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We consider a two-layer network. One layer is the slave layer, which corresponds to the original network over which one wants to impose the desired dynamics (i.e. a given evolution compatible with the equations of motion). The other layer is the master layer, which is identical to the slave layer, but starts from a different initial condition (i.e. the one generating the specific desired dynamics towards which the state of the slave layer is to be steered), and evolves autonomously. In applications, the master layer may just be an experimental recording or a simulation of the original system—as long as it can be coupled to the slave network its physical nature is irrelevant. Our control method consists then in establishing directed inter-layer links from nodes in the master layer to their counterparts in the slave layer. Once they are established, these links remain in place as more nodes are connected in sequential control steps. At each step the selected node is the one whose pinning causes the most rapid approach towards inter-layer synchronization (i.e. the imposition of the evolution followed by the master layer on the slave layer). While the two layers have to be identical, the nodes (i.e. the dynamical units) and links (the coupling structure connecting the dynamical systems) on each layer can be completely different, as we will see below. This is thus a generalization of the method proposed in Ref.6.
We illustrate our method by applying it to networks of identical chaotic oscillators, and leave the applicability to more challenging real-world systems to the next section. Specifically, we consider networks of (N=50) nodes whose topology is that of a mixed random graph, i.e. containing both bidirectional and unidirectional links. These graphs are realizations of the configuration model22 with the in-degree (k_text {in}) (i.e. the number of links pointing to a given node) and the out-degree (k_text {out}) (i.e. the number of links emanating from a given node) uniformly distributed in ({5,6,ldots ,45}). Each node evolves autonomously in time as a chaotic Rössler oscillator, which we simply denote as ({dot{mathbf{r}}} = mathbf{f}(mathbf{r})), where (mathbf{r} = (x,y,z)^text {T}) and ({dot{x}} = -y – z, {dot{y}} = x + a y, {dot{z}} = b + z (x – c)), with parameters (a=0.2), (b=0.2) and (c=7). Nodes are coupled quadratically via their z variables, a nonlinear coupling form that was previously considered in Ref.23.
Before the first control step is applied (prior to the creation of the first inter-layer connection) both master and slave layers evolve spontaneously as follows

$$begin{aligned} {dot{mathbf{r}}}_i = mathbf{f}(mathbf{r}_i) + sigma _1 sum _{j=1}^N D_{ji} (z_j^2-z_i^2) = mathbf{f}(mathbf{r}_i) + sigma _1 sum _{j=1}^N {mathcal {L}}_{ji} z_j^2. end{aligned}$$
(1)

where (D_{ji} = 1) if there is a directed link from node j to node i, and is zero otherwise (for bidirectional links (D_{ij} = D_{ji})). As we do not consider self-links, the diagonal terms vanish, i.e. (D_{ii} = 0 forall , i), and the in-degree of node i is (k_{text {in},i} = sum _{j} D_{ji}). The graph can thus be alternatively represented by the Laplacian matrix ({mathcal {L}}_{ji} =D_{ji} – k_{text {in},i} delta _{ji}). The vector field (mathbf{f}(mathbf{r}_i)) governs the dynamics of node i, which would evolve autonomously (if uncoupled from its neighbors) simply as ({dot{mathbf{r}}}_i = mathbf{f}(mathbf{r}_i)), and the parameter (sigma _1) is the intra-layer coupling strength
When the control procedure starts, each node i in the master network keeps evolving according to the dynamics in Eq. (1), ({dot{mathbf{r}}}^M_i = mathbf{f}(mathbf{r}^M_i) + sigma _1 sum _{j} {mathcal {L}}_{ji} (z^M_j)^2). In the slave layer dynamics, however, one has to consider an additional term which accounts for the inter-layer coupling from the master layer (without loss of generality, we here take a linear coupling through the y variable). One has

$$begin{aligned} {dot{mathbf{r}}}^S_i = mathbf{f}(mathbf{r}^S_i) + sigma _1 sum _{j} {mathcal {L}}_{ji} (z^S_j)^2 + sigma _2 chi _i (y^M_i – y^S_i). end{aligned}$$
(2)

Here (chi _i) is a binary variable that is one if there is a link coupling node i in the master layer to node i in the slave layer (i.e. if the targeting procedure includes a pinning action from master to slave at node i) and is zero otherwise. The parameter (sigma _2) is the inter-layer coupling strength. We emphasize that the coupling that is linear (in fact, diffusive) is the externally-imposed inter-layer coupling, which does not restrict in any way the form of the (intra-layer) couplings between the nodes of the system under study. Such diffusive inter-layer coupling is chosen as it is the simplest form that makes the inter-layer synchronization manifold into an invariant set of the dynamics (for a detailed mathematical treatment of invariant sets and related concepts, see e.g. Ref.24).
Figure 1

Controlling the dynamics of a mixed network with uniform (k_text {in}) and (k_text {out}) distributions comprising (N=50) nonlinearly-coupled Rössler oscillators with intra-layer coupling (sigma _1 = 0.01) and inter-layer coupling (sigma _2 = 1). (Top) Maximum Lyapunov exponent (lambda _text {max}) (main panel) and synchronization error (inset) as functions of the targeting step. (Bottom) Influence index (k_text {out}/k_text {in}) of the node that is pinned at each targeting step. The curves are averages of 20 different network realizations. A 4th-order Runge-Kutta method with a step of 0.01 time units has been employed for the numerical integration of the systems of (3 N=150) ordinary differential equations corresponding to each layer.

