule for the Atlantic Forest, Paraguay, case study. Top shows extant (red lines) and protected (blue lines) species under the optimal (dashed lines) and front-loading strategy (solid lines). The distance between the blue and red lines reflects the size of the extinction debt. Note the different timescales of the plots. Bottom shows the amount of funding available to the manager over time (green), and the amount spent (bars) in yearly increments under the optimal strategy.

Fig. S5.

Gradual disbursement with immediate spending. This figure shows the optimal schedule for the Atlantic Forest, Paraguay, case study, with a higher initial budget ($B_0=10$ billion 2016 AUD, versus $5 billion for Fig. 3, main text) and a higher interest rate (r = 0.1, versus r = 0.036 in Fig. 3, main text). Top shows extant (red lines) and protected (blue lines) species under the optimal (dashed lines) and front-loading strategy (solid lines). The distance between the blue and red lines reflects the size of the extinction debt. Note the different timescales of the plots. Bottom shows the amount of funding available to the manager over time (green), and the amount spent (bars) in yearly increments under the optimal strategy. The figure illustrates that, when initial budgets and interest rates are very high, the optimal disbursement schedule can recommend some immediate spending. However, even in this extreme case, the solution clearly does not recommend front-loading. For the first 6 y, proportional spending remained below 10% of the available budget [i.e., $u\left(t\right)<0.1$], effectively keeping the initial endowment at a constant level. At this level of potential financial return, managers could protect 112 species (compared with 96 in the main text) within 28 y and restore 295,000 km² of forest habitat (compared with 12,000 km² in the main text). The delay would still achieve a 52% reduction in expected extinctions (compared with the 23% reduction in the main text).

Fig. S6.

Atlantic Forest—Treasury bond interest rates. This figure shows the optimal schedule for the Atlantic Forest, Paraguay, case study when the interest rate is set to 1% to represent the more conservative returns available from the 25-y average US Treasury bond rates. Top shows extant (red lines) and protected (blue lines) species under the optimal (dashed lines) and front-loading strategy (solid lines). The distance between the blue and red lines reflects the size of the extinction debt. Note the different timescales of the plots. Bottom shows the amount of funding available to the manager over time (green), and the amount spent (bars) in yearly increments under the optimal strategy.

SI Model of the System Dynamics

Analytic Solution.

We first examined the general conditions for optimality by calculating an analytic solution for the simplest case. In this case, the management strategy is to wait before spending the endowment, until a time $t_s$ when a single intervention is made. Before and after this intervention, no protection is undertaken, and we assume that there are $R_0$ protected areas at the beginning of the scenario. Until the funding is spent, it is either banked or invested, during which time it accumulates compound interest at rate $r$. Thus, by the time the intervention occurs, the total budget will be as follows:

$$B\left(t_s\right)=B_0{e}^{r{t}_s}.$$ [S1]

The number of species remaining extant at that point in time $S\left(t_s\right)$ can be derived from the extinction debt dynamics:

$$\frac{dS\left(t\right)}{dt}=-\theta \left[S\left(t\right)-\alpha R_0^z\right],$$ [S2]

$$\frac{dS\left(t\right)}{dt}+\theta S\left(t\right)=\alpha \theta R_0^z.$$ [S3]

Thus:

$$S\left(t_s\right)=\alpha R_0^z+\left(S_0-\alpha R_0^z\right)e^{-\theta t_s},$$ [S4]

where $S_0$ is the number of species that existed at the beginning of the project [i.e., $S\left(0\right)=S_0$]. For simplicity, we examine the case where there is initially no area protected; thus:

$$S\left(t\right)=S_0{e}^{-\theta t}.$$ [S5]

If we spend all of the funding at some time $t_s$, then at this point we will restore an amount of habitat $R\left(t_s\right)=B\left(t_s\right)/c_R$. This habitat will protect a certain amount of species $S_p$ that is either:

$$S_p\left(t_s\right)=\alpha R{\left(t_s\right)}^z,$$ [S6]

or

$$S_p\left(t_s\right)=S_0{e}^{-\theta t_s}.$$ [S7]

Eq. S6 holds if the decision maker acts quickly, while there are still more extant species than the newly protected land can support. This is the steadily growing capacity that results from investment. Eq. S7 holds when the decision maker acts slowly, and the amount of land they protect can support more species than remain extant in the system. This equation reflects the inexorable extinction of species that are not supported by sufficient habitat.

