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    Using size-weight relationships to estimate biomass of heavily targeted aquarium corals by Australia’s coral harvest fisheries

    Establishing size-weight relationships for heavily targeted coral species is an important first step towards informing sustainable harvest limits19. Placing coral harvests into an ecological context is a core requirement for implementing a defensible stock assessment strategy, and this need is particularly critical given escalating disturbances and widespread reports of coral loss7,17,25. Using these relationships, managers can now easily sample and calculate biomass per unit area. It is important to point out that all sites sampled in our study represent fished locations, and there is no information available to test whether standing biomass has declined due to sustained coral harvesting at these locations. While these data may now provide a critical baseline for assessing the future effects of ongoing fishing, it is also important to sample at comparable locations where fishing is not permitted or has not occurred (where possible), to test for potential effects of recent and historical harvesting.Biomass per unit area data presented herein highlights the highly patchy abundance and biomass of targeted coral species14, which is evident based on the often vastly different mean and median values (Table 2). Examining biomass per unit area estimates for C. jardinei for example, which returned some of the highest biomass estimates, the 33.75 kg·m−2 maximum estimate from a transect stands as an extreme outlier, with 12 of the 16 other transects being below 0.2 kg·m−2. This indicates the challenges of managing species that occur in patchily distributed concentrations, particularly in a management area the size of the QCF. It is also important to note, these estimates are generated only on transects where the target species occurred, and therefore, should technically not be considered as an overall estimate of standing biomass. While the estimation of size-weight relationships is a step towards a standing biomass estimate, many challenges remain in terms of sampling or reliably predicting the occurrence of these patchily distributed species. Bruckner et al.14 attempted to overcome this management challenge in a major coral fishery region of Indonesia by categorising and sampling corals (in terms of coral numbers) in defined habitat types, and then extrapolating to estimated habitat area based on visual surveys and available data. This approach, utilising size-weight relationship derived biomass per unit area estimates (instead of coral numbers), may be a viable method for the QCF, however much more information is needed to understand the habitat associations (e.g., nearshore to offshore), and environmental gradients that influence the size and abundance of individual corals. Fundamentally, it is also clear that much more data is required to effectively assess the standing biomass of aquarium corals in the very large area of operation available to Australian coral fisheries.These corals are found in a range of environments, and it is important to consider available information on life history if attempting to use coral size-weight relationships to inform management strategies via standing biomass estimation. All corals in this study can be found as free living corals (at least post-settlement) in soft-sediment, inter-reefal habitats, from which they are typically harvested by commercial collectors19. However, only four of the 6 species are colonial (C. jardinei, D. axifuga, E. glabrescens, M. lordhowensis) while the remaining two species (H. cf. australis and T. geoffroyi) are more typically monostomatous or solitary. As indicated in previous work24, if larger colonial corals were to be fragmented during harvesting instead of removed entirely, fishery impacts would likely be lessened24. Given the power relationship between coral maximum diameter and weight, larger corals contribute disproportionately to the total available biomass of each species in a given area. The potential environmental benefit of leaving larger colonies (at least partially) intact is not limited to impacts on standing biomass, as this practice would likely be demographically beneficial given the greater reproductive potential (i.e., fecundity) of larger colonies, which also do not need to overcome barriers to replenishment of populations associated with new recruits (i.e., high mortality during and post-settlement26). This conclusion was drawn largely from data on branching taxa (e.g., Acropora), which are relatively resilient to fragmentation and commonly undergo fragmentation as a result of natural processes27,28,29. D. axifuga can be considered to exhibit a relatively similar branching growth form, however, the growth form of E. glabrescens and C. jardinei changes with size, moving from small discrete polyps to large phaceloid and flabello-meandroid colonies, respectively19. While larger colonies of E. glabrescens and C. jardinei may be relatively resilient to harvesting via fragmentation, the same may not be true for smaller colonies, or species with massive growth forms such as M. lordhowensis. Typically, for each species, the average reported weight was quite low, coinciding with the lower end of the sampled maximum diameter range. For colonial species, the harvested smaller maximum diameters (if fragments) are ideal from an ecological perspective as this will have the least impact possible on standing biomass, and may also leave a potentially mature breeding colony intact. Ultimately, in light of these considerations, the development of uniform and standardised industry-wide harvest guidelines to balance economic and ecological outcomes may be necessary. The development of these guidelines would require consultation with commercial harvesters, as well as considerable additional work in measuring ecological impacts and better understanding the cost of these impacts from an economic perspective. Conversely, if whole colonies are collected, which is necessarily the case for solitary species such as H. cf. australis and T. geoffroyi (and potentially smaller colonies of other species such as E. glabrescens and C. jardinei); smaller colonies may be collected before they reach sexual maturity, hindering their ability to contribute to population replenishment. Therefore, collection of small fragments should be encouraged for colonial species; while for monostomatous species where this is not possible, introduction of a minimum harvest size based on sexual maturity should be considered.Additionally, the need for further consideration of the selectivity of ornamental coral harvest fisheries3,4,30 when assessing standing biomass is evident. Due to various desirable traits, the majority of available biomass may not be targeted by collectors. As emphasised in this study, the focus on smaller corals is indicative of the trend towards collection of most of these species at the lower portion of their size range, at least compared to some of the maximum sizes recorded on transects (e.g., see Tables 1 and 2, section b). However, it is also important to consider that transects were conducted in areas subject to commercial collection and are likely to skew results and prevent clear conclusions relating to size selectivity. Sampling of unfished populations (i.e., any residing outside of permitted