Full size image

The results of applying our method to the network of Rössler chaotic oscillators are shown in Fig. 1. Two observables are employed to characterize the inter-layer synchronization between master and slave as more and more inter-layer links are established in successive targeting steps. One is the maximum Lyapunov exponent, (lambda _text {max}), computed from the dynamics of the slave network linearized around that of the master network as in Ref.6. For a review of the theory and numerical computation of Lyapunov spectra, see e.g. Refs.25,26. The other observable is the synchronization error, which is the time average of the Euclidean distance in phase space ({mathbb {R}}^{N m}) (N is the number of nodes, m is the phase space dimensionality of the node dynamics—in our case (N=50), (m = 3)) between the full state of the master layer and that of the slave layer, (lim _{Trightarrow infty } frac{1}{T} int _0^T sqrt{ sum _{i=1}^{N} (x^M_i(t) – x^S_i(t))^2 } dt). In practice, T is finite, but orders of magnitude larger than the characteristic timescales of oscillation (thus the numerical convergence to an asymptotic value is guaranteed). In the top panel of Fig. 1, we show the maximum Lyapunov exponent (lambda _text {max}) as a function of the targeting step, which is seen to progressively decrease as more and more nodes are pinned. Analogous results in terms of the synchronization error are reported in the inset, which shows how the synchronization error becomes zero when (lambda _text {max}) becomes negative.
The maximum Lyapunov exponent is also used to identify the node to be targeted at each step: of all the nodes that remain unconnected to their counterparts in the other layer, the one that, when a master-slave connection is established, leads to the largest decrease in (lambda _text {max}) is targeted next. An exploration of possible correlations between the resulting targeting sequence (i.e. the ordered list of nodes that are targeted at successive steps) and local topological properties yields a remarkable correspondence between the targeting sequence position and the ranking of nodes in terms of their influence index (k_text {out}/k_text {in}), as shown in the lower panel of Fig. 1. This index is large when a node has a privileged position for influencing other nodes, while receiving very little influence from the rest of the network. No such correlations are observed for connectivity indices that are insensitive to the directionality of connections, such as ((k_text {in}+k_text {out})/2), while correlations only based on (k_text {out}) or (k_text {in}) give considerably poorer results that those shown in the figure. Other measures of connection directionality that we have inspected, such as ((k_text {out} – k_text {in})/(k_text {out} + k_text {in})), show weaker correlations with the targeting sequence than the influence index does. While these results are based on networks with uniform distributions of (k_text {in}) and (k_text {out}), which have been chosen precisely because a large variety of possible degree values is desirable, a strong correlation between the influence-index ranking and the targeting ranking is also observed for Barabási-Albert scale-free networks27 and Erdös-Rényi random graph28 topologies, as shown in Sect. A of the Supplementary Information.
This correlation is most clearly seen for small values of the intra-layer coupling strength, such as the value (sigma _1 = 0.01) considered in Fig. 1. For larger values of (sigma _1), which make inter-layer synchronization possible with a very small number of steps, the correlation is less strong, while no obvious correlation between the targeting sequence and local topological properties are found for very large (sigma _1), see again Sect. A of the Supplementary Information. This might be related to the enhanced contribution of next-nearest neighbors and other relative distant nodes as the coupling strength is increased. Despite its limited range of validity, this correlation is nontheless remarkable, as it is very robust, and quite different from the situation observed in undirected networks, where the topological observable correlating with the targeting sequence is the degree6. On the other hand, there is an intriguing parallel between the correlation reported in the lower panel of Fig. 1 and the fact that, in undirected networks, nodes with a higher dynamic vulnerability are those with less influence from the rest of the network, followed by those that have the strongest ability to influence the rest29. In fact, both aspects of a node position are combined in the influence index in the case of directed or mixed networks. More

