A metric driving local action for global biodiversity conservation

Diﬃculties identifying appropriate biodiversity impact metrics remain a major barrier to 13 inclusion of biodiversity considerations in environmentally responsible investment. We propose 14 and analyse a simple science-based local metric: the sum of the proportions that all local 15 populations contribute to their global species abundances, with a correction for species close to 16 extinction. It links mathematically to a widely used global biodiversity indicator, the Living 17 Planet Index, for which we propose an improved formula that directly addresses the known 18 problem of singularities caused by extinctions. We show that, in an ideal market, trade in 19 our metric would lead to near-optimal conservation resource allocation, emphasising support 20 of extinction-threatened species. Barriers to adoption are low, as use of the metric does not 21 require an institutional certiﬁcation system. Used in conjunction with other metrics, potential 22 areas of application include biodiversity related ﬁnancial disclosures and voluntary or legislated 23 no net biodiversity loss policies. ecosystem functioning and services (e.g. via MSA)—and these should not be confounded. We have demonstrated that Biodiversity Stewardship Credits have properties suitable for use in bio- diversity-related disclosures in business and ﬁnancial contexts. The metric can also support voluntary or legislated no-net-loss policies. In all cases, we recommend its use in conjunction with metrics for ecosystem services or natural capital. Many of the attractive properties of BSC reﬂect that this metric is strictly science based, mathematically linked to the LPI and the species conservation objective. Data requirements of BSC are similar to those of existing comparable metrics. Pilot studies are now underway to develop 266 suitable protocols for application of BSC in practice.

The rapid recent growth of markets for responsible investments considering Environmental, Social and 26 Governance (ESG) concerns 1 , and the growing attention biodiversity receives in this context 2,3 , highlight 27 the need for metrics of biodiversity impacts for use by businesses and in financial markets 2,4,5 .

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Requirements on such metrics are different from those for metrics used in the policy context, where 29 some flexibility is desired and a balanced representation of multiple objectives sought 6 . In the business 30 context, metric simplicity is of the essence. Non-expert market participants must understand the impacts 31 their decisions have on a metric 4 . Metric definitions should be rigorous and robust enough to withstand 32 strong financial incentives to game them 4 . Metric values should be comparable across a wide range of 33 business activities 7 . Importantly, biodiversity metrics should support a simple narrative about biodiversity 34 conservation in communication between business, customers, and the general public 4 . To reduce exposure with regards to the protection of biodiversity for its 'intrinsic value' and as a matter of intergenerational 48 justice. Specifically, it is designed to link local action mathematically to its impact on the Living Planet 49 Index (LPI) 11,12 , one of the most widely cited global diversity indicators currently in use. 50 To overcome technical and conceptual problems arising when populations of species entering the LPI 51 approach zero 12,13 , we begin by introducing a variant of this indicator that avoids these problems. For this 52 variant we demonstrate that, under certain simplifying assumptions, it quantifies long-term extinction risk, 53 thus providing a solid scientific basis for the marketable metric (BSC) that we subsequently derive. 54 Extinction risk and regularisation of the Living Plant Index 55 The LPI quantifies the change in the geometric mean of the abundances of species in a given taxonomic 56 or functional group (below: group of species), relative to a baseline year 12 . Geometric mean abundance 57 biodiversity indices of this type stand out by their combined simplicity, favourable statistical properties 14 , 58 intuitive accessibility and ecological plausibility 14 . Not all species are regularly surveyed at sufficient accu-59 racy to compute the LPI directly, but the index can be estimated as an appropriately weighted geometric 60 average from existing survey time series 12 . These estimates are now regularly determined and published 61 under the leadership of the World Wildlife Fund 11 . The LPI is amongst the most frequently cited biodi-62 versity status indicators in the media, evidencing its wide acceptance by policy makers and the public. It 63 can be interpreted as the typical amount by which species abundances have increased or decreased since 64 the baseline. Because geometric mean abundance is highly sensitive to changes in the abundances of rare 65 species 13 , LPI is sensitive to pressures on rare species even when the total biomass of the group of species 66 represented changes only little.

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Mathematically, the LPI is often studied in terms of the sum L of the natural logarithms (notation 68 'log x') of the global abundances N i of all S species i in the group of interest: In place of abundance N i , other measures of population size can be used, such as population biomass or 70 area covered, whichever approximates total reproductive value 15 well.

