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DataPathosystemWMV is widespread in cucurbit crops, mostly in temperate and Mediterranean climatic regions of the world16. WMV has a wide host range including some legumes, orchids and many weeds that can be alternative hosts16. Like other potyviruses, it is non-persistently transmitted by at least 30 aphid species16. In temperate regions, WMV causes summer epidemics on cucurbit crops, and it can overwinter in several common non-cucurbit weeds when no crops are present16,34. WMV has been common in France for more than 40 years, causing mosaics on leaves and fruits in melon, but mostly mild symptoms on zucchini squash. Since 2000, new symptoms were observed in southeastern France on zucchini squash: leaf deformations and mosaics, as well as fruit discoloration and deformations that made them unmarketable. This new agronomic problem was correlated to the introduction of new molecular groups of WMV strains. At least four new groups have emerged since 2000 and they have rapidly replaced the native “classical” strains, causing important problems for the producers35. These new groups, hereafter “emerging strains” (ES) are significantly more related molecularly to worldwide strains than to any other isolates from the French populations36. As emphasised in35, this supports that the new group of emerging strains has arisen through introductions, mostly from Southeastern Asia, rather than through local differentiation.In this study, we focus on the pathosystem corresponding to a classical strain (CS) and four emerging strains (ESk, (k = 1, ldots ,4)) of WMV and their cucurbit hosts.Study area and samplingThe study area, located in Southeastern France, is included in a rectangle of about 25,000 km2 and is bounded on the South by the Mediterranean Sea. Between 2004 and 2008, the presence of WMV had been monitored in collaboration with farmers, farm advisers and seed companies. Each year, cultivated host plants were collected in different fields and at different dates between May 1st and September 30th. In total, more than two thousand plant samples were collected over the entire study area. All plant samples were analyzed in the INRAE Plant Pathology Unit to confirm the presence of WMV and determine the molecular type of the virus strain causing the infection (see35 for detail on field and laboratory protocols). All infected host plants were cucurbits, mostly melon and different squashes (e.g., zucchini, pumpkins).Observations In the absence of individual geographic coordinates, all infected host plants were attributed to the centroid of the municipality (French administrative unit, median size about 10 km2) where they have been collected. Then for one date, one observation corresponded to a municipality in which at least one infected host plant was sampled. Table 1 summarizes for each year, the number of observations (i.e. number of municipalities), the number of infected plants sampled and the proportion of each WMV strain (CS, and ES1 to ES4) found in the infected host plants. Errors in assignment of virus samples to the CS or ES strains was negligible because of the large genetic distance separating them: 5 to 10% nucleotide divergence both in the fragment used in the study and in complete genomes35, also precluding the possibility of local jumps between groups by accumulation of mutations.Table 1 Number of observations and corresponding proportions of classical and emerging strains.Full size tableLandscapeTo approximate the density of WMV host plants over the study area, we used 2006 land use data (i.e. BD Ocsol 2006 PACA and LR) produced by the CRIGE PACA (http://www.crige-paca.org/) and the Association SIG-LR (http://www.siglr.org/lassociation/la-structure.html). Based on satellite images, land use is determined at a spatial resolution of 1/50,000 using an improved three-level hierarchical typology derived from the European Corine Land Cover nomenclature. Here we used the third hierarchical level of the BD Ocsol typology (i.e. 42 land use classes) to classify the entire study area in three habitats: (1) WMV-susceptible crops, (2) habitats unfavorable to WMV host plants (e.g. forests, industrial and commercial units…) and, (3) non-terrestrial habitat (i.e. water). The proportion of WMV-susceptible crops was then computed within all cells of a raster covering the entire study area, with a spatial resolution of (1.4 times 1.4) km2. These proportions were used to approximate host plant density (zleft( {varvec{x}} right)), which was normalized, so that (zleft( {varvec{x}} right) = 0) corresponds to an absence of host plants and (zleft( {varvec{x}} right) = 1) to the maximum density of host plants (Fig. 1).Figure 1Approximated density (zleft( x right)) of the host plants in the study area. The density was normalized, so that (zleft( x right) = zleft( {x_{1} ,x_{2} } right) = 0) corresponds to an absence of cucurbit plants and (zleft( x right) = 1) to the maximum density. The axes (x_{1}) and (x_{2}) correspond to Lambert93 coordinates (in km). The white regions are non-terrestrial habitats (water). Land use data were not available in the gray regions; the host plant density was then computed by interpolation.Full size imageMechanistic-statistical modelThe general modeling strategy is based on a mechanistic-statistical approach12,22,33. This type of approach combines a mechanistic model describing the dynamics under investigation with a probabilistic model conditional on the dynamics, describing how the measurements have been collected. This method that has already proved its theoretical effectiveness in determining dispersal parameters using simulated genetic data12 aims at bridging the gap between the data and the model for the determination of virus dynamics.Here, the mechanistic part of the model describes the spatio-temporal dynamics of the virus strains, given the model parameters (demographic parameters, introduction dates/sites). This allows us to compute the expected proportions of the five types of virus strains (CS and ES1 to ES4) at each date and site of observation. The probabilistic part of the mechanistic-statistical model describes the conditional distribution of the observed proportions of the virus strains, given the expected proportions. Using this approach, it is then possible to derive a numerically tractable formula for the likelihood function associated with the model parameters.Population dynamicsThe model is segmented into two stages: (1) the intra-annual stage describes the dispersal and growth of the five virus strains during the summer epidemics on cucurbit crops, and the competition between them, during a period ranging from May 1st (noted (t = 0)) to September 30 (noted (t = t_{f}), (t_{f} = 153) days); (2) the inter-annual stage describes the winter decay of the different strains when no crops are present and the virus overwinters in weeds. We denote by (c^{n} left( {t,{varvec{x}}} right)) and (e_{k}^{n} left( {t,{varvec{x}}} right)) the densities of classical strain (CS) and emerging strains (ESk, (k = 1, ldots ,4)), at position ({varvec{x}}) and at time (t) of year (n.)Dynamics of the classical strain before the first introduction eventsBefore the introduction of the first emerging strain, the intra-annual dynamics of the population CS is described by a standard diffusion model with logistic growth: (partial_{t} c^{n} = D{Delta }c^{n} + rc^{n} left( {zleft( {varvec{x}} right) – c^{n} } right)). Here, ({Delta }) is the Laplace 2D diffusion operator (sum of the second derivatives with respect to coordinate). This operator describes uncorrelated random walk movements of the viruses, with the coefficient (D) measuring the mobility of the viruses (e.g.,26,37). The term (r zleft( {varvec{x}} right)) is the intrinsic growth rate (i.e., growth rate in the absence of competition) of the population, which depends on the density of host plants (zleft( {varvec{x}} right)) and on a coefficient (r) (intrinsic growth rate at maximum host density). Under these assumptions, the carrying capacity at a position ({varvec{x}}) is equal to (zleft( {varvec{x}} right)), which means that the population densities are expressed in units of the maximum host population density. The model is initialized by setting (c^{1980} left( {0,{varvec{x}}} right) = (1 – m_{c} ) zleft( {varvec{x}} right)), where (m_{c}) is the winter decay rate of the CS (see the description of the inter-annual stage below). In other terms, we assume that the CS density is at the carrying capacity in 1979, i.e., 5 years after its first detection and 20 years before the first detections of ESs38.Introduction eventsThe ESs are introduced during years noted (n_{k} ge 1981), at the beginning of the intra-annual stage (other dates of introduction within the intra-annual stage would lead—at most—to a one-year lag in the dynamics). Their densities are (0) before introduction: (e_{k}^{n} = 0) for (n < n_{k}). Once introduced, the initial density of any ES is assumed to be 1/10th of the carrying capacity at the introduction point (other values have been tested without much effect, see Supplementary Fig. S1), with a decreasing density as the distance from this point increases:$$e_{k}^{{n_{k} }} left( {0,x} right) = frac{{zleft( {varvec{x}} right)}}{10}exp left( { - frac{|{{varvec{x}} - {varvec{X}}_{{varvec{k}}}|^{2} }}{{2sigma^{2} }}} right),$$where ({varvec{X}}_{{varvec{k}}}) is the location of introduction of the strain (k.) In our computations, we took (sigma = 5) km for the standard deviation.Intra-annual dynamics after the first introduction eventIntra-annual