fishing zones) and/or spatial and temporal matching of catch data and transect data across a larger sample of operators will be required to properly address industry size selectivity trends. For instance, only 17.5% of C. jardinei corals measured on transects fell within the diameter range represented by data obtained from collectors, with 81.9% of corals measured on transects exceeding this range. If it is viable to collect fragments from larger colonies (which does appear to be the case for some corals such as C. jardinei), then a larger proportion of standing biomass outside of this size range could be targeted by fishers. As an additional consideration, only desirable colour morphs of these corals will be harvested, and due to lack of appropriate data, the prevalence of these morphs remains unclear. H. cf. australis and M. lordhowensis for example often occur in brown colour morphs, which are far less popular in markets where certain aesthetic qualities (e.g., specific, eye-catching colours or combinations of colours) are desired, such as the ornamental aquarium industry. Even without delving into further considerations such as heritability of phenotypic traits, management conclusions drawn from standing biomass estimates may be ineffective in the absence of efforts to account for selectivity in this fishery.The relationship between size and weight was found to differ between all corals, with the exception of C. jardinei and E. glabrescens. There can be some moderate similarity in skeletal structure between these two species, particularly between small colonies, reflecting the similar maximum diameter range of sampling in the current study. Subsequently, inherent physiological constraints may be imposed on corals that prevent the maintenance of growth rates between corals of smaller and larger sizes, for example, as the surface area to volume ratio declines with growth31. In the current study, all corals, with the exception of C. jardinei, showed evidence of allometric growth, as exhibited by an estimated exponent value different to 3. Sample size for C. jardinei was greatly limited, as this species typically forms extensive beds, and are rarely brought to facilities as whole colonies. Therefore, the lack of evidence for allometric growth may reflect higher error for the species coefficient parameter due to the comparatively small sample size for this species. This suggests that mass would not increase consistently with changes in colony size in 3 dimensions31, which seems likely considering the change in exhibited form described for E. glabrescens and C. jardinei previously. In the current context, this indicates that the estimated ‘a’ and ‘b’ constants are likely to vary as the sample range increases, reflecting the changes in the size-weight relationship between smaller and larger samples of these species. Therefore, ideally, these models should incorporate data that reflect the maximum diameter range of the species in the region of application to allow increased accuracy of biomass estimation. To achieve this will require additional fishery-independent sampling, as large colonies are rarely collected whole, though may be collected as fragments depending on the species. Sampling may be challenging for some species given the difficulty of physically collecting and replacing large whole colonies, particularly for inter-reefal species such as M. lordhowensis, which can occur in deep, soft sediment habitat, subject to strong currents. Importantly, obtaining ex situ or in situ growth rate data should be considered a priority for the management of heavily targeted species. This data is likely to be another necessary component (in conjunction with size-weight relationships) of any stock assessment model developed for LPS corals, and may also eliminate the need to collect large sample colonies to improve estimated size-weight relationships.The disproportionate focus on smaller corals (i.e., corals in the current study averaged between 4.28 and 11.48 cm in maximum diameter) is likely to lead to an underestimation of weight in corals at greater diameters when used as inputs for size-weight models. This may explain the apparent minor underestimation observed in some species (e.g., M. micromussa, T. geoffroyi). In the current context, this represents an added level of conservatism with estimates obtained from these equations. While the relationship between size and weight was particularly strong for some species, (mainly D. axifuga and T. geoffroyi), for other species, such as M. lordhowensis, growth curves tended towards underestimation at larger diameter values. As the mass of a coral is reflective of the amount of carbonate skeleton that has been deposited32, the coral skeleton may increase disproportionately to coral diameter if or when corals start growing vertically. For example, in massive corals such as M. lordhowensis, vertical growth (i.e., skeletal thickening) is often very negligible among smaller colonies, with thickening of the coral skeleton only becoming apparent once the coral has reached a threshold size in terms of horizontal planar area. Additional fisheries-independent sampling outside of the relatively narrow size range of harvested colonies will be required to address this source of error in future applications. Ecological context in the form of fishery independent data on stock size and structure is essential for effective management, especially in ensuring that exploitation levels are sustainable and appropriate limits are in place. Coral harvest fisheries offer managers an ecologically and biologically unique challenge, as the implementation of standard fisheries management techniques and frameworks is hampered by their coloniality and unique biology, as well as a general lack of relevant data for assessing standing biomass and population turnover, not to mention the evolving taxonomy of scleractinian corals33. Similarly, fishery-related management challenges such as the extreme selectivity in terms of targeted size-ranges and colour-morphs, plus the potentially vast difference in the impact of various collection strategies (i.e., whole colony collection vs fragmentation during collection) also complicates the application of typical fisheries stock assessment frameworks. The relationships and equations established in the current work offer an important first step for coral fisheries globally by laying the groundwork for a defensible, ecologically sound management strategy through estimation of standing biomass, thus bridging the gap between weight-based quotas and potential environmental impacts of ongoing harvesting. It is important to note that the species selected for the current work do not represent the extent of heavily targeted LPS corals. For example, Fimbriaphyllia ancora (Veron & Pichon, 1980), Fimbriaphyllia paraancora (Veron, 1990), Cycloseris cyclolites (Lamark, 1815), and Acanthophyllia deshayesiana (Michelin, 1850) are examples of other heavily targeted corals of potential environmental concern19, and management would also benefit from the estimation of size-weight relationships for these species. Moving forward, the next challenge for the coral harvest fisheries will be to comprehensively document and track the standing biomass of heavily targeted and highly vulnerable coral stocks, explicitly accounting for fisheries effects and also non-fisheries threats, especially global climate change. More