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If we denote by L 0 the value of L in the baseline year, current LPI can be computed from L as If one of the S species went extinct since the baseline year, L attains a value of negative infinity and LPI is, for a given group of S species, proportional to the expected number of surviving species after a long time

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T . In view of this intuitive ecological interpretation, we propose use of the regularised quantity L reg in place 95 of L and correspondingly define the regularised Living Planet Index as

Box 1 The regularised Living Planet Index as a measure for extinction risk
We show that with a small modification the LPI can be understood as quantifying long-term extinction risk. We model the change in the population size of a species over a time interval ∆t as where ξ(t) and ξ ′ (t) denote independent standard normal random numbers. Parameters ve and v d represent the strengths of environmental and demographic stochasticity, respectively. Formally taking the limit ∆t → 0, standard procedures lead to an approximation of this process by the Ito stochastic differential equation where Wt represents a Wiener process (Brownian motion). The first term on the right-hand-side describes drift to larger values. It goes back to the fact that the expectation value of the log-normal distribu- (3) is exp(ve∆t/2). If one were to formulate this process in terms of log(N ) rather then N , this term would disappear. However, demographic stochasticity would then generate another drift term instead. To eliminate drift altogether, we express population sizes in terms of x = log(1 + N/N * ), where N * = v d /ve is the population size at which environmental and demographic stochasticity have the same strength (see Eq. (4)). Note that x becomes zero when a species goes extinct (N = 0). Applying Ito's lemma, this change of variables simplifies Eq. (4) to That is, x performs a Brownian motion, represented by dx = √ vedWt, except when x is of the order of one or smaller. For x approaching zero from above, the factor 1 − e −x reduces the magnitude of fluctuations in x, slowing down the random walk. As a result, x can get trapped in the region of low x, and the vicinity of 0 acts similar to an absorbing boundary 20,21 . This effect is reinforced by the breakdown of the diffusion approximation underlying Eqs. (4) and (5) for small x 22 . In reality N and so x reach zero eventually, implying global extinction of that species. We therefore approximate the dynamics of x by simple Brownian motion with an absorbing boundary at x = 0 23 . Now consider the probability that a species starting from x = x 0 will still exist after a time T , i.e., the probability for x to never reach 0 before T . Textbook methods evaluate this to erf(x 0 / √ 2T ve) 24 , where erf denotes the error function 25 . For T not too near in the future That is, for any sufficiently large, fixed observation time T , the current value of x 0 = x = log(1 + N/N * ) is directly proportional to the probability of species survival. In Supplementary Information (S1) we demonstrate validity of Eq. (6) for a model with discrete population sizes N . Data suggest that ve does not usually vary strongly within taxonomic groups 17,18 . When this is so, S i log 1 + N i /N * i is, for a given group of S species, proportional to the expected number of species surviving after a long time T .
with L reg,0 denoting the value of L reg at the baseline year. These regularised quantities naturally avoid the 97 singularity that arises for extinct species (N i = 0). Yet, LPI and LPI reg are nearly identical when all species 98 populations N i are much larger than the corresponding N * i .

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Quantifying local biodiversity impact 100 With global biodiversity quantified in terms of L reg , the biodiversity impact of any intervention can be 101 quantified as the resulting change ∆L reg in this metric. If the changes in population sizes ∆N i resulting 102 from a local impact are just small fractions of population sizes N i , as will often be the case, ∆L reg is well 103 approximated to linear order in ∆N i as From Eq. (9) it is clear that ∆L reg weights impacts on globally rare species higher than impacts on common 105 species, thus plausibly providing a measure of pressure on (∆L reg < 0) or relief to (∆L reg > 0) global 106 biodiversity.

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Any practical determination of ∆L reg would require that the changes in abundances ∆N i entering Eq. (9) 108 are calculated by comparing measured abundances before and not long after an intervention has occurred.

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This might raise concerns that the long-term ecological reverberations of an intervention are yet to unfold and unaccounted for biodiversity loss 7,27 . Impact metrics are therefore usefully defined in terms of change in a 123 local measures of biodiversity state 8 . However, this approach is not directly applicable here, since LPI and 124 LPI reg are difficult to apportion over space. Indices permitting easy spatial disaggregation 28 , on the other 125 hand, tend to have difficulties incentivising the protection of rare species 29 .

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As a proposal to overcome this conundrum, we define, for a given area of land (or water), associated 127 Biodiversity Stewardship Credits as with S, N i and N * i defined as above and n i being the local population size of species i that this area sustains 129 (for migratory species scaled by the proportion of time spent in the area).

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Comparison of Eqs. (9) and (10) shows that, when all n i are small relative to N i , e.g. because the area is 131 small, BSC quantify the short-term impact that would result from removing all species of the group in that 132 area. The metric supports a narrative of biodiversity stewardship because it gives high weight to locally 133 sustained populations of globally rare species. 134 We note the following four properties of BSC that follow directly from Eq. (10): sum of all BSC for a given group of species of size S is close to but never larger than S.

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We are unaware of direct alternatives to BSC that would quantify local contributions to extinction risk.

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Comparable is the recently proposed Species Threat Abatement and Recovery (STAR) metric 44 . However,

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STAR is neither a measure of biodiversity state nor has it been designed for marketisation. In Supplementary   164 Information (S2) we show that the mathematical structure underlying STAR and BSC is similar, suggesting 165 that these metrics support broadly consistent conservation priorities.