Given that the management objective is to maximize the number of extant species following a T year conservation project:

$$\underset{0\le t_s\le T}{\max }{S}_p\left(T\right).$$ [S8]

The optimal decision is to spend the budget when the species protection functions are equal (Fig. 1):

$$\alpha {\left[\frac{B_0{e}^{r{t}_s}}{c_R}\right]}^z=S_0{e}^{-\theta t_s}.$$ [S9]

We can scale this equation by expressing the original number of species present in the region as a proportion of $\left(s=S_0/\alpha \right)$. The optimal disbursment time is then as follows:

$$t_s^{\ast }=\frac{\ln \left(s\right)+z\ln \left(\frac{c_R}{B_0}\right)}{zr+\theta }.$$ [S10]

This formulation describes the critical ecological and habitat dynamics as instantaneous rates (e.g., the loss of species, or the clearing and restoration of habitat), but we acknowledge that in practice they occur as discrete events. However, the objective of this exercise is to understand the general attributes of optimal approaches to funding conservation, as opposed to suggesting specific conservation action. If specific management recommendations are desired, a more detailed model formulation will be necessary.

Parameterization of Mount Lofty Ranges Case Study.

Species model parameters.

For the initial species richness ($S_0$), we chose the number of species extant in 2016 (107 species). This value was calculated by subtracting the eight species that have gone extinct (40) from the 115 historically found in the region (39). We estimated the relaxation rate $\left(\theta \right)$ of species in the region by fitting a nonlinear regression line to records of forest cover and extinction data (Fig. S2) and adjusting the relaxation rate parameter of Eq. 5 until the slope of the curves matched.

Forest loss parameters.

We used estimates of deforestation ($R_t$) in 1945 and 1980 from the chronological snapshot of the AMLR region in the Regional Recovery Plan for Threatened Species and Ecological Communities of Adelaide and the Mount Lofty Ranges, South Australia (ref. 42, Appendixes Part A). The proportion of the landscape that is currently deforested (2016) was obtained from the 2009 Department for Environment and Heritage report (38). We used Ford and Howe’s (39) estimate of 500,000 ha of intact forest before clearing ($R_0$).

Funding parameters.

We estimate annual current spending (B_t) at over $500,000 (2016 AUD). This estimate was obtained from a report on continuing Natural Resources Management (NRM) projects 2011–2012, obtained from the Government of South Australia website that details expenditures on forest recovery initiatives in the region (www.environment.sa.gov.au/files/sharedassets/public/nrm/cons-mnr-nrmfundingcontinuingprojects2010-12.pdf; accessed April 20, 2016).

We set the economic discount rate (e.g., interest rate) to $r=0.058$ based on the geometric mean of the Australian lending interest rate (adjusted for inflation) between 1990 and 2015. World Bank Development Indicators. Retrieved from data.worldbank.org/indicator/FR.INR.RINR on January 9, 2017. We used a restoration cost of $c_R=$ $1,132/ha (2016 AUD) based on mean costs of $1,121/ha for direct seeded environmental plantings across Australia (69) inflation corrected to 2016 AUD using the Reserve Bank of Australia inflation calculator (accessed at www.rba.gov.au/calculator/annualDecimal.html, December 12, 2016).

Parameterization of Atlantic Forest Case Study.

Species model parameters.

We set the number of species extant in 2016—124 species for our starting species richness ($S_0$). This value was obtained from Brooks and Balmford (33) who claim that no recorded forest-dependent species have yet gone extinct in the region. We used Brooks et al.’s (44) previous estimate of relaxation rate for bird species in the region, so $\theta$= − 0.03.

Forest loss parameters.

We set the initial forested area of the region ($R_0$) to 88,050 km² to represent the extent of the World Wildlife Fund’s Atlantic Forest Ecoregion within Panama (70). Current extent of remaining forest ($R_t$) was estimated by Huang et al. (70) using remote sensing to be 25% of the original extent. Based on this study, we set current forest extent to 22,013 km².

Funding parameters.

We estimated the available funding (B₀) for a hypothetical sinking fund for restoration Paraguayan Atlantic Forest to be $500 million (2016 US$). This value is based on the magnitude of assets managed by FUNBIO, which is a similar fund in Brazil (www.funbio.org.br/en/o-funbio/quem-somos/; accessed January 9, 2017).

We assumed that our hypothetical fund was managed by a US nongovernmental organization and therefore set the interest rate to $r=0.036$, based on the geometric mean of the US nominal interest rate (adjusted for inflation) between 1990 and 2015. (World Bank Development Indicators; retrieved from data.worldbank.org/indicator/FR.INR.RINR on January 9, 2017).

We obtained forest restoration costs from a previous study (46). This study suggested that restoration of abandoned agricultural sites in southeastern Brazil costs about $c_R=$ $112,000/km², not including land costs. We converted this estimate to 2016 USD $1,530/ha, using the US Bureau of Labor Statistics Consumer Price Index Inflation Calculator (accessed at www.bls.gov/data/inflation_calculator.htm; December 12, 2016).