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    Diversity enables the jump towards cooperation for the Traveler’s Dilemma

    Game theory is a framework for analysing the outcome of the strategic interaction between decision makers1. The fundamental concept is that of a Nash equilibrium where no player can improve her payoff by a unilateral strategy change. Typically, the Nash equilibrium is considered to be the optimal outcome of a game, however in social dilemmas the individual optimal outcome is at odds with the collective optimal outcome2. This means that one player can improve her payoff at the expense of the other by unilaterally deviating, but if both deviate, they end up with lower payoffs. In this type of games, the mutually beneficial, but non-Nash equilibrium strategy is called cooperation. However, in this context cooperation should not be interpreted as an interest in the welfare of others, as players only aim to secure a high payoff for themselves.In this framework, payoff maximisation is considered to be rational, but when such rational players then seize every opportunity to gain at the opponent’s expense, they may counterintuitively both end up with low payoffs. A game that clearly exhibits this contradiction is the Traveler’s Dilemma. Since its formulation in 1994 by the economist Kaushik Basu3, the game has become one of the most studied in the economics literature. Additionally, it has been discussed in theoretical biology in the context of evolutionary game theory.In general, the dilemma relies on the individuals’ incentive to undercut the opponent. To be more specific, players are motivated to claim a lower value than their opponent to reach a higher payoff at the opponent’s expense. Such incentive leads players to a systematic mutual undercutting until the lowest possible payoff is reached, which is the unique Nash equilibrium. It seems paradoxical that players defined as rational in a game theoretical sense end up with such a poor outcome. Therefore, the question that naturally arises is how can this poor outcome be prevented and how cooperation can be achieved.To address these questions, it can be helpful to better understand price wars, which consist in the mutual undercutting of prices to gain market share. In addition, it can provide information about human behaviour, because economic experiments have shown that individuals prefer to choose the cooperative high payoff action, instead of the Nash equilibrium4.Our analysis focuses on showing that the Traveler’s Dilemma can be decomposed into a local and a global game. If the payoff optimisation is constrained to the local game, then players will inevitably end up in the Nash equilibrium. However, if players escape the local maximisation and optimise their payoff for the global game, they can reach the cooperative high payoff equilibrium.Here, we show that the cooperative equilibrium can be reached in a game like the Traveler’s Dilemma due to diversity, which we define as the presence of suboptimal strategies. The appearance of strategies far from those of the residents allows for the local maximisation process to be escaped, such that an optimisation at a global level takes place. Overall this can lead to cooperation because by considering “suboptimal strategies” that play against each other it is possible to reach higher payoffs, both collectively and individually.GameThe Traveler’s Dilemma is a two-player game. Player i has to choose a claim, (n_i), from the action space, consisting of all integers on the interval [L, U], where (0 le L < U). The payoffs are determined as follows: If both players, i and j, choose the same value ((n_i = n_j)), both get paid that value. There is a reward parameter (R >1), such that if (n_i < n_j), then i receives (n_i + R) and j gets (n_i- R) Thus, the payoff of player i playing against player j is$$begin{aligned} pi _{ij} = {left{ begin{array}{ll} n_i& text { if } n_i = n_j\ n_i + R& text { if } n_i < n_j\ n_j - R& text { if } n_i > n_j end{array}right. } end{aligned}$$
    (1)
    Thus, a player is better off by choosing a slightly lower value than the opponent: when j plays (n_j), then it is best for i to play (n_j-1). The iteration of this reasoning, which we will call the stairway to hell, leads to the only Nash equilibrium of the game, ({L,L}), where both players choose the lowest possible claim. The classical game theory method to arrive to this equilibrium is called iterative elimination of dominated strategies5.The game can be visualised through its payoff matrix (Fig. 1). For simplicity, we use the values from the original formulation: (L=2), (U=100) and (R=2). The payoff matrix shows that the Traveler’s Dilemma can be decomposed into a local and a global game. Let us begin with the local game. When the action space of the game is reduced to two adjacent actions n and (n+1) (black boxes in Fig. 1), the Traveler’s Dilemma with (R=2) is equivalent to the Prisoner’s Dilemma6. In general, for any value of R, the Traveler’s Dilemma becomes a Prisoner’s Dilemma for any pair of actions n and (n+s), where ( 1 le s le R-1 ). For example, for (R=4) the pair of actions n and (n+1), n and (n+2), n and (n+3) follow the same game structure as the Prisoner’s Dilemma. Therefore, the Traveler’s Dilemma consists of many embedded Prisoners’ Dilemmas. This means that at a local level the game is a Prisoner’s Dilemma.If we now consider actions that are distant from each other in the action space, e.g. 2 and 100, we can observe a coordination game structure (gray boxes in Fig. 1), where ({100,100}) is payoff and risk dominant7,8. In general, any pair of actions n and (n+s), where ( R le s le U-n), construct a coordination game. As a result, the Traveler’s Dilemma becomes a coordination game at a global level, which has different equilibria than the local game.Figure 1Payoff matrix of the Traveler’s Dilemma. Visualisation of the payoff scheme described by Eq. (1). For simplicity, the action space is ( {n_i in {mathbb {N}} mid 2 le n_i le 100}) and the reward parameter is (R=2). The Traveler’s Dilemma can be decomposed into a local and a global game. At a local level the game is a Prisoner’s Dilemma (black boxes). This happens when the action space is reduced to any pair of actions n and (n+s), where ( 1 le s le R-1 ). While at the global level, we can observe a coordination game (gray boxes). This level is defined as any pair of actions n and (n+s), where ( R le s le U-n).Full size imageParadoxSocial dilemmas appear paradoxical in the sense that self-interested competing players, when rationally playing the Nash equilibrium, end up with a payoff that clearly goes against their self-interest. But with the Traveler’s Dilemma, the paradox goes further, as suggested in its original formulation3. Classical game theory proposes ({L,L}) as the Nash equilibrium of the game. However, it seems unlikely and implausible that, with R being moderately low, say (R=2), for individuals to play the Nash equilibrium. This has been confirmed in economic experiments where individuals rather choose values close to the upper bound of the interval. Such experiments have also shown that the chosen value depends on the reward parameter (R), where an increasing value of R shifts players’ decision towards ({L,L})4. Nonetheless, classical game theory states that the Nash equilibrium of the game is independent of R.Consequently, the aim of this paper is to seek and explore simple mechanisms through which the apparent non-rational cooperative behaviour can come about. We also examine the effect of the reward parameter on the game’s outcome. Given that the Traveler’s Dilemma paradox emerges in the classical game theory framework, we analyse the game using evolutionary game theory tools5,9,10. This dynamical approach allows us to explore adaptive behaviour outside of the stationary classical game theory framework. To be more precise, for this approach individuals dynamically adjust their actions according to their payoffs.The key point of course is to understand how the system can converge to high claims. We show that this behaviour is possible because the Traveler’s Dilemma can be decomposed into a local and a global game. If the payoff maximisation is constrained to the local level, then the stairway to hell leads the system to the Nash equilibrium; given that locally the game is a Prisoner’s Dilemma. On the other hand, at a global level the game follows a coordination game structure, where the high claim actions are payoff dominant. Thus, for the system to reach a high claim equilibrium the maximisation process needs to jump from the local to the global level.Our analysis led us to identify the mechanism of diversity as responsible for enabling this jump and preventing players from going down the stairway to hell. This mechanism works on the idea that to reach a high claim equilibrium, players have to benefit from playing a high claim. For a population setting, it means that players need to have the chance to encounter opponents also playing high. From a learning model point of view, it refers to the belief that the opponent will also play high, at least with a certain probability. If the belief is shared by both players, they should both play high and reach the cooperative equilibrium. Here, we explore these two types of models to unveil the mechanism leading to cooperation.Population based models unveil diversity as the cooperative mechanism via the effect of mutations on the game’s outcome. This is shown for the replicator-mutator equation and the Wright–Fisher model. Similarly, a two-player learning model approach, more in line with human reasoning, shows that if players are free to adopt a higher payoff action from a diverse action set during their introspection process, they can reach the cooperative equilibrium. This result is obtained using introspection dynamics.Finally, we explain how diversity is the underlying mechanism that enables the convergence to high claims in previously proposed models. To be more precise, we show that diversity is required because it allows for the maximisation process to jump from the local to the global level. More