A waterborne, abiotic, and other indirectly transmitted (WAIT) model for the dynamics of emergent viral outbreaks
Several models have been engineered to explore aspects of COVID-19 dynamics. For example, models have been used to investigate the role of social distancing20,42, social mixing43, the importance of undocumented infections44, the role of mobility in the early spread of disease in China45, and the potential for contact tracing as a solution46. Only a few notable models of SARS-CoV-2 transmission incorporate features of indirect or environmental transmission40,46 and none consider the dynamical properties of viral free-living survival in the environment. Such a model structure would provide an avenue towards exploring how variation in free-living survival influences disease outbreaks. Indirect transmission includes those routes where pathogen is spread through means other than from person to person, and includes transmission through environmental reservoirs. Environmental transmission models are aplenty in the literature and serve as a theoretical foundation for exploring similar concepts in newer, emerging viruses1,2,3,4,5,6,7,8,9,10.
Here, we parameterize and validate an SEIR-W model: Susceptible (S), Exposed (E), Infectious (I), Recovered (R), and WAIT (W) model. Here W represents the environmental component of the transmission cycle during the early stage of the SARS CoV-2 pandemic. This component introduces more opportunities for infection, and complex dynamics resulting from viral persistence in the environment. In this framework, both indirect and direct transmission occur via mass-action, “random” encounters.
This model is derived from a previously developed framework called “WAIT”—which stands for Waterborne, Abiotic, and other Indirectly Transmitted—that incorporates an environmental reservoir where a pathogen remains in the environment and “waits” for hosts to interact with it11,12. The supplementary information contains a much more rigorous discussion of the modeling details. In the main text, we provide select details.
Building the SEIR-W model framework for SARS-CoV-2
Here W represents the environmental component of the early stage of the SARS CoV-2 pandemic (Fig. S1). This environmental compartment refers to reservoirs that people may have contact with on a daily basis, such as doorknobs, appliances, and non-circulating air indoors. The W compartment of our model represents the fraction of these environmental reservoirs that house some sufficiently transmissible amount of infectious virus. We emphasize that the W compartment is meant to only represent reservoirs that are common sites for interaction with people. Thus, inclusion of the W compartment allows us to investigate the degree to which the environment is infectious at any given point, and its impact on the transmission dynamics of SARS CoV-2.
Model parameters are described in detail in Table 1. The system of equations in the proposed mathematical model corresponding to these dynamics are defined in Eqs. (1)–(6):
Table 1 Model population definitions and initial values denoted with subscript 0 for each state variable.
Full size table

$$frac{dS}{dt}=mu (N-S)-left(frac{{beta }_{A}{I}_{A}+{beta }_{S}{I}_{S}}{N}+{beta }_{W}Wright)S$$
(1)

$$frac{dE}{dt}=left(frac{{beta }_{A}{I}_{A}+{beta }_{S}{I}_{S}}{N}+{beta }_{W}Wright)S-(epsilon +mu )E$$
(2)

$$frac{d{I}_{A}}{dt}=epsilon E-(omega +mu ){I}_{A}$$
(3)

$$frac{d{I}_{S}}{dt}=(1-p)omega {I}_{A}-(nu +{mu }_{S}){I}_{S}$$
(4)

$$frac{dR}{dt}=pomega {I}_{A}+nu {I}_{S}-mu R$$
(5)

$$frac{dW}{dt}=left(frac{{sigma }_{A}{I}_{A}+{sigma }_{S}{I}_{S}}{N}right)left(1-Wright)-kW$$
(6)