Numerical Optimization Process.

Operationally, we calculated the optimal disbursement solution using FMINCON, a constrained numerical optimization routine in Matlab. We discretize the control function $u\left(t\right)$ into a vector of 51 different control variables $0\le u_i\le 1$, which we distribute evenly across the project of total duration T. Thus, $u_1$ is the proportion of the funds spend on direct conservation actions between $0\le t<T/50$, $u_2$ is the proportion of the funds spent between $T/50\le t<2T/50$, and so on. We choose a value of T with increasing length, until the all funds have been expended well before the end of the simulation. We then allow the numerical optimization routine to choose values for $u_i$ that maximize the number of extant species at T. We initialize each $u_i$ at a random value between 0 and 1. To increase the probability that the algorithm is finding the global optimum, we repeatedly ran the algorithm from multiple initial conditions. For both case studies, the algorithm found the same optimal schedule from each set of initial conditions, suggesting that the optimum was globally attracting.

Sensitivity Test of Case Study Results with Treasury Bond Interest Rates.

Our analysis used inflation-adjusted lending interest rates to estimate the return on investments, but this may be an optimistic expectation of what conservation organizations could secure. We performed a sensitivity test in which we substituted relevant Treasury bond rates for interest rates in the analyses. These are securities issued by national governments which carry a fixed annual rate of interest over the medium to long term. They provide low-risk, predictable returns, but generally at a relatively low rate.

Mount Lofty Ranges.

Australian 10-y Treasury bonds averaged (geometric mean) an annual rate of return of 7.17% over the 25-y period of 1988–2013, according to historical data from the Australian Reserve Bank (accessed at www.rba.gov.au/statistics/historical-data.html, June 2, 2017). Over this same time period, year-end average inflation was 3.15% (www.rba.gov.au/inflation/measures-cpi.html#quarterly). We used these values to estimate a conservative inflation-adjusted bond return of 4.02% for investment in the Mount Lofty Ranges project. At this level of financial return, the extinction debt could be negated within 100 y, with 140,000 ha of habitat restored (Fig. S4). The optimal spending schedule eventually reduces the number of extinctions by 41% over business as usual (compared with 51% in the main text) and allows the extinction debt to be eliminated within 100 y.

Atlantic Forest.

The US 10-y monthly government bond rates averaged (geometric mean) 4.46% annual yield over the 25-y period between 1990 and 2015. These data were collated from historical data provided by the Federal Reserve [Board of Governors of the Federal Reserve System (US), 10-y Treasury Constant Maturity Rate [DGS10], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/DGS10, June 6, 2017]. Over this same time period, mean yearly inflation was 2.53% (World Bank, inflation, consumer prices for the United States [FPCPITOTLZGUSA], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/FPCPITOTLZGUSA, June 6, 2017). We calculated annual estimated bond returns, minus inflation, to obtain an inflation-adjusted bond return of 1.02% for the Atlantic Forest investments. At this level of potential financial return, managers could protect 90 species within 92 y and restore 0.6 million hectares of forest habitat (Fig. S6). The delay would still achieve a 7% reduction in expected extinctions (compared with the 23% reduction in the main text).

Discussion

Our results emphasize an important additional dimension in systematic conservation planning by demonstrating that there are optimal times to take action, in addition to optimal locations. When conservation capacity can increase faster than the irreversible rate of biodiversity decline, a delay of finite duration allows projects to leverage additional advantage from conservation funds. Strategic delay can thus allow decision makers to protect more species—even to complete projects more rapidly—suggesting that front-loaded conservation projects may be overlooking a substantial source of efficiency in the temporal dimension. The optimal amount of delay will balance the rate of biodiversity loss with the rate at which waiting increases managers’ capacity to act. In our specific examples, we use financial interest to represent the benefits of waiting because it is conceptually simple, and extinction debts to represent the incentive to act immediately because they offer a well-understood link between the loss of habitat and the loss of species. However, competing rates of biodiversity loss and capacity accumulation are, in some form, present in all conservation problems.

