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Detecting alternative attractors in ecosystem dynamicsWe use empirical dynamical modeling, a set of equation-free tools for analyzing non-linear time series (for a review and assumptions see25,26, respectively), to test if the temporal dynamics of alternative dynamical regimes are qualitatively different. Empirical dynamic modeling builds fundamentally on Takens embedding theorem, which shows that attractors of multi-dimensional dynamical systems can be reconstructed using higher order lags of its embedded time series27. However, if a dynamical system has gone through a bifurcation, or switched to an alternative basin of attraction, attractors are qualitative dissimilar in the two regimes. Theoretically, this infers that it should be possible to reconstruct the attractor of one regime using information from the same regime, but not from the other regime. In practice, this implies that if a model (attractor reconstruction) based on one dynamical regime is used to predict the dynamics of variables from the same dynamical regime predictions should be accurate (i.e. low prediction errors), whereas if an attractor reconstruction based on one dynamical regime is used to predict the dynamics of variables of another attractor predictions should be less accurate (i.e. high prediction errors). We make use of this idea by specifically testing if prediction errors of across and within regime predictions are different. As explained below this idea can be used for both univariate and multivariate time series data.Univariate approachUnivariate attractor reconstructions can be found using the simplex algorithm28,29. First, for a given dynamical regime, a time series can be split into a library of vectors, and each vector is described by$${underline{y}}_{A}(t)= < {Y}_{A}(t),{Y}_{A}(t-1),{Y}_{A}(t-2),ldots ,{Y}_{A}(t-(E-1)) > ,$$
(1)
where ({Y}_{A}(t)) is an observation of variable Y at time t in dynamical regime A and E is the reconstructed attractors embedding dimension. Using the simplex projection algorithm, a one-step ahead forecast is produced as follows:$${hat{Y}}_{A}(t+1)|{{{{{{boldsymbol{M}}}}}}}_{B}=mathop{sum}limits_{m=1ldots E+1}{w}_{m}{Y}_{B}({t}_{m}+1),$$
(2)
where tm is a time index of an observation in dynamical regime B, E is the embedding dimension of regime B, and wm is an exponential weighting described by:$${w}_{m}={u}_{m}/mathop{sum}limits_{n=1,ldots ,E+1}{u}_{n},$$
(3)
where n and m belongs to the set of the E+1 nearest neighbors of vector ({underline{y}}_{A}(t)) in the set of vectors ({{underline{y}}_{B}({t}_{m})}), ({u}_{m}=exp {-d[{underline{y}}_{A}(t),{underline{y}}_{B}({t}_{m})]/d[{underline{y}}_{A}(t),{underline{y}}_{B}({t}_{1})]}), and (d[{underline{y}}_{A}(t),{underline{y}}_{B}({t}_{1})],)is the Euclidean distance between the prediction vector ({underline{y}}_{A}(t)) and its nearest neighbor ({underline{y}}_{B}({t}_{1})) in the set ({{underline{y}}_{B}({t}_{m})}).The only parameter that is estimated using the simplex algorithm is the embedding dimension E. This parameter is estimated by optimizing the correlation between observations (({Y}_{A}(t+1))) and predictions (({hat{Y}}_{A}(t+1)|{{{{{{boldsymbol{M}}}}}}}_{A})) using a leave-one-out cross validation approach (See Supplementary Discussion). The embedding dimension E and its corresponding set of E-dimensional vectors (Eq. 1) constitutes the reconstructed attractor, MA, of a given dynamical regime A. This reconstructed attractor (MA) is then used to predict data for both the same dynamical regime (({hat{Y}}_{A}(t+1)|{{{{{{boldsymbol{M}}}}}}}_{A})), and the contrasting dynamical regime ({hat{Y}}_{B}(t+1)|{{{{{{boldsymbol{M}}}}}}}_{A}). Likewise, the reconstructed attractor MB can be used to predict time series dynamics from both dynamical regimes; that is, ({hat{Y}}_{A}(t+1)|{{{{{{boldsymbol{M}}}}}}}_{B}) and ({hat{Y}}_{B}(t+1)|{{{{{{boldsymbol{M}}}}}}}_{B}), respectively.Multivariate approachA multivariate time series describes a number of simultaneously evolving variables. For example, a bivariate time series can be described by variables X and Y. For such time series, Sugihara et al.30 developed an approach for testing if two variables (time series) are dynamically coupled. Their methodology builds on the fact that a reconstructed attractor should map 1:1 to the original attractor on which the reconstruction is based. This infers that two attractor reconstructions (based on two different variables) should also map 1:1 to each other30. Practically, this