Infection trajectories
In addition to including a compartment for the environment (W), our model also deviates from traditional SEIR form by splitting the infectious compartment into an IA-compartment (A for asymptomatic), and an IS-compartment (S for symptomatic). As we discuss below, including asymptomatic (or sub-clinical) transmission is both essential for understanding how environmental—as opposed to simply unobserved or hidden—transmission affects the ecological dynamics of pathogens and also for analyzing SARS-CoV-2. The former represents an initial infectious stage (following the non-infectious, exposed stage), from which individuals will either move on to recovery directly (representing those individuals who experienced mild to no symptoms) or move on to the IS-compartment (representing those with a more severe response). Finally, individuals in the IS-compartment will either move on to recovery or death due to the infection. This splitting of the traditional infectious compartment is motivated by mounting evidence of asymptomatic transmission of SARS CoV-244,47,48,49,50. Thus, we consider two trajectories for the course of the disease, similar to those employed in prior studies42: (1) E → IA → R and (2) E → IA → IS → R (or death). More precisely, once in the E state, an individual will transition to the infectious state IA, at a per-person rate of ε. A proportion p will move from IA to the recovered state R (at a rate of p ⍵). A proportion (1—p) of individuals in the IA state will develop more severe systems and transition to Is (at a rate of (1—p) ⍵). Individuals in the Is state recover at a per-person rate of ν or die at a per-person rate μS. In each state, normal mortality of the individual occurs at the per-person rate μ and newly susceptible (S) individuals enter the population at a rate μN. The important differences between these two trajectories are in how likely an individual is to move down one path or another, how infectious individuals are (both for people and for the environment), how long individuals spend in each trajectory, and how likely death is along each trajectory.
Clarification on the interactions between hosts and reservoirs
The model couples the environment and people in two ways: (1) people can deposit the infectious virus onto environmental reservoirs (e.g. physical surfaces, and in the case of aerosols, the ambient air) and (2) people can become infected by interacting with these reservoirs. While most of our study is focused on physical surfaces, we also include data and analysis of SARS-CoV-2 survival in aerosols. While aerosols likely play a more significant role in person-to-person transmission, they also facilitate an indirect means of transmitting. For example, because SARS-CoV-2 can remain suspended in the air, other individuals can become infected without ever having to be in especially close physical proximity to the aerosol emitter (only requires that they interact with the same stagnant air, containing infectious aerosol particles)51. That is, a hypothetical infectious person A may produce aerosols, leave a setting, and those aerosols may infect a susceptible individual B who was never in close proximity to person A. In the transmission event between person A and person B, aerosol transmission functions in a similar fashion to surface transmission, where aerosols may be exchanged in the same room where infected individuals were, rather than exchanging infectious particles on a surface.
In our model, indirect infection via aerosols is encoded into the terms associated with the W component, just as the different physical surfaces are. Alternatively, aerosol transmission that leads to direct infection between hosts is encoded in the terms associated with direct infection between susceptible individuals and those infected (see section entitled Infection Trajectories).
Environmental reservoirs infect people through the βW term (Eqs. 1 and 2), a proxy for a standard transmission coefficient, corresponding specifically to the probability of successful infectious transmission from the environment reservoir to a susceptible individual (the full rate term being βWW·S). Hence, the βW factor is defined as the fraction of people who interact with the environment daily, per fraction of the environment, times the probability of transmitting infection from environmental reservoir to people. The factor βWW (where W is the fraction of environmental reservoirs infected) represents the daily fraction of people that will interact with the infected portion of the environment and become infected themselves. The full term βWW·S is thus the total number of infections caused by the environment per day.
In an analogous manner, we model the spread of infection to the environment with the two terms σAIA·(1—W) / N and σSIS·(1—W) / N representing deposition of infection to the environment by asymptomatic individuals, in the former, and symptomatic individuals, in the latter. In this case, σA (and analogously for σS) gives the fraction of surfaces/reservoirs that interact with people at least once per day, times the probability that a person (depending on whether they are in the IA or the IS compartment) will deposit an infectious viral load to the reservoir. Thus, σAIA/ N and σSIS/ N (where N is the total population of people) represent the daily fraction of the environment that interacts with asymptomatic and symptomatic individuals, respectively. Lastly, the additional factor of (1—W) gives the fraction of reservoirs in the environment that have the potential for becoming infected, and so σAIA·(1—W) / N (and analogously for IS) gives the fraction of the environment that becomes infected by people each day. We use W to represent a fraction of the environment, although one could also have multiplied the W equation by a value representing the total number of reservoirs in the environment (expected to remain constant throughout the course of the epidemic, assuming no intervention strategies).
Parameter values estimation
Table 1 displays information on the population definitions and initial values in the model. Tables 2 and 3 contain the fixed and estimated values and their sources (respectively). The model’s estimated parameters are based on model fits to 17 countries with the highest cumulative COVID-19 cases (of the 181 total countries affected) as of 03/30/2020, who have endured outbreaks that had developed for at least 30 days following the first day with ≥ 10 cumulative infected cases within each country14 (See supplementary information Tables S1–S3). In addition, we compare country fits of the SEIR-W model to fits with a standard SEIR model. Lastly, we compare how various iterations of these mathematical models compare to one another with regards to the general model dynamics. For additional details, see the supplementary information.
Table 2 Fixed parameter values estimated based on available published literature.
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Table 3 Estimated parameter values, averaged across countries.
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Basic reproductive ratios (({mathcal{R}}_{0}))
We can express the ({mathcal{R}}_{0}) (Eq. 7) in a form that makes explicit the contributions from the environment and from person-to-person interactions. In this way, the full ({mathcal{R}}_{0}) is observed to comprise two ({mathcal{R}}_{0}) sub-components: one the number of secondary infections caused by a single infected person through person-to-person contact alone (Rp) and the other is the number of secondary infections caused by exchanging infection with the environment (Re).

$${R}_{0}=frac{{R}_{p} + sqrt{{R}_{p}^{2} + 4 {R}_{e}^{2}}}{2}$$
(7)

where Rp and Re are defined in Eqs. (8a) and (8b)

$${R}_{p}=frac{varepsilon ({beta }_{A} ({mu }_{S }+ nu ) + {beta }_{S}(1 – p) omega )}{(mu + varepsilon )(mu + omega )({mu }_{S} + nu )}$$
(8a)

$${R}_{e}^{2}=frac{varepsilon { beta }_{W} ({sigma }_{A} ({mu }_{S }+ nu ) + {sigma }_{S}(1 – p) omega )}{k (mu + varepsilon )(mu + omega )({mu }_{S} + nu )}$$
(8b)