Our central conclusion is that a focus on future capacity, at the expense of immediate action, can deliver superior outcomes over relevant timescales. Many conservation organizations clearly understand this concept and focus on activities that deliver delayed, long-term benefits (47). Prominent examples include environmental education and awareness initiatives, climate change adaptation projects, and conservation policy think tanks. However, our work demonstrates that delayed action can also deliver efficiencies at the scale of individual projects, which are primarily concerned with direct conservation actions such as restoration. This conclusion runs counter to standard conservation practice at a project level: many sources of one-off funding (e.g., grants, offsets) ask projects to deliver and report on outcomes within short time frames. Similarly, philanthropic rating schemes tend to encourage organizations to spend more of their income immediately, on direct actions (48, 49). Just as spatially constrained conservation planning cannot deliver the best outcomes (50, 51), temporal constraints incur opportunity costs, by restricting managers’ freedom to act at the most effective point in time.

Previous conservation theory has shown how prioritizing actions across time can result in superior outcomes (e.g., refs. 52–57). However, these dynamic optimizations still treat conservation as a race against time—they essentially identify where to front-load spending. In contrast, we take advantage of variation along the temporal dimension to identify efficient periods of time in which to act. An assessment of the shadow values of alternative actions confirms the existence of periods of high relative efficiency (Fig. S7).

Fig. S7.

Shadow values of different manager strategies. We calculated the shadow values of both spending and investing an additional dollar of resources, at all points in time throughout the optimal schedule, to better understand the MLR case study solution. The shadow value measures the increase in extant species at the end of the project (i.e., the increase in the objective function) that would result from expending one additional dollar for each conservation action (i.e., spending or saving) at each point in time. The shadow values associated with each action suggest that additional spending has almost no benefit for decades, while the action of investing additional funds provides a relatively large increase in outcomes. The shadow value of investing is much higher than that of spending for about the first 40 y but decreases over time because funding accumulation due to compound interest becomes less valuable as the optimal solution prepares to disburse. After this point, the shadow values become approximately equal—indicating that the optimal decision will be a proportional allocation between the two decisions [i.e., $0\le u\left(t\right)\le 1$]. The shadow values decline and remain approximately equal as the optimal strategy expends all available resources.

Our ideas also have close analogues in environmental and resource economics, but with important differences. Real options analysis has long understood that delaying irreversible actions can be an optimal strategy, because waiting allows decision makers to learn more about the situation, and to respond better to uncertain future events (58–60). Our results provide a parallel justification for waiting, but where the returns to management actions are known with certainty because the system dynamics are deterministic. Our conclusions are closely related to Hotelling’s rule, which states that nonrenewable resources should initially be strategically underexploited to increase future revenue (61). Similarly, Fisher’s rule illustrates how the timing of logging in production forests should reflect the timber growth rate, price, and economic discount rate (62). However, the conservation problem is the inverse of these resource extraction problems. Rather than maximizing the revenue generated by exploiting a limited resource, we seek to minimize the loss of a declining resource, while operating under tight budget constraints. The result is that the discount rate works against resource managers and for conservation managers (because it acts to increase their capacity). Nevertheless, because they are pursuing opposite goals, in both cases lower interest rates encourage longer strategic delays.

We made a series of simplifying assumptions to clearly highlight the competing rates that drive the optimal delays. Our ecological and economic models are deterministic, whereas both systems experience variability and shocks. Thus, both the costs (e.g., land prices) and benefits of delay will become more uncertain in more distant futures. These sources of uncertainty will change our model recommendations and are a critical reason why our results—particularly those that recommend decades of delay—should not be used prescriptively. However, these factors do not change the central conclusion, that strategic delay can deliver conservation benefits. Ecologically, our models assume that restoration rapidly and predictably restores habitat, but functional habitat can lag restoration actions by many decades, and restored habitat may never be fully equivalent (63). However, more complex alternative models that include time lags and uncertainty in restoration do not affect conclusions in comparable models (34).

The ability of conservation organizations to capitalize on temporal opportunities will be subject to constraints on institutions’ financial flexibility. The benefits of investment for future spending will be unavailable to agencies that operate under short budget cycles, and penalize surpluses (64), or to funding instruments (e.g., grants) that expect spending to occur within defined time frames. Our results suggest that there is an advantage to developing conservation institutions and instruments that are free of these constraints. For instance, a growing number of organizations are establishing long-term conservation trust funds (65, 66) [e.g., The Nature Conservancy’s Land Preservation Fund (67)] or are using investments to secure perpetual management income (e.g., Tasmanian Land Conservancy Management Endowment). Sinking funds, which we modeled in our Atlantic Forest case study (Fig. 3), are another vehicle for leveraging interest from conservation resources and for avoiding front-loaded spending. These funds are becoming increasingly common [e.g., EcoFundo in Ecuador, or FUNBIO in Brazil (68)] and are well-suited to one-off funding sources such as large endowments or offsets. Our results highlight the importance of the temporal flexibility offered by these financial instruments but emphasize how their disbursement time frames should reflect the ecological dynamics of the target ecosystems.