means that if two variables are dynamically coupled one-time series should be predictable based on an attractor reconstruction of another variable. However, if a dynamical system has gone through a bifurcation, or potentially switched to an alternative basin of attraction, a new set of rules will govern the dynamics of the system. Hence, a new attractor should have emerged. Now, since this new attractor is most likely governed by a new set of rules it should be difficult to predict the dynamics of this new alternative attractor based on information from the former attractor. Thus, if one variable in one dynamical regime is used to predict another variable in another dynamical regime, predictions should be biased. Yet, if one variable from one dynamical regime is used to predict another variable from the same regime predictions should be more accurate.The simplex algorithm can be used to make predictions of a variable Y using a time series of another variable X30. Predictions are produced as follows:$${hat{Y}}_{{{{{{boldsymbol{A}}}}}}}(t)|{{{{{{boldsymbol{M}}}}}}}_{B}=mathop{sum}limits_{m=1ldots E+1}{w}_{m}{Y}_{B}({t}_{m}),$$
(4)
where tm is the time series index of a vector of variable X of dynamical regime B, wm is an exponential weighting based on variable X:$${w}_{m}={u}_{m}/mathop{sum}limits_{n=1,ldots ,E+1}{u}_{n},$$
(5)
where n and m belongs to the set of the E+1 nearest neighbors of ({underline{x}}_{A}(t)) in ({{underline{x}}_{B}({t}_{m})}), ({u}_{m}=exp {-d[{underline{x}}_{A}(t),{underline{x}}_{B}({t}_{m})]/d[{underline{x}}_{A}(t),{underline{x}}_{B}({t}_{1})]}), and (d[{underline{x}}_{A}(t),{underline{x}}_{B}({t}_{1})],)is the Euclidean distance between the prediction vector(,{underline{x}}_{A}(t)) and its nearest neighbor ({underline{x}}_{B}({t}_{1})) in dynamical regime (B).The reconstructed attractors, MA and MB, for each variable and regime are found using the univariate simplex algorithm described above28,29,30. Similar to the univariate case, the reconstructed attractor (MA) is used to predict data from the same dynamical regime (({hat{Y}}_{{{{{{boldsymbol{A}}}}}}}(t)|{{{{{{boldsymbol{M}}}}}}}_{A})), and to predict time series of a contrasting dynamical regime (({hat{Y}}_{{{{{{boldsymbol{A}}}}}}}(t)|{{{{{{boldsymbol{M}}}}}}}_{B})). Yet, it is important to stress that MA here reflects an attractor reconstruction based on a variable that is not being predicted (that is, variable X is used to predict variable Y). This prediction approach thus infers that predictions are made on data that was not used to fit the model (X predicts Y and vice versa). Thus, neither across nor within regime predictions are made on data used to fit a model.Test statisticWe used mean absolute prediction errors to test for difference between across and within regime predictions. Alternative metrics, such as mean sum of square errors, can also be used. However, since our approach gives skewed prediction errors we used mean absolute prediction errors to reduce the impact of extreme values. Further, since the absolute prediction errors are non-normally distributed we used a permutation test. The null hypothesis that is tested reads:$$H0:{{{{{rm{MAP{E}}}}}}}_{A} < {{{{{rm{MAP{E}}}}}}}_{w},$$ (6) where MAPEA is the mean absolute prediction error for across regime predictions (that is, ({{{{{rm{MAP{E}}}}}}}_{A}=frac{1}{n}mathop{sum}limits_{t=1:n}{{{{{rm{abs}}}}}}({hat{Y}}_{{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{A}}}}}}}}(t)|{{{{{{boldsymbol{M}}}}}}}_{B}-{Y}_{{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{A}}}}}}}}(t))), and ({{{{{rm{MAP{E}}}}}}}_{w}) is the mean absolute prediction error for within regime predictions (that is, ({{{{{rm{MAP{E}}}}}}}_{w}=frac{1}{n}mathop{sum}limits_{t=1:n}{{{{{rm{abs}}}}}}({hat{Y}}_{{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{A}}}}}}}}(t)|{{{{{{boldsymbol{M}}}}}}}_{A}-{Y}_{{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{A}}}}}}}}(t))). A test is consider significant if observed difference in across and within regime mean prediction errors is larger than the 95th percentile of 1000 permuted data sets.Food-chain modelWe used a food-chain model parameterized as in McCann and Yodzis31 to simulate food-chain dynamics:$$frac{{{{{{rm{d}}}}}}R}{{{{{{rm{d}}}}}}t}=Rleft(1-frac{R}{K}right)-frac{{x}_{c}{y}_{c}CR}{R+{R}_{0}}$$ (7) $$frac{{{{{{rm{d}}}}}}C}{{{{{{rm{d}}}}}}t}={x}_{c}Cleft(-1+frac{{y}_{C}R}{R+{R}_{0}}right)-frac{{x}_{P}{y}_{P}PC}{C+{C}_{0}}$$$$,frac{{{{{{rm{d}}}}}}P}{{{{{{rm{d}}}}}}t}={x}_{P}Pleft(-1+frac{{y}_{P}C}{C+{C}_{0}}right),$$where