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Incentive structure generated by BSC 167 We envision that organisations can usefully include regular BSC accounts in environmental impact reports.

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The fact that BSC are always positive encourages reporting as complete as logistically possible. The bonuses 169 that environmentally friendly assets currently attract on markets then imply that BSC on their own have where µ α,i and λ are the Karush-Kuhn-Tucker multipliers. It leads to the stationarity condition that A local population size of n i = N * i , for example, earns this market participant 0.5 BSC (Fig. 1). This is 202 plenty and worth preserving, considering that populations of size N * i tend to be comparatively small and . As illustrated by the dashed lines, BSC gained by increasing the species' population by a single individual are largest when it is closest to extinction. However, to achieve full BSC, the species' population must be lifted well above N * , the population size below which random birth and death events cause large population fluctuations.
that the global sum of available BSC is limited to S. In fact, market incentives to rebuild the population of 204 i are highest at the lowest population size (Fig. 1).

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On the second point, note that market participants that sustain a smaller population of a species i on 206 their land benefit more from increasing this population than those holding a larger population. The reason 207 is that the latter incur a higher penalty from reducing the BSC of their current holding by increasing N i 208 in the denominator Eq. (10). Hence BSC disincentivise dominant holdings of species. We therefore expect 209 that market misalignment due to the difference between conditions Eq. (11) and (13) will be harmless in 210 most cases, implying optimal resource allocation as explained above.

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The role of BSC in biodiversity conservation 212 To illustrate the role BSC can play in biodiversity conservation and its limitations, we consider a hypothetical 213 use case. Assume that Organisation X has committed to a net gain in BSC on the land it holds. Conversion

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Critics might argue that a low-cost, small-scale conservation project cannot compensate the large-scale 226 biodiversity loss due to solar farm construction. Yet, X's use of BSC was as intended by the design of the 227 metric. Capital was directed to a conservation cause where it made a substantial difference at moderate cost.

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A. nanchuanensis was saved from extinction. Overall, global LPI for plants has increased as a result of X's 229 action. Other early adopters would find similar opportunities to acquire BSC at low cost. As the market for 230 BSC matures, organisations committed to net biodiversity gain would seek more efficient schemes to raise 231 BSC (e.g. by restoring entire ecosystems), thus further improving resource allocation to conservation.

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To address the concern noted above, consider, for example, the Mean Species Abundances (MSA) metric, 233 originally called the Biodiversity Intactness Index 28 , a well-established metric which is based on similar data. Step B is not required for a principled approach to conservation. suitable protocols for application of BSC in practice.

Box 2 Example of BSC calculation
Schoolchildren at Northgate High School and Sixth Form in Ipswich, UK, participating in the Big Schools' Birdwatch organised by the Royal Society for the Protection of Birds (RSPB), counted the birds listed below i in the school's Wildlife and Wellbeing Garden on 11 December 2020. Taking these counts at face value (despite some methodological limitations) and combining them with published global abundance estimates 30 , we calculate (table) that the garden holds around 1.3 × 10 −6 species worth of Bird-BSC. More than half of BSC come from only two species (pie chart): black headed gulls that were abundant in the garden; and blue tits, due to their relatively low global abundance. To demonstrate that we are justified in disregarding N * in these calculations, consider dunnock, the globally rarest species in the sample. From the worst-case literature estimates 31,32 v e = 0.017 yr −1 and v d = 0.5 yr −1 , we get N * ≈ 29, negligible by a wide margin compared to the estimated global abundance N ≈ 9,000,000.
with ξ(t) denoting a standard normal random number and Poisson(µ) a Poisson-distributed random number The definition of STAR has a similar structure, but with two additional factors in the sum. Firstly, the 290 numeric red-list category W i of species i, an integer ranging from 0 (Least Concern) to 4 (Critically Endan-291 gered). Secondly, an index C i,t quantifying either the relative contribution of threat t to the extinction risk 292 of that species, or the contribution threat abatement could make to i's recovery. Hence The factor C i,t apportioning impacts over threats adds semantics to STAR that BSC do not carry (though 294 the idea might be transferable). The technical difference between STAR and BSC thus lies mainly in the 295 threat-level factor W i . We have argued above that extinction risk is already captured by n i /(N * i + N i ), 296 approximated by P i , suggesting that the W i factor might duplicate this information. Indeed, the extent of 297 occurrence of a species, closely related to AoH 45 , is a central red-listing criterion. Besides, inclusion of the 298 factor W i implies choices that may not have an obvious justifications: for example, why not use instead a 299 factor W 3 i or W 1/3 i ? However, since the range of values that W i can attain is rather narrow, the overall effect 300 this factor has on incentive structure might be small. We therefore expect BSC and STAR to be largely 301 aligned with each other.