dynamics were described by a neutral competition model with diffusion (properties of this model have been analyzed in [54]):$$left{ {begin{array}{*{20}c} {partial_{t} c^{n} left( {t,x} right) = DDelta c^{n} + rc^{n} left( {zleft( {varvec{x}} right) - c^{n} - mathop sum limits_{i = 1}^{4} e_{i}^{n} left( {t,{varvec{x}}} right)} right)} \ {partial_{t} e_{k}^{n} left( {t,x} right) = DDelta e_{k}^{n} + re_{k}^{n} left( {zleft( {varvec{x}} right) - c^{n} - mathop sum limits_{i = 1}^{4} e_{i}^{n} left( {t,{varvec{x}}} right)} right)} \ end{array} } right.,$$for (t = 0 ldots t_{f}) and for all introduced emerging strains, i.e. all (k) such that (n ge n_{k} .) We assume reflecting boundary conditions, meaning that the population flows vanish at the boundary of the study area, due to truly reflecting boundaries (e.g., sea coast in the southern part of the site) or symmetric inward and outward fluxes26. In addition, in order to limit the number of unknown parameters, and in the absence of precise knowledge on the differences between the strains, we assume here that the diffusion, competition and growth coefficients are common to all the strains during the intra-annual stage (see the discussion for more details on this assumption).Inter-annual dynamicsThe population densities at time (t = 0) of year (n) are connected with those of year (n - 1,) at time (t = t_{f} ,) through the following formulas:$$left{ {begin{array}{*{20}c} {c^{n} left( {0,{varvec{x}}} right) = left( {1 - m_{c} } right)c^{n - 1} left( {t_{f} ,{varvec{x}}} right) hbox{ for } n ge 1981} \ {e_{k}^{n} left( {0,{varvec{x}}} right) = left( {1 - m_{e} } right)e_{k}^{n - 1} left( {t_{f} ,{varvec{x}}} right) hbox{ for }n ge n_{k} + 1} \ end{array} } right.$$with (m_{c}) and (m_{e}) the winter decay rates of the CS and ESs strains (note that (m_{e}) is common to all of the ESs). Estimation of CS and ES decay rates provides an assessment of the competitive advantage of one type of strain vs the other.Numerical computationsThe intra-annual dynamics were solved using Comsol Multiphysics time-dependent solver, which is based on a finite element method (FEM). The triangular mesh which was used for our computations is available as Supplementary Fig. S2.Probabilistic model for the observation processDuring the years (n = 2004, ldots ,2008), (I_{n}) observations were made (see Section Observations above and Table 1). They consist in counting data, that we denote by (C_{i}) and (E_{k,i}) for (k = 1, ldots ,4) and (i = 1, ldots ,I_{n}), corresponding to the number of samples infected by the CS and ESs strains, respectively, at each date of observation and location (left( {t_{i} ,{varvec{x}}_{i} } right)). Note that these dates and locations depend on the year of observation (n). More generally, the above quantities should be noted (C_{i}^{n} , E_{k,i}^{n} , t_{i}^{n} , {varvec{x}}_{i}^{n} ;) for simplicity, the index (n) is omitted in the sequel, unless necessary.We denote by (V_{i} = C_{i} + mathop sum nolimits_{k = 1}^{4} E_{k,i}) the total number of infected samples observed at (left( {t_{i} ,{varvec{x}}_{i} } right)). The conditional distribution of the vector (left( {C_{i} ,E_{1,i} ,E_{2,i} ,E_{3,i} ,E_{4,i} } right)), given (V_{i}) can be described by a multinomial distribution ({mathcal{M}}left( {V_{i} ,{varvec{p}}_{i} } right)) with ({varvec{p}}_{i} = left( {p_{i}^{c} ,p_{i}^{{e_{1} }} ,p_{i}^{{e_{2} }} ,p_{i}^{{e_{3} }} ,p_{i}^{{e_{4} }} } right)) the vector of the respective proportions of each strain in the population at (left( {t_{i} ,{varvec{x}}_{i} } right)). We chose to work conditionally to (V_{i}) because the sample sizes are not related to the density of WMV.Statistical inferenceUnknown parametersWe denote by ({{varvec{Theta}}}) the vector of unknown parameters: the diffusion coefficient (D,) the intrinsic growth rate at maximum host density (r), the winter decay rates ((m_{c} , m_{e} )) and the locations ((x_{k} in {mathbb{R}}^{2})) and years ((n_{k})) of introduction, for (k = 1, ldots ,4.) Thus ({{varvec{Theta}}} in {mathbb{R}}^{16} .)Computation of a likelihoodGiven the set of parameters ({{varvec{Theta}}}), the densities (c^{n} left( {t,{varvec{x}}|{{varvec{Theta}}}} right)) and (e_{k}^{n} left( {t,{varvec{x}}|{{varvec{Theta}}}} right)) can be computed explicitly with the mechanistic model for population dynamics. Thus, at a given year (n), at (left( {t_{i} ,x_{i} } right)), the parameter ({varvec{p}}_{i}) of the multinomial distribution ({mathcal{M}}left( {V_{i} ,{varvec{p}}_{i} } right)) writes:$$p_{i}^{c} left( {{varvec{Theta}}} right) = frac{{c^{n} left( {t_{i} ,{varvec{x}}_{i} |{{varvec{Theta}}}} right)}}{{c^{n} left( {t_{i} ,{varvec{x}}_{i} |{{varvec{Theta}}}} right) + mathop sum nolimits_{i = 1}^{4} e_{i}^{n} left( {t_{i} ,{varvec{x}}_{i} {|}{{varvec{Theta}}}} right)}}, p_{i}^{{e_{k} }} left( {{varvec{Theta}}} right) = frac{{e_{k}^{n} left( {t_{i} ,{varvec{x}}_{i} |{{varvec{Theta}}}} right)}}{{c^{n} left( {t_{i} ,{varvec{x}}_{i} |{{varvec{Theta}}}} right) + mathop sum nolimits_{i = 1}^{4} e_{i}^{n} (t_{i} ,{varvec{x}}_{i} |{{varvec{Theta}}})}}.$$The probability (P(C_{i} ,E_{1,i} ,E_{2,i} ,E_{3,i} ,E_{4,i} |{{varvec{Theta}}},{text{V}}_{{text{i}}} )) of the observed outcome (C_{i} ,E_{1,i} ,E_{2,i} ,E_{3,i} ,E_{4,i}) is then$$Pleft( {C_{i} ,E_{1,i} ,E_{2,i} ,E_{3,i} ,E_{4,i} {|}{{varvec{Theta}}},{text{V}}_{{text{i}}} } right) = frac{{left( {V_{i} } right)!}}{{C_{i} ! mathop prod nolimits_{k = 1}^{4} E_{k,i} !}}left( {p_{i}^{c} left( {{varvec{Theta}}} right)} right)^{{C_{i} }} mathop prod limits_{k = 1}^{4} (p_{i}^{{e_{k} }} left( {{varvec{Theta}}} right))^{{E_{k,i} }} .$$Assuming that the observations during each year and at each date/location are independent from each other conditionally on the virus strain proportions, we get the following formula for the likelihood:$${mathcal{L}}left( {{varvec{Theta}}} right) = mathop prod limits_{n = 2004}^{2008} mathop prod limits_{{i = 1, ldots , I_{n} }} Pleft( {C_{i} ,E_{1,i} ,E_{2,i} ,E_{3,i} ,E_{4,i} {|}{{varvec{Theta}}},{text{V}}_{{text{i}}} } right).$$A priori constraints on the parameters By definition and for biological reasons, the parameter vector ({{varvec{Theta}}}) satisfies some constraints. First, (D in left( {10^{ - 4} ,10} right){text{ km}}^{2} /{text{day}}), (r in left( {0.1,1} right) {text{day}}^{ - 1} ,) and (m_{c} , m_{e} in left{ {0,0.1,0.2, ldots ,0.9} right},) (see Supplementary Note S7 for a biological interpretation of these values). Second, we assumed that the locations of introductions ({varvec{X}}_{{varvec{k}}}) belong to the study area. To facilitate the estimation procedure, the points ({varvec{X}}_{{varvec{k}}}) were searched in a regular grid with 20 points (see Supplementary Fig. S3), and the dates of introduction (n_{k}) were searched in (left{ {1985,1990,1995,2000} right}.)Inference procedureDue to the important computation time (4 min in average for one simulation of the model on an Intel(R) Core(R) CPU i7-4790 @ 3.60 GHz), we were not able to compute an a posteriori distribution of the parameters in a Bayesian framework. Rather, we used a simulated annealing algorithm to compute the maximum likelihood estimate (MLE), i.e., the parameter ({{varvec{Theta}}}^{*}) which leads to the highest log-likelihood. This is an iterative algorithm, which constructs a sequence (({{varvec{Theta}}}_{j} )_{j ge 1}) converging in probability towards a MLE. It is based on an acceptance-rejection procedure, where the acceptance rate depends on the current iteration (j) through a "cooling rate" ((alpha )). Empirically, a good trade-off between quality of optimization and time required for computation (number of iterations) is obtained with exponential cooling rates of the type (T_{0} alpha^{j}) with (0 < alpha < 1) and some constant (T_{0} gg 1) (this cooling schedule was first proposed in= 39 = 39). Too rapid cooling ((alpha ll 1)) results in a system frozen into a state far from the optimal one, whereas too slow cooling ((alpha approx 1)) leads to important computation times due to very slow convergence. Here, we ran (6) parallel sequences with cooling rates (alpha in left{ {0.995,0.999,0.9995} right}). For this type of algorithm, there are no general rules for the choice of the stopping criterion [HenJac03], which should be heuristically adapted to the considered optimization problem. Here, our stopping criterion was that ({{varvec{Theta}}}_{j}) remained unchanged during 500 iterations. The computations took about 100 days (CPU time).Confidence intervals and goodness-of-fitTo assess the model’s goodness-of-fit, 95% confidence regions were computed for the observations (left( {C_{i} ,E_{1,i} ,E_{2,i} ,E_{3,i} ,E_{4,i} } right)) at each date/location (left( {t_{i} ,{varvec{x}}_{i} } right),) and for each year of observation. The confidence regions were computed by assessing the probability of each possible outcome of the observation process, at each date/location, based on the computed proportions ({varvec{p}}_{i} = left( {p_{i}^{c} ,p_{i}^{{e_{1} }} ,p_{i}^{{e_{2} }} ,p_{i}^{{e_{3} }} ,p_{i}^{{e_{4} }} } right)), corresponding to the output of the mechanistic model using the MLE ({{varvec{Theta}}}^{user2{*}}) and given the total number of infected samples (V_{i}). Then, we checked if the observations belonged to the 95% most probable outcomes. More