Note that when Rp = 0, the ({mathcal{R}}_{0}) reduces to Re and when Re = 0, the ({mathcal{R}}_{0}) reduces to Rp. Thus, when person-to-person transmission is set to zero, the ({mathcal{R}}_{0}) consists only of terms associated with transmission from the environment, and when transmission from the environment is set to zero, the ({mathcal{R}}_{0}) consists only of infection directly between people. When both routes of transmission are turned on, the two ({mathcal{R}}_{0})-components combine in the manner in Eq. (7).
While Re represents the component of the ({mathcal{R}}_{0}) formula associated with infection from the environment, the square of this quantity Re2 represents the expected number of people who become infected in the two-step infection process: people → environment → people, representing the flow of infection from people to the environment, and then from the environment to people. Thus, while Rp gives the expected number of people infected by a single infected person when the environmental transmission is turned off, Re2 gives the expected number of people infected by a single infected person by way of the environmental route exclusively, with no direct person-to-person transmission. Also note that Re2/(Re2 + Rp) can be used to measure the extent of transmission that is mediated by the environment exclusively. This proportion can be used as a proxy for how important environmental transmission is in a given setting. Elaboration on formulas 8a–b—and associated derivation-discussions—appear in the supplementary information. More

Case 1: spectacled eiders, from Wang et al.42
The experiment
This case is based on captive feeding trials conducted on 8 adult spectacled eiders, Somateria fischeri, which were maintained on an initial diet containing 1% Atlantic surf clam, 3% Antarctic krill, 88% Mazuri sea duck formula, 4% blue mussel and 4% Atlantic silverside, for 69 days prior to the start of the feeding experiment. After this, on day 0, a biopsy sample of the synsacral adipose tissue was obtained from each eider. With the FA data of the adipose tissue the authors calculated CCs. Feeding trials started on Day 0, and spectacled eiders were switched to diet A, consisting of 56% krill and 44% Mazuri sea duck formula for 21 days. On Day 21, eiders were biopsied again and switched to diet B consisting of 48% Mazuri formula and 52% silverside. On Day 50, a final biopsy sample was collected (Fig. 2A). FA turnover was considered near complete by 69 days.
Figure 2

Spectacled eiders case. (A) Feeding experiment: spectacled eiders (n = 8) spent 69 days on the initial diet described in the figure; after this, on day 0 eiders were biopsied and switched to diet A. On day 21 they were biopsied again and switched to diet B. After 29 days on diet B, eiders were biopsied on day 50. (B) Plots for diet estimations of spectacled eiders fed different combined diets. The true diet is indicated in each plot by the blue asterisks. CCs were calculated using FA data of day 0. Images by Alicia Guerrero.

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The model
This analysis is based on FA data from day 0, 21 and 50. We used the CCs calculated in this study after eiders were maintained on the same initial diet for 69 days. All sources were significantly different from each other (PERMANOVA, F4 = 9936.9, P = 0.001). ‘Day’ was set as fixed factor in the model.
Diet predictions
Based on FA data of day 0, MixSIAR estimated a contribution of 0.1% clam, 4% krill, 88% Mazuri, 6% mussel, and 1% silverside (Fig. 2B). After the first shift of diet, MixSIAR estimations changed to 28% krill and 62% Mazuri for FA data obtained on day 21. The final biopsy sample on day 50 produced estimations of 62% Mazuri, and 27% silverside.
Case 2: Steller’s eiders, from Wang et al.42
The experiment
This case corresponds to a feeding trial conducted simultaneously to the previous case by Wang et al.42, although the diet of Steller’s eiders, Polysticta stelleri, differed slightly. For 69 days prior to the start of the feeding trial, 8 adult Steller’s eiders were maintained on an initial diet containing 1% clam, 1% Antarctic krill, 88% Mazuri sea duck formula, 7% mussel, and 3% silverside. CCs were calculated after a biopsy was extracted to each eider on day 0. At the start of the feeding experiment, on day 0, Steller’s eiders were switched to diet A, containing 66% krill and 34% Mazuri formula. Then, on day 21, they were switched to diet B, consisting of 34% Mazuri formula and 66% silverside. Biopsy samples were collected on days 0, 21 and 50 (Fig. 3A). FA turnover was considered near complete by 69 days.
Figure 3

Steller’s eiders case. (A) Feeding experiment: eiders (n = 8) were maintained on the initial diet described in the figure for 69 days after which biopsy samples were collected (day 0) and eiders were switched to diet A. After 21 days, eiders were biopsied again and switched to diet B. On day 50, after 29 days on diet B, eiders were biopsied one last time. (B) Plots of diet estimation for Steller’s eiders fed different combined diets. Diet estimates are based on biopsy samples collected on days 0, 21 and 50. The true diet is indicated in each plot by red asterisks. CCs were calculated using FA data of day 0. Images by Alicia Guerrero.