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    Bimodality and alternative equilibria do not help explain long-term patterns in shallow lake chlorophyll-a

    Real-world dataThe dataset consisted of 2986 observations from 902 freshwater shallow lakes in Denmark and North America (data extracted from the LAGOSNE database on 22 February 2022 via R LAGOSNE package version 2.0.2)56 (Supplementary Fig. 9). The Danish lakes were sampled for one or several years from 1984 to 2020 (data extracted in October 2021 from https://odaforalle.au.dk/main.aspx) (Supplementary Fig. 10). Prerequisites for inclusion in the analysis were that lakes had been sampled for physical and chemical variables at least four times or at least three times over the growing season (May to September) for the Danish or North American lakes, respectively, had a mean depth of less than 3 m and were freshwater. Water chemistry samples were analysed using standard methods and data for total phosphorus (TP), total nitrogen (TN) and chlorophyll-a are included here57. The mean and range of TP, TN and chlorophyll-a for the combined sites is given in Table 1, along with the values for each region separately.To gain a longer-term perspective on the relationship between nutrients and chlorophyll-a, we calculated the across-year averages of the summer means of TP, TN and chlorophyll-a, sequentially increasing numbers of years included in the mean up to a total of a five-year mean, at which point there were only 99 lakes left in the dataset. In calculating the multi-year means we allowed a maximum gap of 2 years between observations (i.e. two observations could cover 3 years) to avoid including time series with too many missing years in between. Hence, only lakes with sufficient numbers of sequential data were included, resulting in a large drop in lake numbers as the length of the multi-year mean increased (Table 2).Numerical methodsDiagnostic tests or proxies of alternative equilibriaWe modelled the response of chlorophyll-a to TP and TN using generalised-linear models58 with Gamma distribution and an identity link on untransformed data for single-year and multiple-year means up to 5-year means. We used the Gamma distribution, as chlorophyll fit this significantly better than a normal or log-normal distribution. We used psuedo R2 of the model along with the patterns of residuals, and finally, we plotted the kernel density of the chlorophyll-a values as diagnostics of the presence, absence or prevalence of alternative equilibria in the simulated and real work data.To test how appropriate these diagnostics or proxies of alternative stable states in terms of how well they identify the existence of alternative stable states in randomly sampled multi-year data, we