R is the resource density, C consumer density, and P predator density. All parameters, except half-saturation constants R0 (here set to 0.16129) and C0 (here set to 0.5), and resource carrying capacity K, are derived from bioenergetics and body size allometry30 (xc = 0.4, yc = 2.009, yp = 2.876, R0, r = 1, xp = 0.08).This model can display a rich set of dynamics depending on parameter values31. Here we alter resource carrying capacity K in order to simulate the dynamics (using the deSolve package32 in R) of qualitatively different attractors (See Supplementary Fig. 1; K = 0.78, equilibrium; K = 0.85; two-point limit cycle; K = 0.92, four-point limit cycle; K = 0.997, chaotic dynamics). Every fifth time step of the simulated dynamics, corresponding to a sampling frequency of ≈10 samples per cycle for the 2-point limit cycle, was sampled. Observation noise was thereafter added to the deterministic dynamics produced by the model:$${N}_{l}(t)={N}_{l}^{prime}(t)+rho * e(t);e(t) sim N(0,{sigma }_{N^{prime_{l}}}),$$ (8) where (N_{l}^{prime}(t)) is the abundance of species l (P, C or R) simulated by the food-chain model at time point t, (rho) is the level of observation noise and ({sigma }_{N_{l}^{prime}}) is the standard deviation of the deterministic dynamics of species l produced by the food chain model.In order to investigate how time series length and observation noise affects the probability of detecting alternative attractors we derived probability landscapes. These were derived by testing the null-hypothesis (H0:(|{hat{{{{{{boldsymbol{Y}}}}}}}}_{{{{{{boldsymbol{i}}}}}}}|{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{i}}}}}}}-{{{{{{boldsymbol{Y}}}}}}}_{{{{{{boldsymbol{i}}}}}}}| > |{hat{{{{{{boldsymbol{Y}}}}}}}}_{{{{{{boldsymbol{i}}}}}}}|{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{j}}}}}}}-{{{{{{boldsymbol{Y}}}}}}}_{{{{{{boldsymbol{i}}}}}}}|); See Test statistic above) across 100 replicates for each combination of time series length and level of observation noise, (rho). Time-series length was varied from 10 to 100 in steps of 10, and observation noise, (rho), was varied from 0.01 to 0.3 in steps of 0.01, in total yielding 300 combinations of observation noise and time series length, for each combination of dynamical regimes i and j. Predator dynamics was used to predict consumer and resource dynamics using the multivariate approach described above (results for the cases where consumer or resource dynamics are used to predict the other species´ dynamics are presented in Supplementary Figs. 2, 3). All time series were standardized ((mu =0;sd=1)) prior testing for dynamical difference.Experimental data setThe experimental data set was given by Fussman et al.7. This data set contains 14 time series of a predator Brachionus calyciflorus and its prey Chlorella vulgaris derived from chemostat experiments. Time series for different dilution rates were produced by keeping the dilution rate fixed in different chemostats (Supplementary Figs. 3–11). Brachionus calyciflorus and Chlorella vulgaris time series were used to predict Chlorella vulgaris and Brachionus calyciflorus time series, respectively, using the multivariate approach described above. We tested for qualitative difference in the temporal dynamics across all time series, which were standardized ((mu =0;sd=1)) prior testing.Alternative stable state modelWe used a stochastic version of a well-known alternative stable state model4,33 to produce alternative stochastic dynamical regimes. The model is described by:$${{{{{rm{d}}}}}}x=left(xleft(1-frac{x}{{{{{{rm{K}}}}}}}right)+frac{c{x}^{2}}{1-{x}^{2}}right){{{{{rm{d}}}}}}t+sigma {{{{{rm{d}}}}}}w,$$
(9)
where K is the carrying capacity (here set to 11), c is a harvest rate, and σ (here set to 0.01) is the magnitude of noise which is described by a Wiener process (dw).The model was simulated for fixed harvest rates (c) assuming that the system state resides in either of its two basins of attraction. The initial value for the simulation was set to the equilibrium of the noise-free model skeleton for fixed harvest rates c, and σ is set low in order to avoid stochastic flips, so-called flickering, between alternative basins of attraction. Dynamics was integrated (Δt = 0.01) using the matlab-package SDE-Tools34.In order to investigate how time-series length and harvest rate, c, affects the probability of detecting alternative attractors in stochastic regimes we derived probability landscapes.These were derived by testing the