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Two species unidirectional coupling ecosystemThis bio-inspired ecosystem S((beta _{xy}=0), (beta _{yx})) describing the unidirectional coupling is run for 1000 time steps for reaching stationarity, generating a set of 1000 points long time-series dependent on (beta _{yx}). This means that species X has an increasing effect on Y with the increase of (beta _{yx}), but Y has no effect on X. Both CCM and the proposed OIF model are separately used to quantify the potential causality between species X and Y. The inferred causality dependent on (beta _{yx}) only (as a physical interaction) is shown in Fig. 2A. (beta), (rho) and TE have different units; specifically (beta) and (rho) are dimensionless while TE is measured in bits or nats (a logarithmic unit of information or entropy). Therefore, any comparison is done considering gradients of change when these variables vary together rather than making comparisons between absolute values which are meaningless. Figure 2A shows that under the condition of (beta _{xy})=0, results of “Y to X” (i.e. the estimated effect on Y on X) is close to 0 for the OIF model ((TE_{Y rightarrow X}(beta _{yx}))) that precisely describe the no-effect of Y on X. “X to Y” ((TE_{X rightarrow Y}(beta _{yx}))) well tracks the increasing strength of the effect of X on Y for increasing values of the physical interaction (beta _{yx}) embedded into the mathematical model. However, considering results of the CCM model, “Y to X” ((rho _{Y rightarrow X}(beta _{yx}))) presents non obvious (and likely wrong) non-zero values with higher fluctuations compared to (TE_{Y rightarrow X}(beta _{yx})) especially for lower values of (beta _{yx}). This erroneous estimates of CCM is likely related to the need of CCM for convergence. For CCM, “X to Y” (((rho _{X rightarrow Y}(beta _{yx})))) shows an increasing trend for increasing values of (beta _{yx}) and decreasing when (beta _{yx}) is greater than (sim)0.5 non-trivially. In consideration of these results for the unidirectional coupling ecosystem, the OIF model performs better over CCM in terms of unidirectional causality inference.Two species bidirectional coupling ecosystemIn this case, the effect between two species is bidirectional. Species X has an effect on species Y and vice versa. The univariate dynamical systems S(0.2/0.5/0.8, (beta _{yx})) are run for 1000 time steps under the same conditions determined by (beta _{xy}). Certainly this situation is fictional since in real ecosystems the interaction strength is changing when other interacting species change their interactions.Thus, keeping one interaction fixed around one value is a strong unrealistic simplification (analogous of one-factor at-a-time sensitivity analyses) but it is a toy model that allows to verify the power of network inference models. These models generate three sets of 1000 points long time-series dependent of (beta _{yx}) for each fixed (beta _{xy}). OIF and CCM are used to infer “causality” between X and Y—in the form of (rho) and TE—and compare that against the real embedded interaction (beta _{yx}) and (beta _{xy}) shown in Fig. 2B,C,D. Considering all results of Fig. 2 corresponding to fixed (beta _{xy})s, the correlation coefficient (rho) yielded from CCM and TE from OIF are both able to track the strength of causal trajectories. However, TE seems to perform better in term of ability to infer fine-scale changes in interactions. In particular, considering Fig. 2D (right plot), higher (TE_{yx}) higher for low (beta _{yx}) makes sense because (beta _{xy} > beta _{yx}) that means Y has a larger influence on X than vice versa and then Y is able to predict X. Additionally, TE does not suffer of convergence problems; specifically, considering Fig. 2A (left plot), higher (rho) for small (beta _{yx}) is not sensical and that is likely related to convergence problems of CCM.Considering all results of Fig. 2 corresponding to fixed (beta _{xy})s, the correlation coefficient (rho) yielded from CCM and TE from OIF are both able to track the strength of causal trajectories. Ideally, the causality from Y to X is a constant since (beta _{xy}) is a fixed value for each case. In this figure, the red curve in the right panel representing the OIF-inferred (TE-based) causality from Y to X is higher for greater (beta _{xy})s, while red curves representing CCM-inferred ((rho)) causality in the left panel present higher fluctuations especially for lower (beta _{yx}). For the causality from X to Y determined by (beta _{yx}) in the mathematical model, theoretically speaking, the causality from X to Y should monotonously grow when (beta _{yx}) increases from 0 to 1. In Fig. 2, blue curves in the right panel representing the OIF-inferred (TE-based) causality from X to Y present monotonously increasing features as a whole with the increasing (beta _{yx}), while those from CCM model ((rho)) do not and show considerable fluctuations.Therefore, OIF outperforms CCM in terms of the ability to infer the fine-scale changes in causality. In particular, considering Fig. 2D (right plot), higher (TE_{yx}) higher for low (beta _{yx}) makes sense because (beta _{xy} > beta _{yx}) that means Y has a larger influence on X than vice versa and then Y is able to predict X. Additionally, TE does not suffer of convergence problems; specifically, considering Fig. 2A (left plot), higher (rho) for small (beta _{yx}) is not sensical and that is likely related to convergence problems of CCM.Additionally, (rho _{Y rightarrow X}(beta _{yx})) shows higher fluctuations on average especially for the condition of lower (beta _{yx})s compared to (TE_{Y rightarrow X}(beta _{yx})). When considering the effect of X on Y that is a function of (beta _{yx}) for CCM, (rho _{X rightarrow Y}) reaches an extreme value at around (beta _{yx}) = 0.5 and then declines for larger values of (beta _{yx}). This is not consistent with the expected effect of X on Y that should be proportional to (beta _{yx}) embedded into the mathematical model. The ability of (rho) to reflect the proportional relationship between the effect of X on Y (manifested by (beta _{yx})) vanishes for high (beta _{xy})s due to unexpected and somewhat inconspicuous changes in (rho _{X rightarrow Y}) for larger (beta _{yx}). In simple words, the expected increasing trend of (rho) is lost for larger (beta _{xy}) that is counterintuitive. On the other side, (TE_{X rightarrow Y}(beta _{yx})) invariably maintains an increasing trend for increasing values of (beta _{xy}). OIF is also performing better than CCM when predicting higher average values of (TE_{Y rightarrow X}) for increasing values of (beta _{xy}) (red curves in Fig. 2A–D, right plots) as expected by the fixed effect in the mathematical model of Y on X. These results suggest that when compared to (rho) of CCM, TE can track well the causal interactions over (beta _{yx}) with higher performance and without considering the convergence requirement of CCM. CCM needs to consider the length of time series that makes (rho _{X rightarrow Y}(beta _{yx})) convergent to a stable value, but uncertain for large differences in time-series length of (X,Y) and sensitive to short time series.In more realistic settings for real ecosystems (and in analogy to global sensitivity analyses) when (beta _{xy}) and (beta _{yx}) are both considered as arguments of the two-variable (X,Y) bio-inspired model, the simulated ecosystem becomes a truly bivariate system, yet yielding complexity but more interest into the causality inference (Fig. 3). The dynamical system S((beta _{xy}), (beta _{yx})) was generated for 800 time steps under the same conditions mentioned above. We generated the datasets that allowed us to study linear and non-linear predictability indicators for inferring the embedded physical interactions. Specifically, we measure undirected linear correlation coefficient (corr_{X;Y}(beta _{xy},beta _{yx})), non-linear undirected mutual information (MI_{X;Y}(beta _{xy},beta _{yx})), directed non-linear correlation coefficient (rho _{X rightarrow Y}(beta _{xy},beta _{yx})) and (rho _{Y rightarrow X}(beta _{xy},beta _{yx})), and non-linear directed transfer entropy (TE_{X rightarrow Y}(beta _{xy},beta _{yx})) and (TE_{Y rightarrow X}(beta _{xy},beta _{yx})) as shown in Fig. 3. These 2D phase-space maps in Fig. 3 show strikingly similar patterns for classical linear correlation coefficients, MI, (rho) of CCM and TE of OIF which underline the fact that all methods are able to infer the interdependence patterns of interacting variables explicitly defined by (beta _{xy}) and (beta _{yx}). The color of phase-space maps is proportional to the inferred interaction between X and Y when the mutual physical interactions are varying according to the mathematical model in Eq. (1). In Fig. 3, even though phase-space maps of undirected (corr_{X;Y}(beta _{xy},beta _{yx})) and (MI_{X;Y}(beta _{xy},beta _{yx})) present similar patterns (in value organization and not value range) to those of directed (rho) and TE, neither (corr_{X;Y}((beta _{xy},beta _{yx})) and (MI_{X;Y}((beta _{xy},beta _{yx})) provide information about the direction of causality. As expected MI shows the opposite pattern of the average TE due to the fact that MI is the amount of shared information (or similarity) versus the amount of divergent information (divergence and asynchronicity) between X and Y.Figure 3Phase-space maps of normalized coupling predictive causation via correlation, mutual information, CCM and OIF for varying true causal interactions. Both true causal interactions (beta _{xy}) and (beta _{yx}) are free varying within the range [0, 1], indicating a bivariate model S((beta _{xy}),(beta _{yx})) where both species (or variables more generally) are interacting with each other with different strength. (A) normalized correlation coefficient, (B) normalized mutual information, (C) and (E) normalized CCM correlation coefficient ((rho)) for interaction directions of (X rightarrow Y) and (Y rightarrow X), (D) and (F) normalized transfer entropy (TE) from OIF model for interaction directions of (X rightarrow Y) and (Y rightarrow X).Full size imageFigure 4Dynamics of abundance and predictability for the bidirectional two species ecosystem model. (A) plots refer to the species abundance in time for the mathematical model in Eq. 1 for different predictability regimes associated to different interaction dynamics from low to high complexity ecosystem associated to low and high predictability. Blue, green and red refer to a range of predictable interactions as in Fig. 3: specifically, Blue is for ((beta _yx), (beta _xy))=(0.18, 0.39) (small mutual interaction, and predominant effect of Y on X), Green is for (0.64, 0.57) (high mutual interactions, and slightly predominant effect of X on Y), and Red for (0.94, 0.34) (high mutual interactions, and predominant effect of X on Y). (B) phase-space plots showing the non-time delayed associations between X and Y corresponding to synchronous and homogeneous, mildly asynchronous and divergent, and asynchronous and divergent dynamics. The transition from synchronous/small interactions to asynchronous/high interaction leads to a transition from modular to nested ecosystem interactions when more than one species exist (Fig. 6).Full size imageIn a biological sense TE should be interpreted as the probability of likely uncooperative dynamics (leading to or driven by environmental or biological heterogeneity) while MI as the probability of cooperative dynamics (leading to or driven by homogeneity). Here we refer to cooperative and uncooperative interactions based on the similarity or dissimilarity in pair dynamics manifested by species abundance fluctuations. For instance divergence and asynchronicity (that define TE) in pair species dynamics manifest uncooperative interactions. The balance of cooperative and uncooperative interactions can result into net interactions at the ecosystem scale manifesting neutral patterns, or net interactions may lead to niche patterns biased toward strong environmental or biological factors52. Certainly, cooperation in a biological sense should be interpreted on a case by case basis. In a broader uncertainty propagation perspective49, “cooperation” between variables means that variables contribute similarly to the uncertainty propagation, while “competition” means that one variable is predominant over the other in terms of magnitude of effects since TE is proportional to the magnitude rather than the frequency of effects. For the former case the total entropy of the system is higher than the latter case. Interestingly, correlation (text{ corr }(beta )), (rho) and TE show similar patterns in both organization and value range (but not in singular values of course), which sheds some important conclusions about the similarity and divergence of these methods as well as their capacity and limitations in characterizing non-linear systems.When comparing the phase-space patterns from CCM and OIF (displaying (rho) and TE) a more colorful and informative pattern is revealed by OIF. This means that TE gives a better gradient when tracking the increasing strength of causality for increasing values of (beta _{xy}) and (beta _{yx}). When comparing the phase-space patterns for the two causal directions of (“X rightarrow Y”) and (“Y rightarrow X”), phase-space maps from CCM are very similar, while those from TE present apparent differences in the strength of effects for the two opposite direction of interaction. Therefore, OIF is more sensitive to the direction of interaction compared to CCM when detecting directional causality.These results imply that TE performs better to distinguish directional embedded physical interactions (that are dependent on direct interactions (beta)-s, species growth rate (r_x) and (r_y), and contingent values X(t) and Y(t) determining the total interaction as seen in the model of Eq. (1)) in the species causal relationships. It should be emphasized how all linear and non-linear interaction indicators are inferring the total interaction and not only those exerted by (beta)-s. In a broad uncertainty purview49 the importance of these three factors ((beta)-s, r-s and X(t)/Y(t)) depends on their values and probability distributions that define the dynamics of the system; dynamics such as defined by the regions identified by patterns in Fig. 3 for the predator-prey system in Eq. (1). In principle, the higher the difference between these three interaction factors in the species considered, the higher the predictability and sensitivity of OIF. Figure 4 highlights three different dynamics corresponding to the TE blue, green and red regions in Fig. 3.In all dynamical states represented by Fig. 3, species are interacting with different magnitudes and this defines distinct network topologies. Three prototypical dynamics are shown in Fig. 4 with colors representative of (rho) and TE in Fig. 3. The “blue” deterministic dynamics has very high synchronicity and no divergence considering variable fluctuation range (the gap is deterministic and related to the numerically imposed (u=1)), as well as no linear correlation between non-lagged variables. In perfect synchrony one would have one point in the phase-space. Thus, absence of correlation does not imply complete decoupling of species but it can be a sign of small interactions. The “green” dynamics shows a relatively high synchronicity and medium divergence. In the phase-space of synchronous values of X and Y a correlation is observed with relatively small fluctuations because the divergence is small. Lastly, the “red” dynamics shows a relatively high asynchronicity and divergence. The stochasticity is higher than previous dynamics and the “mirage correlation” in the phase space has higher variance. Time-dependent mirage correlations in sign and magnitude mean that correlation (that may suggest common dynamics in a linear framework) does not imply similarity in dynamics for the two species. Non-linearity is higher from blue to red dynamics as well as predictability but lower absolute information entropy. Then, it is safe to say that linear dynamics (or small stochasticity) does not imply higher predictability.Real-world sardine–anchovy-temperature ecosystemCCM and proposed OIF model are also used for a real-world fishery ecosystem to infer potential causal interactions between Pacific sardines (Sardinops sagax) landings, Northern anchovies (Engraulis mordax) and sea-surface temperature (SST) recorded at Scripps Pier and Newport Pier, California. Sardines and anchovies do not interact physically (or the interaction is low in number), while both of them are influenced by the external environmental SST that is the external forcing. To quantify the likely causal interactions between species and SST based on real data, we use CCM considering the length of time series for convergence of (rho), as well as OIF considering a set of time delays for acquiring stable values of inferred interactions TEs.Figure 5Inferred predictive causality for the sardine-anchovy-Sea Surface Temperature ecosystem. CCM correlation coefficient ((rho)) and OIF predictor (TE) are shown in the left and middle plots for different pairs considered (sardine–anchovy, sardine and SST, anchovy and SST from top to bottom).Full size imageResults from CCM in Fig. 5A (plots from top to bottom) show that no significant interaction can be claimed between sardines and anchovies, as well as from sardines or anchovies in the SST manifold which expectedly indicates that neither sardines nor anchovies affect SST. This latter results, considering its biological plausibility should be taken as one validation criteria of predictive models, or complimentary as a test for anomaly detection of spurious interactions. The reverse effect of SST on sardines and anchovies can be quantitatively detected with the correlation coefficient (rho) as well as TE. Although the calculated causations between SST and sardines or anchovies are moderate, CCM is able to provide a good performance in causality inference when the length of time series used is long enough due to convergence requirement.Figure 5B shows OIF’s results of inferred causal interactions between sardines, anchovies and SST dependent on the time delay u. For sardines and anchovies, OIF exposes bidirectional interactions that are actually biologically plausible, especially when both populations coexist in the same habitat, versus the results of CCM that infer (rho =0). Ecologically speaking, even though fish populations do not directly influence sea temperature, we can find some clues about SST in fish populations influenced by SST. These clues can be interpreted as information of SST encoded in fish populations over abundance time records. So, observations of fish populations can be used to inversely predict the change of SST; this can be interpreted as “reverse predictability” (or “biological hindcasting”) in a similar way of when predicting historical climate change from ice cores. This information is captured by OIF, leading to nonzero values of TE from fish populations to SST. In this regard, we emphasize the distinction between direct and indirect (reverse) information flow, where direct information flow is most of the time larger and signifies causality (e.g. of SST for sardine and anchovies), and indirect (reverse) information flow that is typically smaller and signifies predictability (e.g. sardine and anchovies for ocean