    1.

    Simulated two scenarios for the main manuscript, with and without alternative stable states in the data, which were as close to the real-life data as possible. The results of these scenarios appear in the main text (please see details below in the “Data Simulation” section).

    2.

    We provide multiple scenarios with different degrees, or prevalence, of alternative stable states in the data, see simulations of alternative stable state scenarios. The results of these scenarios appear in Supplementary note 2.

    Hierarchical bootstrap approachThere are a large number of permutations of data, both real-world and simulated, that can provide a mean of the two to five sequential years from each lake in the time series data. It was vital to have a method that selects the data for analysis that provides a valid and comparable representation of both real work and simulated data and the models’ errors. In order to provide this we used a non-parametric hierarchical bootstrap procedure38. The flowchart shows the data preparation and data analysis steps of the hierarchical bootstrap procedure (Fig. 4). In the first step (step 1 in Fig. 4), all possible longer-term means are calculated for each lake. To keep as much data as possible, we decided to allow for up to 2 years of gap in the data between years. Taking the 5-year mean data as an example, if data from a lake existed for the years 1991 and 1994−1997, a 5-year mean would be calculated for the years 1991, 1994, 1995, 1996 and 1997. However, if the time series would contain a larger gap, e.g. data would only exist for the years 1991 and 1995–1998, no 5-year mean could be calculated. After the selection procedure, all the 2-year, 3-year and 5-year means are transferred into a new table (step 2 in Fig. 4).Fig. 4Data preparation and analysis steps of the hierarchical bootstrap procedure.Full size imageThe procedure is the same for each temporal scale from 2-year means to 5-year means. For the example of 5 mean years, lakes are randomly sampled from the full 5-year mean dataset in step 2 (Fig. 4) with replacement up to the number of lakes as in the original dataset, for the 5-year means 99 (step 3a). Here, the same lake can appear multiple times or not at all. This step is common for every bootstrap procedure59. However, since we have nested data (5-year means within lakes), we need a second step, in which for every resampled lake in step 3a, one 5-year mean is chosen (step 3b in Fig. 4). Then the three GLM models are produced from the randomly selected data in step 3c (Fig. 4). These steps are then repeated 1000 times to get a good representation of the uncertainties of the model. To ensure a fair comparison between single-year data and their equivalent multi-year mean data, we repeated the bootstrap procedure with single years only using only the lakes for which we also calculated multi-year means. To take the five-year mean as an example, there were 99 lakes where we could calculate at least one 5-year mean observation. First, we ran the bootstrap procedure to calculate 5-year mean values of TP, TN and chlorophyll-a (1000 times) and then took single years’ values of TP, TN and chlorophyll-a (1000 times) from exactly the same 99 lakes. With this approach, exactly the same datasets with the same lakes and observations within lakes are used for the calculation of the multi-year means and their single-year counterparts, making for a robust analysis. The GLM models did not always converge. If either the TP, TN or TP*TN model with interaction did not converge, the iteration was not used in further analysis. The number of converging models equal for each iteration of random samples is given in the results.The described hierarchical approach is the best way to reflect the structure of the original data. A simple, non-hierarchical bootstrap would favour lakes with more five-year means over lakes with fewer five-year means, simply because these make up a larger part of the data. Furthermore, sampling without replacement at the lake level would result in five-year means from lakes with few data dominating the produced random dataset, as every lake would be sampled every time, which then would result in high model leverage of 5-year means from lakes with less data. In contrast, the hierarchical procedure ensures that every lake has the same chance to end up in the randomly sampled bootstrap, in the second step, it ensures that of each sampled lake, every 5-year mean has the same chance to end up in the random dataset. These notions are in agreement with the findings of an assessment on how to properly resample hierarchical data by non-parametric bootstrap38.Data simulationGeneral approach of simulation assumptions and proceduresWe generated random scatter for the generalised-linear model based on Gamma distributions for two different “populations” of lakes with two different intercepts and slopes. At first, we calculated the linear equations for the two populations:For each population i and j, 99 samples (equalling the number of lakes in real-life data with 5-year means, nlake) were generated with a specific number of data points depending on the scenario (nyear) each, hence nlake = 1−99 for each population of lakes, e.g. with 20 years (nyear = 20) each.We found the real nutrient data to be normally distributed, with total nitrogen (TN) having a range between 0.33 and 4.93 mg/L and a constant coefficient of variation (CV, with a mean CV of 0.35) across this range (the same is true for total phosphorus (TP) at a shorter range). Hence, for each nlake, the x for the nyear = 20 were generated based on the mean range (mean per lake of the real-life data) and CV (0.35) from the real-life TN concentration data, hence with a range of 0.33 to 4.33 mg/L. Therefore the values and random variability of x in the simulations are close to the true values of the TN concentrations. The x is then fed into the linear equations above.To the resulting yi and yj we added random noise based on the Gamma distribution (using the rgamma function in R). We used a Gamma distribution because the Chlorophyll-a concentration also follows a Gamma distribution. The variability of a Gamma distribution is expressed by the shape variable. The variability of chlorophyll-a, its shape value, equals 2.63. This shape value was used in the Gamma distribution of yi and yj. The final calculated yi and yj had therefore a random rate calculated as shape/yi or shape/yj. Hence, their variability in the y dimension was close to the true chlorophyll-a variability.The data from both lake populations were then pooled and randomly sampled using the same hierarchical bootstrap procedure with 500 iterations for the scenarios in the supplementary materials and with 1000 iterations for main text simulation scenarios, which is identical to what was done for the real-world data.Simulation scenarios based on characteristics of real-world dataThe real-world 5-year mean data consisted of 99 lakes with 5–20 years of data for each lake. For the simulation scenario in the main text, we therefore randomly sampled between 5 and 20 data points for each of the 99 simulated lakes based on the x distribution described above. Intercepts and slopes of the simulation, resembled the range of the true data (see scatter plots in Fig. 2 of the main manuscript).In the alternative stable state scenario, we chose two slopes and intercepts for different populations of lakes:

    Population i: ai = 0, bi = 40

    Population j: aj = 50, bj = 120

    We based the slopes and intercepts of the ASS scenarios on the diagnostic combination defined by Scheffer and Carpenter7 which propose an abrupt shift in (a) the time series, (b) the multimodal distribution of states and (c) the dual relationship to a controlling factor. Here, the idea is that an ecosystem will jump from one state to the next at the same (nutrient) conditions (different intercept and/or slope, condition a within ref. 7), where any change in the nutrient will have different effects on algae or macrophytes (best represented by different slope, condition c), resulting in a multimodal distribution of the response (condition b). Hence, simulations are in line with what is predicted for ASS, but we took great lengths to also show other possibilities with the simulations in the Supplementary information to ensure we did not overlook any occasional occurrence of alternative equilibria.Here, the appearance of alternative stable states in the data could happen at any point in the time series of a single lake, or the entire time series could include only one of the two alternative stable states. To accommodate these alternative stable state constellations (for each of which we made a separate simulation scenario, (see Supplementary Note 2, “Simulations of alternative stable state constellations”), we forced the alternative stable state scenario to be constructed of 1/3 of data with one state, 1/3 of data with the second state and 1/3 of data where both alternative states could occur. In the latter case, the alternative stable state appeared after the first 20% but before the last 20% of the time series. Since the variability and range of x (nutrient) and y (chlorophyll -a response) is simulated as close as possible to the real-world data in all scenarios, the measures taken here (variable time series and combination of different alternative stable state scenario constellations) produce a simulation as close to the real-world data as possible. Specifically, we found the real-world nutrient data to be normally distributed, with total nitrogen (TN) having a range between 0.33 and 4.93 mg/L and a constant coefficient of variation (CV, with a mean CV of 0.35) across this range (the same is true for total phosphorus (TP) at a shorter range). Hence, for each simulated lake, the x were generated based on this mean range and CV. Furthermore, the resulting yi and yj were randomised by using a Gamma distribution (using the rgamma function in R). We used a Gamma distribution because the chlorophyll-a concentration also follows a Gamma distribution. The variability of a Gamma distribution is expressed by the shape variable. The variability of chlorophyll-a, its shape value, equals 2.63. This shape value was used in the Gamma distribution of yi and y. The final calculated yi and yj had, therefore a random rate calculated as shape/yi or shape/y. Hence, their variability in the y dimension was close to the true chlorophyll-a variability.For the scenario without alternative stable states, both populations of data had the same intercept and slope:

    Population i: ai = 0, bi = 40

    Population j: aj = 0, bj = 40.

    Please see Supplementary Note 2 for further simulations of different potential constellations of alternative states. There we show that our approach finds alternative stable states in response to nutrient concentration, even if they appear in time series from different lakes.Assessment of diagnostic tests or proxies of alternative equilibriaWe modelled the response of chlorophyll-a to TP and TN using generalised-linear models3 with Gamma distribution and an identity link on untransformed data for single-year and multiple-year means up to 5-year means. We used the Gamma distribution, as chlorophyll fit this significantly better than a normal or log-normal distribution. We used R2 of the model along with the patterns of residuals, and finally, we plotted the kernel density of the chlorophyll-a values as diagnostics of the presence, absence or prevalence of alternative equilibria in the simulated and real work data.The comparison of how the diagnostics/proxies of alternative stable states respond to the variation in the prevalence of alternative equilibria in the simulated datasets provides a robust assessment of their ability to identify both the presence and absence of alternative equilibria. It is the response of these diagnostic tests over time, with the increase in the temporal perspective as more years are added to the mean values of TP, TN and chlorophyll-a, that are key to the identification of the presence and or absence of alternative equilibria in a given dataset. The simulations show that a dataset which contains alternative equilibria will show (1) no improvement in R2 as the temporal perspective of the data increases (more years in the multi-year mean); (2) an increased bimodality in the residuals of the models of nutrients predicting chlorophyll-a will increase as more years are added to the multi-year mean and (3) the kernel density function of chlorophyll-a will display increasingly bimodality as more years are added to the mean. In the absence of alternative equilibria, the patterns differ with an R2, and increase in unimodality of residuals and a consistent unimodal pattern in the kernel density function. Thus, the diagnostic tests provide a robust test of both the presence and absence of alternative equilibria in a given dataset.Alternative stable state assessment for real data with limited data rangeIt could be the case that alternative stable states do not appear in the full dataset but only in a limited TN and TP concentration range. We filtered and re-analyzed the data, only keeping data points within the following two ranges: – TN concentration = 0.5−2 mg/L–TP concentration = 0.05−0.4 mg/L. In the filtered data, 1329 out of the original 2876 single-year data points, 289 out of 1028 3-year mean data points and 212 out of the 864 five-mean year data points remained. The filtered data consisted of data points from 550, 48 and 27 lakes for the single-year data, 3-year means and 5-year means, respectively. The smaller range resulted in lower R² of the models, yet the pattern that multi-year means result in higher R² compared to single-year data was largely consistent, apart from the 5-year mean TN models for which both, the single-year and mean data resulted in very low R² (Supplementary Fig. 6). Furthermore, due to the lower number of samples, the errors of all proxies are higher, making conclusions more difficult than for the full data. Still, we do not see any clear indication of alternative stable states in the scatter plots (two groups of dots are not appearing (Supplementary Fig. 5), the Kernel density plots (or model residuals (Supplementary Fig. 6)). i.e. no signs of bimodality in residuals or Kernel density plots. Please see details on this analysis in the supplementary material.Details and the R code for the steps for the random multi-year sampling can be found in the supplementary materials.Reporting summaryFurther information on research design is available in the Nature Portfolio Reporting Summary linked to this article. More