null-hypothesis H0:(|{hat{{{{{{boldsymbol{Y}}}}}}}}_{{{{{{boldsymbol{i}}}}}}}|{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{i}}}}}}}-{{{{{{boldsymbol{Y}}}}}}}_{{{{{{boldsymbol{i}}}}}}}| > |{hat{{{{{{boldsymbol{Y}}}}}}}}_{{{{{{boldsymbol{i}}}}}}}|{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{j}}}}}}}-{{{{{{boldsymbol{Y}}}}}}}_{{{{{{boldsymbol{i}}}}}}}|) (permutation test p = 0.05) across 100 simulated data sets for each combination of time series length and harvest rate, c. Time-series length was varied between 50 and 150 in steps of 10, and c was varied between 1.83 and 2.73 in steps of 0.05, in total yielding 209 combinations of time series length and harvest rate. Each time series was standardized ((mu =0;sd=1)) prior testing for difference in temporal dynamics of contrasting regimes.Natural time-series dataIn a previous study on early warning signals of impending regime shifts, Gsell et al.18 used breakpoint analysis to identify two potential alternative dynamical regimes. We here test if these two-time series segments constitute alternative dynamical attractors. Prior analysis, we imputed a few missing observations (n = 24) using a kalman smoother35. The two time series segments, i.e. pre- and post-breakpoint time series, were standardized ((mu =0;sd=1)) prior testing for dynamical difference.Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

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When introducing ARs as a fisheries management tool to Senegal, the Japanese management had the mindset of Japanese stakeholders, i.e., introducing fishing rights. However, after discussions with Senegalese stakeholders, it was decided that no-take areas would be delineated around ARs because the establishment of a strong fishing rights regime was not socially acceptable to the Senegalese fishing community. Japanese governance is based on the acceptance and respect of fishers towards individual, private AR concessions. In contrast, fishers in Senegal, and more widely in West Africa, are characterized by high mobility, particularly in the context of climate change and overexploitation18,19. Consequently, respect for local management regulations is lower, with open access being generally assumed. The basic concept of implementing a no-take area on the AR was not easily accepted by fishers. The immersion of AR concrete blocks was set as a top priority by managers at the expense of more complex socio-economic considerations, such as consciousness-raising activities and self-sustaining participative monitoring of the AR.The clear contradiction between the ecological knowledge of fishers and their behavior was explained by the well-known effects of open access resources on individual behavior. This phenomenon was also observed in our mathematical model. The processes in the mathematical model are in accordance with those perceived by the fishers, so that the results are also those expected by fisher’s local ecological knowledge. It is interesting to notice that the theoretical results presented here are the mathematical solutions of the model at equilibrium between fishing effort and fish population growth, i.e. after an oscillation period. It is obvious that short-term effect of fishing on the AR is always to increase the catch, but many fishers did perceive the longer-term effect of decreasing catches. The potential negative effect of the AR on catch when there is high fish attraction combined with high fishing pressure on the AR might explain the reluctance of a part of the fishers community to AR deployment (Fig. 2). In particular, the model illustrates that the AR attraction effect strongly determines the impact of the management. In general, fish attraction is the most immediate effect perceived after AR deployment11, as was true for our study16. Though the AR volume was relatively small (70 m3), the empty space between the higher blocks also contributes approximately 280 to 570 m3 of good habitat/refuge for schooling fish; therefore, it is actually difficult to accurately describe the volume that affects fish. Thus, it is difficult to say whether this AR is below or above the forecasted optimal volume in absence of fishing (120m3 with model parameters). The existence of an optimal volume for AR was also suggested by field studies as a trade off between food supply and refuge20, in line with our results. For management purposes, it is interesting to determine whether the AR is above or below this optimal level because if the volume is too small, the