fluctuations). It is possible—especially for linear systems where an effect is observed immediately after a change—that information of SST encoded in fish populations is high if the interdependence, represented by the functional time delay u, of the environment-biota is small. However, for highly non-linear systems such as fishes and the ocean, changes in temperature may take a while before being encoded into fish population abundance53. Thus, it is correct that the highest values of TE are for high u. Values of TE for small u-s are numerical artifacts related to systematic errors leading to overestimation of interactions that are time-delayed eventually. One way to circumvent this problem, largely present for short time series, would be to extend time series by conserving their dynamics (see Li and Convertino39) or to bound the calculation of TE only for the u that maximizes the Mutual Information; this would provide an average u within a range where TE is approximately invariant. Thus, for the effect of external SST on sardines and anchovies, OIF model gives unstable causal interactions with bias for lower time delays due to known dependencies of TE on u (such as cross-correlation for instance) that establishes the temporal lag on which the dependency between X and Y is evaluated. In a sense, plots in Fig. 5B are like cross-variograms for the pairs of variables considered. TE becomes stable when the time delay is located in an appropriate range. It means that OIF requires an optimal time delay that makes results of the causality inference robust and that is related to optimal TEs (as highlighted in Li and Convertino39 and Servadio and Convertino49) that defines the most likely interdependency between variables for the u with the highest predictability. The fact that TE of sardine and anchovies to SST is high for same small ranges of u may be also a byproduct of data sampling, i.e., fish and SST sampling locations are different (fish abundance is actually about fish landings) and that can introduce spurious correlations/causation. Overall, these findings suggest that the OIF model provides more plausible results, but it requires careful selection of optimal time delays.Figure S1 shows the relationships between normalized (rho) and TE estimated for all selected values of L and u of pairs in Fig. 5 (sardine-anchovy, sardine and SST, anchovy and SST). These plots show opposite results than the proportionality between (rho) and TE in Fig. 3 because non-optimal values are used, that is non-convergent (rho)-s and suboptimal TE during the interaction inference procedure (Fig. S1). TE for too small u-s determines overestimation of interactions due to the implicit assumptions that variables have an immediate effect on each other and that is not always the case as highlighted by the vast time-lagged determined non-linear regions in Fig. 3. If “transitory” values of (rho) for small L are disregarded, as well as TEs for small u-s, the relationship between (rho) and TE shows a correct linear proportionality.Real-world multispecies ecosystemInteractions between fish species living in the Maizuru bay are intimately related to external environmental factors of the ecosystem where they live, the number of species living in this region considering also the unreported ones and biological species interactions, which lead to a complex dynamical nonlinear system. In Fig. 6 the network of observed fish species (Table S1) is reported where only the interactions considered in Ushio et al.37 for the CCM are reported. This is because the goal is to compare the CCM inferred network to the TE-based one based on abundance. Figure 7 shows the temporal fluctuations of abundance and the functional interaction matrices of (rho) and TE. In this paper we study and compare average ecosystem networks for the whole time period considered but dynamical networks can also be extracted via time-fluctuating (rho) and TE as shown in Fig. S3. These dynamical networks can be useful for studying how diversity is changing over time and ecosystem stability (Figs. S4, S6–S7) as well as understanding the relationship between (rho) and TE (Fig. S5). In the network of Fig. 6 the color and width of links are proportional to the magnitude of TE (Table S2); for the former a red-blue scale is adopted where the red/blue is for the highest/lowest TEs. The diameter is proportional to the Shannon entropy of the species abundance pdf (Table S3). The color of nodes is proportional to the structural node degree, i.e. how many species are interconnected to others. Therefore, the network in Fig. 6 is focusing on uncooperative species whose divergence and/or asynchronicity (that is a predominant factor in determining TE over divergence) is large. Yet, the connected species are rarely but strongly interacting in magnitude rather than frequently and weakly (i.e., cooperative or similar dynamics). Additionally, the species with the smallest variance in abundance are characterized by the smallest Shannon entropy (smallest nodes) and more power-law distribution although the latter is not a stringent requirement since both pdf shape and abundance range (in particular maximum abundance) play a role in the magnitude of entropy. Average entropy such as average abundance are quantities with limited utility in understanding the dynamics of an ecosystem as well as ecological function. Nonetheless, species with high average abundance (e.g. species 5) have a very regular seasonal oscillations and the largest number of interactions with divergent species. This dynamics is expected considering the population size of these dominant species and their synchrony with regular environmental fluctuations.Figure 6Part of the estimated species interaction network for the Maizuru Bay ecosystem. Species properties are reported in Table S1. The color and width of links are proportional to the magnitude of TE (Table S2); for the former a red-blue scale is adopted where the red/blue is for the highest/lowest TEs. The diameter is proportional to the Shannon entropy of the species abundance (Table S3) that is directly proportional to the degree of uniformity of the abundance pdf and the diversity of abundance values (e.g., the higher the zero abundance instances the lower the entropy). The color of nodes is proportional to the structural node degree, i.e. how many species are interconnected to others after considering only the CCM derived largest interactions (see Ushio et al.37 and Fig. 7). Other interactions exist between species as reported in Fig. 7. TE is on average proportional to (rho) (Figs. S4 and S5). Freely available fish images are from FishBase https://www.fishbase.in/search.php; the network was created in Matlab and the composition of network and images was made in Adobe Illustrator version 21 (2017) https://www.adobe.com/products/illustrator.html.Full size imageFigure 7Normalized species interactions matrices inferred by CCM and OIF models for Maizuru Bay ecosystem. In the census of the aquatic community, 15 fish species were counted in total. Interaction inferential models use time lagged abundance magnitude (CCM) or pdfs of abundance (OIF) shown in (A). (B) normalized CCM correlation coefficients ((rho)) between all possible pairs of species. (C) normalized transfer entropies (TEs) between all pairs of species from the OIF model. Both CCM and OIF predict that the most interacting species (in terms of magnitude rather than frequency) are 7, 8 and 9 on average. Thus, interaction matrices are more proportional to the asynchronicity than the divergence of species in terms of abundance pdf, although abundance value range defines the uncertainty (and diversity) for each species that ultimately affects entropy and interactions (e.g., if one species have many zero abundance instances or many equivalent values, such as species 2, TEs of that species are expected to be low due to lower uncertainty despite the asynchrony and divergence).Full size imageFigure S2 shows that the strongest linear correlation is for the most divergent and asynchronous species (from species 4–9) for which both (rho) and TE are the highest (Fig. 7B,C). This confirms the results of Fig. 3 and the fact that competition (or dynamical diversity more generally) increases predictability. This also highlights the fact that linear correlation among state variables does not imply synchronicity or dynamic similarity as commonly assumed. The interaction matrices in Fig. 7B, C confirm that TE has the ability to infer a larger gradient of interactions than (rho) and the total entropy of the TE matrix is lower than (rho). Pairwise the inferred interaction values by CCM and OIF are different but (rho) and TE patterns appear clearly similar and yet proportional to each other.CCM and OIF models are applied to calculate the potential interactions between all pairs of species. Figure 7B,C show interaction matrices describing the normalized (rho) from CCM and TE from OIF model of all pairwise species, respectively. The greater the strength of likely interaction, the warmer the color. These results demonstrate that CCM and OIF model present similar patterns for the interaction matrices in terms of interaction distribution, gradient and magnitude in order of similarity. This indicates that both CCM and OIF are able to infer the potentially causal relationships between species. Compared to the CCM interaction heatmap the OIF heatmap presents larger gradients of inferred interactions that highlight the divergence and asynchrony in fish populations of species 4–9 from other species. This difference can be