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The model
We used the CCs calculated in the same study after Steller’s eiders were maintained on the same diet for 69 days. All sources were significantly different from each other (PERMANOVA, F4 = 10,079, P = 0.001). ‘Day’ was set as fixed factor in the model.
Diet predictions
Based on FA data of Day 0, MixSIAR estimated a contribution of 0.2% clam, 2% krill, 90% Mazuri, 4% mussel, and 3% silverside (Fig. 3B). After the first diet switch, MixSIAR estimations changed to 46% krill and 27% Mazuri. The final biopsy sample on day 50 produced estimations of 9% krill, 28% Mazuri, and 46% silverside.
Case 3: Atlantic salmon, from Budge et al.43
The experiment
For 22 weeks, tank-reared juvenile Atlantic salmon, Salmo salar (n = 132), were fed one of four different formulated feeds based on two marine oils: 100% krill oil, 100% herring oil, a mixture of 70:30 herring to krill oil, or a mixture of 30:70 herring to krill oil. Muscle samples were analysed for FAs after the 22-week experiment, which allowed the calculation of CCs. In this experiment, two sets of CCs were calculated: one derived from salmon fed the diet based on 100% herring oil and another from salmon fed the diet based on 100% krill oil (Fig. 4A). Unlike the previous two cases, where CCs were derived from consumers eating a mixed diet, here CCs were obtained from consumers feeding on a single type of source: either herring or krill oil. This allowed us to evaluate whether the source used to calculate the CCs affected dietary predictions. Additionally, we calculated a combined CC (an average value between CCs derived from krill and herring diets) and run a separate model. FA turnover was considered complete after 22 weeks on the same diet.
Figure 4

Atlantic salmon case. (A) Feeding experiment: Atlantic salmon fed formulated feeds based on either solely herring (n = 36) or krill oil (n = 28), or in proportions of 70:30 or 30:70 herring to krill oil (n = 34 each), for 22 weeks. (B) Plots of diet proportions estimated using MixSIAR for models using CCs derived from salmon fed a herring-oil diet (HO-CC), a krill-oil diet (KO-CC) or CCs averaged from these two treatments (Combined-CC). The true diet is indicated in each plot by black asterisks. Salmon image designed by Creazilla (https://creazilla.com).

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The model
We ran three independent models to estimate the diet of salmon fed each of the four diets: one using the set of CCs derived from herring oil, another using the CCs derived from krill oil, and one using the combined CCs. For each model, we used krill oil, herring oil and initial diet (commercial feed given to salmon prior to the experiment) as sources, and set ‘diet group’ as fixed factor. Significantly different FA compositions were found for the three sets of sources: those multiplied by herring oil CCs (PERMANOVA, F2 = 4008.7, P = 0.003), those multiplied by krill oil CCs (PERMANOVA, F2 = 3354.7, P = 0.002), and those multiplied by the combined CCs (PERMANOVA, F2 = 3677.2, P = 0.003).
Diet predictions
When we used CCs derived from salmon fed on a diet supplemented with herring oil only, MixSIAR correctly estimated the contribution of herring oil in the consumers (salmon) diet (98%). However, the contribution of herring was slightly overestimated where the consumers had been fed a mixture of herring and krill oil (Fig. 4B), and where the salmon’s diet was based on krill oil the contribution of krill oil was slightly underestimated (89%).
We found the opposite trend when we used CCs derived from salmon that had been fed a diet where krill oil had been the lipid source. Here, MixSIAR correctly estimated the contribution of krill oil in the salmon’s diet when they had been fed a diet based on krill oil (98%) or a mixture of 70:30 herring to krill oil (33% krill) or 30:70 herring to krill oil (70% krill); however, when herring oil had been the only dietary source this dietary contribution was underestimated (81%) (Fig. 4B).
Our dietary estimates were less biased when we used CC values that had been derived from the average between the herring- and krill-oil treatments (Combined-CCs). For example, we estimated herring contribution to be 90% of the diet when the actual diet was supplemented only with herring oil, and an estimate of 95% krill contribution when the actual diet was supplemented with krill oil only, and when the actual diet was a combination of herring (70%) and krill (30%), the estimations were 71% and 27%, and where the actual contribution of herring was 30% and 70% of krill, the estimated diets were 34% herring and 65% krill (Fig. 4B).
Case 4: tufted puffin nestlings, from Williams et al. 44
The experiment
Tufted puffin, Fratercula cirrhata, nestlings (n = 6) underwent an experimental feeding trial in their own burrows. Chicks were fed by their parents for approximately 10 days since hatching. When they were estimated to be 10-days old, the access to the burrows was blocked, so adults could not continue feeding their chicks. Through another access hole excavated by the researchers, chicks began being fed Pacific herring once a day, for 27 days. To infer the diet of free-living puffin nestlings during the first 10 days after hatching, wire screens were placed at burrow entrances to collect whole fish dropped by the parents. The species identified, in descending order (by mass), were Pacific sandlance, capelin, Pacific sandfish, salmonid and Pacific cod. An adipose tissue sample was collected on days 10 (start of the feeding trial), 19, 28 and 37. On day 37, assuming complete FA turnover after 27 days on a single prey diet (herring), the researchers calculated CCs (Fig. 5A). The data used to run this model included day 10, which represents the unknown diet provided by the parents, days 19, 28, and 37 which represent the herring diet at different extents. FA turnover was considered “close to, but not entirely complete” after 27 days44.
Figure 5