model predicted that any level of fishing on the AR would, in the long term, decrease the catch in the considered area. On the other hand, if the volume is above the optimal level, a small fishing effort on the AR could be authorized and would increase the total catch in the area.Field observation showed that the fish attraction effect was strong16 but precise estimation of this parameter cannot be inferred, as this would need, ideally, individual fish trajectories. Future field research on the attraction effect may permit estimating the AR attraction parameters. The model sensitivity test showed that the stronger the attraction parameter, the better the impact of the AR for the fisheries in case of no or small fishing effort on the AR (Fig. 3). But at the same time, the attraction is a strong incentive for fishers to fish on the AR, and the predicted benefit for fisheries in the fishing area rapidly vanishes when fishing effort on AR increases. This in turn provides further incentive for fishers to fish the AR, challenging the surveillance capacity. If fish attractiveness is strong and too many fishers fish on the AR, catch in the area will be concentrated on the AR, while the adjacent fishing area will be depleted, with catch levels lower than those prior to AR deployment.Specifically, in the context of generalized overfishing in Senegal21, deciding not to fish on the AR represents significant individual loss, despite being recognized as beneficial, globally22. It has been argued that this situation would rarely occur in small-scale fisheries, due to existing arrangements between individuals23. However, in the context of the highly mobile Senegalese artisanal fishing fleet and its overcapacity, as soon as the AR in Yenne was no longer subject to surveillance, it rapidly attracted fishers from other villages. Also, pre-existing arrangements between fishers might be overruled when new ARs are created, changing the structure of existing fishing grounds.At the time of the survey, the surveillance system set up by the co-management entities was not operational in our case study, because it was dependent on temporally limited external financing. These limitations are typical of short-term projects that focus on a single restricted area for a pre-determined duration, usually up to two years (e.g., NGOs, World Bank). Local fishers perceptions were globally in line with the model prediction that this AR fails to improve fisheries yield when surveillance is not in place to ensure AR regulations are observed, despite effective fish attraction and production existing in the AR.The model predicted that enhanced production on ARs could not keep pace with unrestricted access, which might be particularly true in Senegal where fishing effort rapidly reorganizes itself according to local yields24. Enhanced production due to the AR largely increases the catch if the fishing pressure on the AR remains null or very low, but it has no effect on the catch for higher fishing pressures on the AR (Fig. 3). These results were stable even if fish population growth, fish catchability, mobility and economic parameters could modulate the predicted amplitude of the catch and AR optimal volume. These results are consistent with existing theoretical studies of the impact of fisher movement to high production areas in and around MPAs25. Taking into account several species and their interactions (predation, competition) would lead to a very complex ecosystem model specific to the area (e.g. 26), with necessarily more assumptions. This model would necessarily be more difficult to share with fishers and other stakeholders. Both to simplify model structure and facilitate communication of results to stakeholders, we assumed in our model that the balance of entries exits and is in equilibrium, so that the migratory species did not affect the long-term equilibrium between fishing effort and fish abundance.The design of ARs could be adjusted to reduce the effect of illegal fishing by passively preventing both industrial and artisanal fishing activity. Complex structures are more effective for fish production and attraction27. We showed that, although production might have a limited effect on total catch, attraction can largely increase AR efficiency (total catch) if the rate of illegal fishing rate is very low or absent. Complex structures protect fish more effectively from small scale fishing gear28, including divers (Pers. Comm., Mamadou Sarr, Ouakam fishers committee). Thus, ARs should be appropriately