observed in Figure S2 that shows the strongest linear correlations for the most divergent and asynchronous species (4–9). It is worth noting that species 4-8 are all native species (See Table SS11). Therefore, despite the patterns of interactions of CCM and TE are similar, CCM allows one a better identification of clusters of species with similar or distinct interaction ranges. Additionally TE estimates some weak observed interactions such as of species 2 (E. japonicus) with others, while CCM essentially considers null interactions for these species.Precisely, the most interacting species (4–9) are the most divergent and asynchronous species (with respect to the whole community) as well as diverse in terms of values of abundance; these species form the ”collective core” that is likely determining the stability of the ecosystem. Interestingly the number of these species is relatively small and it confirms results of other studies (see39, 54,55,56,57) showing that the number of species with weak interactions is much larger. Theory suggests that this pattern promotes stability as weak interactors dampen the destabilizing potential of strong interactors54. Mediated cooperation (e.g. by many “weakly” interacting competitors) as shown by Tu et al.52 promotes biodiversity and diversity increases stability. When considering abundance values (at same time steps) of collective core species (Fig. S2) these species are linearly related and this increases their mutual predictability by either using linear or non-linear models based on correlation coefficient and TE. This proves that non-linearity increases predictability.The choice of the optimal u that maximizes MI leads to the optimal TE model and resultant interaction network. The observed u over time is really small (Fig. S9) and this signifies how likely the ecosystem has small memory and responds quickly to rapid changes, or the information of change is carried over time by ecosystem’s interactions which lead to accurate short-term forecasting. In other words, temperature-induced changes may take long time but the information of change is replicated at short time periods. The chosen time delay (u =1) corresponds to the species sampling of two weeks. Note that values of u are also dependent on the data resolution and they are strongly related to fluctuations rather than absolute (alpha)-diversity value. Thus, while biodiversity may fluctuate rapidly in time, value of (alpha)-diversity for seasons or longer time periods can be more stable and manifesting higher memory (representative of u for the whole ecosystem) than the one between species pairs (related to pair’s u). Short-term catastrophic dynamics (for instance related to dramatic habitat change, sudden invasions, extinctions or rapid adaptations) may lead to irreversible shifts in interactions (strength and sign); this, in turn can affect biodiversity patterns that are completely uninformed by past dynamics. Thus, there is certainly a limit to predictability and to the validity of time delays which can change very rapidly. However, we insist in emphasizing that models are predictive tools and predictions are not necessarily causality reflecting the many and highly complex underlying processes. Yet, interpretation of results must be done with care.We also study temporally dynamical networks for the fish ecosystem community (see “Real-world sardine–anchovy-temperature ecosystem” section). CCM and OIF model are applied to quantify the causality between all possible pairs of species at each time period by calculating (rho) and TE, respectively. Estimated effective (alpha)-diversity (Eq. S1.2) from CCM- and TE-based inferred networks at each time point can be obtained and then compared to the taxonomic (or ”real”) (alpha)-diversity. Results are shown in Fig. 8 and Fig. S6. In the whole time period, the estimated (alpha)-diversity from CCM is constant, whereas the global trend of the estimated (alpha)-diversity from OIF model slightly decreases over time that is consistent with the global trend of real (alpha)-diversity. CCM always predicts a non-zero interaction for all species (including negative values) whereas OIF predicts zero interactions for some species that are then not making part of the estimated effective (alpha)-diversity.Figure 8Predicted (alpha) -diversity via optimal interaction threshold for CCM’s (rho) and OIF’s TE versus taxonomic diversity. Effective (alpha)-diversity from CCM and OIF are shown (blue and red) for an optimal threshold of (rho) and TE (i.e., 0.2 and 0.3) that maximizes the correlation coefficient and mutual information (MI) between (alpha _{CCM}) or (alpha _{TE}) and the taxonomic (alpha), respectively. The maximization of the correlation coefficient and MI guarantees that the estimated effective (alpha) are the closest to the taxonomic (alpha). g is the resolution of the network inference determined by the minimum number of points required to construct pdfs and infer TE robustly (see Supplementary Information section S1.3).Full size imageFigure 8 shows the effective (alpha) diversity from CCM and OIF for an optimal threshold of (rho) and TE (i.e., 0.2 and 0.3) that maximizes the correlation coefficient and Mutual Information (MI) between (alpha _{CCM}) or (alpha _{TE}) and the taxonomic (alpha), respectively. The maximization of the correlation coefficient and MI guarantees that the estimated effective (alpha) are the closest to the taxonomic (alpha). Figure S6 shows effective (alpha) for unthresholded interactions and other thresholds. Note that the threshold on TE does not coincide with the value of TE that maximizes the total network entropy (Fig. 8) and then some of the reported species may not be part of the ecosystem strongly. Thus, this threshold method is also useful to identify species that are truly forming local diversity versus transient species. Considering the pattern of fluctuations of effective (alpha)-diversity from CCM, they are poorly unrelated to the real (alpha)-diversity, while those from OIF are much more synchronous with seasonal fluctuations of real (alpha)-diversity. However, (alpha _{TE}) is a bit higher than the average taxonomic (alpha). Both CCM and TE predict a decrease in (alpha) in time that corresponds to an increase in SST. As shown in Fig S6, OIF is attributing higher sensitivity to SST for small interaction species because (alpha) fluctuations show seasonality that happens when species follow environmental dynamics closely. Vice versa, CCM predicts a broader sensitivity for all positively interacting species. These results reveal that OIF gives an effective tool to measure meaningful interdependence relationships between species for constructing temporally dynamical networks where the number of nodes over time [estimated (alpha (t))] can reflect closely the taxonomic (alpha)-diversity. This allows us to find more reliably how changes of environmental factors (e.g. SST) affect biodiversity in ecosystems. The establishment of thresholds on interactions is also useful for exploring ranges of interdependencies and associated effective (alpha)-diversity with respect to the average taxonomic diversity. (beta) effective diversity is another very important macroecological indicator informing about ecosystem changes; for instance in Li and Convertino39 (beta)-diversity identified distinct ecosystem health states. However, (alpha) and (beta) (effective diversity) variability are highly linked to each other and yet looking into one or another would provide equivalent results. The difference between taxonomic and effective (beta)-diversity may provide some information about invasive or rare species have weak influence on the ecosystem since they are characterized by low TEs. Supplementary Information contains further elaborations on results. More

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We demonstrate our approach using an agricultural field under active commercial cultivation of soybeans located in the Arkansas Delta. For this study, we acquired time-series datasets of UAV and geophysical data during the 2018 growing season. This provided the opportunity to investigate the relationship between plant growth and spatially heterogeneous soil physical properties, as well as to analyse the temporal dynamics of such a relationship. All the methodology and data collection comply with relevant institutional and national legislation.Site description and crop managementThe investigated study area (34° 24.458′ N, 91° 40.462′ W), located in Humphrey, Arkansas, is a crop field of about 20 hectars that is under active commercial cultivation of soybeans (Fig. 1).Figure 1Study area located in Humphrey, Arkansas: (a) map produced by cropScape (USDA) showing part of the Delta region with the main agricultural land covers; (b) map of the field under study reporting the locations of the ground data collection; (c) map showing the topographic elevation derived by an UAV-based DEM, acquired acquisition prior to planting; (d) gSSurgo map provided by USDA, showing the soil types. Black vertical lines separate the east, center, and west sections used in the statistical analysis. Maps made with QGIS (v 3.6, https://www.qgis.org)36.Full size imageThe study area is part of the Lower Mississippi River Basin’s (LMRB) alluvial plain37. This region is characterized by a wide range of cropland, such as soybean, corn, rice, grass, and pasture, supported by a humid subtropical climate with hot, humid summers and mild, slightly drier winters.The crop field is adjacent to the oxbow Glenwood Lake (shown in Fig. 1) and is characterized by