source even though it was not part of the nestlings’ diet. The red asterisks in each plot represent the potential diet fed by the parents (and used as priors in the first model). (D) Plots of diet estimations for tufted puffins fed herring, based on their FA profiles of days 19, 28 and 37. The true diet is indicated in each plot by red asterisks. CCs were calculated from tufted puffins fed herring, using FAs from day 37. Images by Alicia Guerrero.

Tufted puffins case. (A) Nestlings (n = 6) were fed by their parents for approximately 10 days since hatching. After this, they were fed herring for another 27 days as the entrance to their burrows was blocked and parents could not feed their chicks. (B) Non-metric multidimensional scaling plots for FAs obtained from chicks at different stages of the experiment and their sources. When FAs of sources were multiplied by their respective CCs, source (herring) and consumer (chicks, day 37) overlap in the plot. (C) Plots of the three models run to estimate the diet of nestlings on day 10. From left to right: Model using informative priors based on meals brought by the parents after the burrows were blocked; the same model without informative priors; and a third model without informative priors but including herring as

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The model
We used the CCs derived from these chicks feeding on Pacific herring for 27 days. For biopsy samples collected on day 10, we conducted three separate analyses: the first model excluded herring as potential prey, and incorporated informative priors based on the amount of different fish (% by mass) dropped by the parents at the burrow entrances; the second model was exactly the same but without prior information; and the third model was run without priors, and included herring as potential prey in order to determine whether this prey could be identified as absent.
For days 19, 28 and 37, we ran another analysis and included herring as source and no informative priors. ‘Day’ was set as a fixed factor in this model. All sources had significantly different FA compositions (PERMANOVA, F5 = 430.9, P = 0.001).
Diet predictions
In this example we included a non-metric dimensional scaling plot (Fig. 5B) to evaluate the effect of applying CCs to sources. Day 0 biopsies show greater variation than those of successive days, in both plots. When using original sources and consumer FA values, chicks are segregated from all the sources, although the similarity of the FAs increases toward the FA of herring as days pass, but they do not match. When CCs were applied to sources, the FA values of herring and chicks from day 37 overlap, indicating that they have the same FA compositions.
For day 0 (Fig. 5C), the estimated diet contributions were similar to meals brought by the parents when the model included informative priors, where the main dietary sources were sandlance (72%) and capelin (15%). Whereas estimates from the model without informative priors misrepresented the diet, as capelin was wrongly identified as the main dietary source (61%), cod the second most important prey (26%), and sandlance was incorrectly estimated to be only 6% of the diet. The third model including herring again identified capelin and cod as the main contributors (56% and 23%, respectively) whereas herring was identified as the least important prey, with 0.9% of contribution.
For day 19 (Fig. 5D), herring was identified as the main source, with 60% of contribution, followed by capelin and sandlance although with greater variation. From day 19 to 37, the contribution of herring increases from 60 to 97%, respectively.
Case 5. Harp seals, from Kirsch et al.45
The experiment
This study evaluated the effect of a low-fat diet on blubber FAs of harp seals, Pagophilus groenlandicus. Only for this experiment, the fat content of the different sources was available. Juvenile harp seals (n = 5) had been maintained on a diet of Atlantic herring (≥ 9% fat) for approximately 1 year prior to the feeding trial. On day 0, a full-depth blubber sample was collected from the posterior flank of each animal. For 30 days, seals were kept on a diet consisting solely of Atlantic pollock (1.7% fat). Blubber biopsies were taken again on days 14 and 30 (Fig. 6A). FA turnover was not considered complete after 30 days on the same diet, and authors suggest that a longer period on the diet, or higher intakes of fat, would be needed to accomplish it.
Figure 6

Diet estimates for juvenile harp seals fed a low-fat prey. (A) Feeding experiment: For a year prior to the feeding experiment, harp seals (n = 5) had been eating only Atlantic herring, a prey with a high-fat content. During the feeding experiment, seals were fed Atlantic pollock, a low-fat prey, for 30 days. (B) Plots of estimates derived from MixSIAR models based on FA data of whole blubber cores from days 0, 14 and 30. The black asterisks in the plots indicate the true diet. CCs correspond to harp seals fed herring, calculated using FAs from day 0. Images: fish by Lukas Guerrero Zambra, harp seal by Alicia Guerrero.