designed to help mitigate potential issues28. Such designs might be more costly, and do not exclude the need for surveillance, but would enhance fisheries management, especially when surveillance cannot capture low levels of illegal fishing.Finally, if socio-economic and governance conditions are not met, well-intentioned AR projects will likely disturb the existing equilibrium among fishers that have different levels of access to the AR. Poor governance of marine resources has previously been described in West Africa, particularly in Senegal29, as has the failure of AR projects in a number of other developing countries9, which further deteriorate fishers trust and management plans efficiency30. In order to avoid that, NGO and governmental agencies driving ARs projects must consider that AR management induces collective costs before providing potentially collective gains. Thus, co-management that involves governmental institutions and fisher communities is required. Future management and adaptation plans for fishers, particularly in developing countries, should, therefore, focus efforts on raising long-term awareness of actors in both government institutions and fishing communities. At the level of institutional or development partners, long-term management costs should be included in the set-up of AR projects. For example, the local fishers committee of Yenne recently reported the establishment of a collective ship chandler whose profits are used to finance AR surveillance during the daytime. Subsequently, fishers noted an improvement in catches around the AR, even though illegal fishing likely continues on the AR at night (Pers. Comm. chair of local fishers committee). These observations support model predictions that low levels of illegal fishing might not disturb the positive impact of the AR. Alternatively surveillance effort could be supported by the community if benefits were managed according to ancestral traditions. Indeed, “no take area” regime on the AR would be in line with some past West African tribal laws, applied before the colonization era, which set marine area where fishing activities were restricted for occasional community celebrations. Collective processes where fishers and other stakeholders can design temporary no-take zones around the AR could increase fishers trust and compliance to the rules, fostering a positive socio-ecological feedback loop30.Hybridization of local and scientific knowledge, through the integration of natural sciences and social sciences, is key point for governance setting31,32,33. Indeed, the communication of the resulting hybrid knowledge in specific events gathering local stakeholders helps strengthen fisheries co-management for the establishment of surveillance and regulatory frameworks. This phenomenon was experienced during the public restitution of the present study with the community, fishers, children’s from local schools and governmental stakeholders. Science popularization of the study results was in French and local language (Wolof) retransmitted on national news (available at https://www.youtube.com/watch?v=yQqFU2P4XZU). Posters were exposed during the event, including pictures of local fishers interviewed and statements reflecting their own perception of how the artificial reef interacts with ecological processes and fisheries dynamics. Straightaway, stakeholders and local promoters of AR publicly expressed their concern and willingness to prioritize the setting up an efficient AR surveillance independent from external resources prior to increase AR deployments. Knowledge hybridization could produce more specific models that could be used for warning and advice, for example by considering potential impacts of ARs on species compositions3,34,35, environmental parameters36, and cascade effects on the trophic food web37. However this approach would need to be adapted to local social-ecological governance, which might require dedicated political-anthropological studies (see concept of adaptive co-management32).In summary, best practices should involve all stakeholders, consider local specificities, such as site configuration, governance, ecosystem, availability of ad hoc human and financial resources for AR surveillance, and define AR volume and design accordingly to these parameters. Thus, if plans exist to deploy ARs at large scales we recommend that legislation is strengthened, with detailed Environmental and social Impact Assessments38 to implement ARs, including considerations of long-term governance. More