an alternating of Hebert silt-loam and Rilla silt-loam, according to the description provided by the gSSURGO map38. The Hebert profile consists of a very deep, poorly drained, moderately slowly permeable soil that formed in silty alluvium, while the Rilla profile consists of a very deep, well drained, moderately permeable soils that formed in reddish silty and loamy alluvium.High-quality soybean seed requires three appropriate conditions for germination—soil moisture, temperature and oxygen. Provided that the soil is not saturated, oxygen concentration is not a limitation, but germination will not occur in flooded soil dueto lack of oxygen. Research shows that soybeans are in general very susceptible to soil saturation and anoxic conditions, which would provoke damage to the root system and in some case the termination of the plant growth39.Planting started on April 21st in rows (raised beds) with a spacing of 0.9 m. Plants reached their first vegetative stage of emergence (VE) on the first days of May and developed their first node with fully developed unifoliate leaves around mid-May (V1). The stage of initial bloom, which represents the starting of the reproductive stage (R1–R8) after the last vegetative stage, started in mid-June, reaching the full maturity (R8) about the end of September. Subsequent to R8, the leaves dry then fall from the plant at the time of harvest, which was performed on October 24th.A common channeled (furrowed) surface irrigation approach was used as an irrigation system, which consists of a plastic pipe (polytube) located along one side of the field to deliver water into the channels between the rows40. At the site, the plastic pipe was located on the east side of the field with the rows and furrows running east to west following the main slope gradient (Fig. 1) from east to west, in order to facilitate the water flow. Smoothing of the field’s ground surface was performed prior to the planting to ensure a good surface drainage. Irrigation started on June 12th, after the rain period, and performed every 14 days.Temperature and precipitation were measured by a weather station that we installed in the southest corner of the field (Fig. 1). Based on the data collected, the average daily temperature increased from 14.6 to 24.7 °C between planting and the first vegetative stage (April–May), reached 27 °C during the vegetative period through the initial reproductive period (June–July), and decreased from 27 to 24 °C during the final period of maturity (August–September). The 2018 growing season was characterized by sporadic heavy-rain events, especially during the vegetative period (from April to mid-June), with a total cumulative precipitation (period April–September) of about 650 mm (Supplementary Figure S1). As a result of rain events, the soil moisture in April ranged from 0.17 (m3/m3) to 0.3 (m3/m3) (Supplementary Figure S2).Instrumentation, data acquisition, and preprocessingUAV-data acquisitionMultitemporal image acquisitions were performed over the field using UAV-mounted sensors. The images were acquired on: May 8th, a few days after planting, May 28th, during the early vegetative development, and June 25th, July 23rd, September 19th, the beginning, middle, and end of the reproductive period.We used the DJI Matrice 600 flying platform with the DJI FC350 on-board RGB (red, green, blue) camera as the primary image acquisition sensor. A flight plan to acquire high-resolution images at centimeter resolution was designed using this acquisition system. Flights were performed during the maximum Sun’s elevation and with clear sky to avoid possible changes in light conditions, at an altitude of about 60 m, with a 75% image overlap, reaching a ground resolution of 2 cm.In order to minimize the impact of possible variations in illumination conditions during each acquisition, two normalized mosaics were computed by using the pictures of a reference panel taken before and after each flight. The final mosaic was obtained by averaging the two normalized mosaics.A total of 13 Ground Control Points (GCP) were positioned along the field perimeter and within the field for the mosaic reconstruction (Fig. 1). These consisted of flat targets and were surveyed at each acquisition with a Topcon real‐time kinematic differential global positioning system (Hyper-V RTK-GPS). The system is composed of a rover and a fixed base station, ensuring high-resolution geolocation information at centimeter resolution. All the collected geolocation data were post-processed obtaining an average planar error of about 2 cm and a vertical error of about 3 cm.We used the commercial PhotoScan (Agisoft) software for photogrammetry to construct the final mosaics. The software uses photogrammetry techniques based on structure from motion from multi-view stereo (SfM-MVS) images42. The processing produced mosaics with spatial resolution less than 10 cm, but the mosaics used in this work have a have a pixel size of 10 cm. Additionally, we used the software to compute digital elevation models (DEMs) from the RGB images with a final vertical resolution less than 5 cm.Electromagnetic Induction (EMI) systemSoil ECa was measured with an electromagnetic induction (EMI) system (CMD Mini-Explorer, GF Instruments), which consists of a transmitter and three receiver coils, allowing an effective acquisition of soil apparent electrical conductivity (ECa) averaged across a depth range of 0.5 m, 1.0 m, and 1.8 m from the surface. The instrument provides soil ECa measurements in milliSiemes per meter (mS/m) and the in-phase in parts per thousand (not considered in this study). The system was mounted in a sled pulled by an all-terrain vehicle and supported by a Differential Global Positioning System (DGPS) in order to tag each measurement with high-resolution positioning information. Time-lapse soil ECa data collection was performed four times, including before and during the growing season: April 21st, May 12th , June 11th, and July 9th. EMI acquisitons were planned according to the irrigation plan and UAV acquisitions. Since the EMI instrument is sensitive to changes in soil moisture, we avoided the collection during or right after irrigation events. The data were acquired consistently every 10 rows with a distance between traverses of about 9 m.In situ soil monitoring, ground imagery, and soil samplingSoil sensors (5TE, METER) were used for continuous acquisition of volumetric water content (VWC) and soil temperature. The sensors were deployed in five areas (A, B …, E, see Fig. 1) based on two factors: a) spatial varibaility of geophysical properties, b) differences in yield during the 2017 growing season (historical data available for this study). Each area has one logger that accomodates 4 sensors within a maximum radius of 4 m. To capture both shallower and deeper soil dynamics, we positioned 2 sensors vertically aligned at the depths of 12 cm and 25 cm on both the south (side1) and north (side2) of the logger. The areas A and D were positioned in high and low soil EC zones, respectively, while B was in an area of transition. Sensors A, B, and D were positioned along the same rows to forms two transects, one on the south and one on the north. Sensor E was positioned within a a very low 2017 yield in the west side of the field but at a mid-range of soil EC values. The logger C was positioned across a soil EC boundarie, but it stopped recording data at early season and was discarded from this analysis. The collected temprature data are used to correct the EMI acquisitions, whereas soil moisture is used to track the irrigation events and possible saturations. As a support system for the sensors’ data transmission, we used EM60G (METER) data loggers and ZENTRA cloud (METER) for cloud data storage.Ground RGB images were taken at several plots (Fig. 1) for ground-based plant spatial abundance assessment. We used a wooden-frame of 1 m by 1 m to delimit the plot’s area and took a picture from above (1.4 m from the ground). This assessment will be used to validate UAV-based plant spatial abundance estimates.Soil samples were collected at twelve locations (Fig. 1). Physical laboratory testing was performed at the Fayetteville Agriculture Diagnostic Laboratory of the University fo Arkanas to characterize soil properties, such as clay density, porosity, dry bulk density, volumetric water content, and organic matter. Such information was used for statistical analysis and to provide an interpretation of the soil ECa measurements. No plant tissues or seeds were collected in this study.Yield data from a combine harvesterCrop yield was harvested on October 24th. The combine harvester recorded yield information as cloud-point data in bushels per acre (bu/ac) and reported here in kilogram per hectare (kg/ha), and the relativre geolocation information. The combine harvester is able to collect 11 rows at once and records a point measurement every 2 s. Areas with artefacts were determined based on the swath parameter recorded during the harvesting. The wrong choice of the swath parameter on the combine harvester would produce unreliable yield values. Such areas were not considered in further analysis.MethodologyWe designed a data analysis framework to combine the UAV optical images and soil geophysical measurements for characterizing plant phenological and physiological properties and soil spatial heterogeneity, respectively. We then performed statistical analysis of their co-variability to quantify the influence of soil properties on plant development. This framework is constructed by the following three steps:

1.