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The model
Since seals had been feeding on the same source for a year, we calculated CCs using FA data from day 0. Thus, consumer values were divided by Atlantic herring values producing CCs that were applied to both herring and pollock FA data. Sources were significantly different (PERMANOVA, F1 = 3680.4, P = 0.001) and ‘day’ was set as fixed factor in the model.
Diet predictions
For all three data sets (day 0, 14 and 30) the main contributor to the diet was Atlantic herring. The predicted proportion of Atlantic herring decreased only slightly from 99% on day 0 to 95% on day 30. Consequently, the contribution of pollock increased from 1% on day 0, to 5% on day 30 (Fig. 6B).
Case 6: Harbour seals, from Nordstrom et al.46
The experiment
Estimations are based on a feeding experiment by Nordstrom et al.46. Prior to the feeding study, juvenile harbour seals, Phoca vitulina, were fed a homogenate of 2:1 Pacific herring to salmon oil for approximately three weeks, and then fed only Pacific herring for four to six days. The feeding experiment consisted of three diets: one group of seals was fed only herring for 42 days (n = 3); the second group was fed only surf smelt for the same period (n = 6); and a third group (n = 7) was fed smelt for 21 days and then only herring for 21 days (Fig, 7A). Whole blubber core samples were collected on days 0, 21 and 42 for each group. Complete FA turnover was estimated to occur after at least 55 days on the same diet, although it could extend well beyond if turnover rate slowed with time.
The model
Since prey FA data was not provided in the same study, we used Pacific herring, surf smelt, and salmon FA values from Huynh and Kitts47, which had significantly different FA composition (PERMANOVA, F2 = 87.7, P = 0.001). CCs were derived from other harbour seals on a Pacific herring diet for over a year, in the same study46. We estimated the diet of the three groups of harbour seals, based on samples collected on day 42, setting ‘diet group’ as fixed factor in our model.
Diet predictions
Overall, diet differences were evident among groups, and the direction of the change was consistent with the shifts in diet. Estimates for harbour seals fed exclusively Pacific herring for 42 days, correctly showed that diet was predominantly based on herring (94.7%). For seals fed surf smelt for 42 days; however, estimates showed that surf smelt only accounted for 26.6% of the diet whereas herring remained to be the main component. For seals fed surf smelt for 21 days and then herring for another 21 days, MixSIAR again identified herring as the main component, with 90.9%, whereas surf smelt was only 3% (Fig. 7B).
Figure 7

Diet estimates for harbour seals. (A) Feeding experiment: For 42 days, seals were fed the following diets: solely herring (n = 3), solely surf smelt (n = 6), or surf smelt for the first 21 days and then herring for the remaining 21 days (n = 7). Prior to the feeding experiments they had been fed a mixture of herring and salmon. (B) Plots for MixSIAR diet estimations, based on blubber FAs obtained on day 42. The red asterisks indicate the true diet. CCs were obtained from harbour seals (other individuals) fed herring for over a year. Image by Alicia Guerrero.

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Influence on zoospore attachment
A slide glass with various sub-micro particles was deposited in a container (outer diameter 61.8 mm, height 125.2 mm) filled with seawater. Zoospores were poured from the surface of the water, and the number of zoospores that had attached to the slide glass was counted. The effect of the particles on attachment was investigated. Here, particles A, B and C were used (silicon carbide–SiC–particles with different size distributions) as the sediment particles. Particles A and B had one peak in the size distribution, and average particle sizes of 1.1 µm and 3.9 µm, respectively. Particle C had two peaks at 0.090 µm and 4.6 µm, and the average particle size was 1.5 µm (Supplementary Fig. S1 online).
When about 5 × 104 of E. bicyclis zoospores were placed in the container, after 12 h an average attachment of 13.5 ind./mm2 was observed on the slide glass without sediment particles. The relationship between the attachment percentage (%) of zoospores and amount of sediment particles of SiC is shown in Fig. 1a. The attachment percentage, expressed as the number of attached zoospores without sediments, was 100%.
Figure 1

Negative influences of sediment on zoospore attachment and gametophyte survival; (a,b) zoospore attachment percentage and gametophyte survival percentage, respectively.

Full size image

In the case of particle A (mean diameter 1.1 µm), which had one peak in the size distribution, the zoospore attachment percentage (mean ± SD) at 0.05 mg/cm2 and 0.1 mg/cm2 of sediments were 25.9 ± 14.2% and 10.2 ± 6.17%, respectively (Fig. 1a, Supplementary Table S1 online). In the case of particle B (mean diameter 3.9 µm), the attachment percentage was 53.9 ± 24.8% at 0.05 mg/cm2 and 41.1 ± 23.1% at 0.1 mg/cm2. In the case of particle A, few attachments were found at sediment levels of 0.3 mg/cm2.
The attachment percentage decreased exponentially as the amount of sediment on the substrate increased at any particle size. A significant negative correlation (Spearman’s rank correlation, p  More