UAV-based monitoring of plant dynamics: This step presents the data processing pipeline used for computing UAV products to characterize plant phenology and structure, such as plant spatial abundance, plant-specific vigor, and plant height.

2.

EMI-based monitoring of soil characterization: This step focuses on quantifying the soil spatial variability. We perform statistical analyses to identify the relationship between soil ECa signal and textural information derived by soil samples collected during the ground data acquisition.

3.

Quantifying soil–plant spatiotemporal co-variability: This step focuses on the statistical analysis of plant physiological and structural properties and near-surface properties, as well as identifying key environmental factors.

UAV-based plant monitoringFigure 2 depicts the data processing pipeline developed to assess the spatial variability of plant characteristics during the growing season. The steps to obtain the plant spatial abundance and plant vigor maps are presented in detail in the following sections.Figure 2Pipeline developed for the UAV data processing.Full size imagePlant spatial abundancePlant spatial abundance is estimated by the RGB mosaics. In the first step of the procedure, a vegetation map was obtained by performing a binary classification to discriminate the two classes of vegetation and soil. We used a supervised spectral-spatial image classification approach43 that combines both the spectral and the contextual (i.e., geometrical) information. The contextual information represents the pixel spatial arrangement and the spatial reletionship between adjacent pixels. By modelling such infromation, we can extract structures within the scene. For this step, we use mathematical operators, such as attibute profiles44,45,46 that belong to the image processing sub-field of mathematical morpohlogy. Such operators are 2D filters that act on regions of connected pixels based on the evaluation of an attribute (e.g., scale, defined as number of pixel composing the region). The advantage of using this operator is the ability to preserve the geometrical detail of the structures in the scene. In a filtered image (or feature), regions with similar properties (i.e., scale) are preserved or otherwise merged to their surroundings (i.e., filtered). A multiscale contextual characterization is obtained by applying a filter recursively and relaxing the scale parameter of the filter at each iteration, resulting in a stack of filtered images. An automatic procedure developed in45 is used to perform such a characterization. Such operation serves to extract homogeneous structures present in the scene at different spatial scales, allowing us to better delineate the soybean rows. The original RGB image, together with the filtered images, compose the feature space used as input to a support vector machine (SVM) algorithm with a radial basis function (RBF) kernel. The algorithm is based on the LIBSVM library47 developed for the MATLAB environment, using a one-against-one multiclass strategy. In a second step, we compute the spatial abundance of the detected plants as the percentage of pixels that belong to the vegetation class within a grid unit. In our study we chose a grid unit of 2 by 2 m, which would allow to obtain a good approximation of the local change, while capturing large scale patterns. The quality of the plant spatial abundance map is mainly dependent on the performance of the classification algorithm used to separate the plants from the soil in the background. To validate this product, we computed the classification performance based on ground-truth data and compared the UAV-based estimates of plant abundance to ground-based assessments. The ground assessment of the plant spatial abundance was performed by selecting 31 plots in conjunction of the UAV acquisition occurring in May 28th, 2018. The plots were located mainly along the east and west sides of the field within a distance that ranges between 12 and 25 m from the edge of the field towards the field center; and some plots were collected along the south and north sides within a distance of 50 m from the field edge. These plots were selected to captured representative spatial abundances, ranging from 0% (almost bare) to 60%. Ground images were systematically taken at a height of 1.4 m from the ground. From these images, we computed the GCC and identified a threshold to separate the plant covered area from the soil backgroun. The plant spatial abundance was then computed as the percentage of pixels identified as soybean over the entire area of the plot. All the ground-data were geolocated using the RTK-GPS system previously described.Plant-specific vigor estimationRemote-sensing-derived VIs have been extensively used as a proxy to estimate plant vigor and investigate plant physiology and response to possible ecosystem changes. Widely used VIs for plant characterization are in general derived by the near-infrared (NIR) channel (e.g., NDVI, SAVI, etc.), which is sensitive to changes in chlorophyll and leaf area. However, RGB-based indices have been developed and used in several phenology studies, showing their effectiveness in estimating plant characteristics compared to NIR-based VIs48. The GCC was chosen based on authors’ recent studies, showing that this RGB-based VI is less affected by saturation compared to NDVI31. GCC is formally defined as follows:$${text{GCC}} = frac{{{text{G}}_{{{text{DN}}}} }}{{{text{B}}_{{{text{DN}}}} + {text{G}}_{{{text{DN}}}} + {text{R}}_{{{text{DN}}}} }},$$
(1)

where R, G, and B are the red, green, and blue channels, and DN refers to the pixel intensity values as digital numbers41. The plant-specific vigor was computed considering only those pixels identified as vegetation, the information for which is provided by the binary classification. In order to have products spatially comparable, the plant-specific vigor was transformed into a 2 by 2 m grid by computing the average within each grid unit. This procedure has the advantage to minimize the soil component in estimating plant vigor, avoiding the nonuniqueness of lower productivity versus increased soil coverage. Specifically, it allows a more accurate way of capturing the spatial variability of the plant’s vigor, in particular during the early vegetative stage, when the soil component is overabundant compared to the plant spatial abundance. This is a clear advantage over a satellite-derived VI with a coarser resolution that would provide an underestimation of plant vigor due to soil coverage.Plant height estimationPlant height was computed by subtracting the digital elevation model (DEM) of bare soil (i.e., reference DEM) from the digital surface model computed at each acquisition. The reference DEM was computed from acquisition occurred on May 8th, which was planned a week after planting so that it is not affected by surface disturbances by the planter. DEMs were computed using the PhotoScan software (Agisoft) during the mosaic reconstruction phase. We assume that the ground surface elevation stays the same during the growing season.EMI-based soil characterizationWe used the ECa data to evaluate the spatial heterogeneity of soil properties and the temporal variability in soil moisture. We primarily used the EMI measurements associated with the 1 m effective depth, which is the depth range expected to encompass the soybean roots. Temperature corrections were applied to the point-cloud data for correcting the measured soil ECa to the reference temperature of 25 °C. We employed the exponential model presented in Corwin and Lesch49 and developed by Sheets and Hendrickx50, and formally described as follows:$${text{E}}C_{25} = {text{E}}C_{a} cdot left[ {0.4470 + 1.4034e^{{left( { – T/26.815} right)}} } right],$$
(2)

where ({text{E}}C_left{ {25} right}) represents the corrected soil ECa at the reference temperature of 25 °C, ({text{E}}C_a) represents the measured soil ECa, and ({text{T}}) represents the soil temperature at the time of the acquisition. The model was chosen based on the on the methodological comparison51, which shows that the exponential model provides very similar estimations compared to the widely used ratio model31,52 in the temperature range of 3–43 °C, but with a smaller total residual.A spatial correction was applied to the data to resolve a spatial shift caused by the distance from the EMI instrument and the actual position of the GPS antenna, which were 4.5 m apart due to the system setup. We performed statistical analysis between EMI data and soil samples collected during the April data acquisition to identify the main control on the spatial variability of soil ECa.Co-variability among plant and soil signaturesWe performed a suite of statistical analyses, based on the high-resolution data layers, on soil and plant characteristics (i.e., plant spatial abundance, plant vigor, plant height, soil ECa, and crop yield) to investigate the impact of soil properties on plant development during the growing season. For the analysis, we considered the yield point-cloud data as geo-reference.Since the EMI and crop-yield data had a lower vertical resolution (along the y axis) of about 10 m (distance between rows selected for the data collection), we computed the average of each metric within a window of 20 m by 20 m. The exploratory data analysis included run charts and scatter plots to investigate the temporal evolution of plant development during the growing season, and the co-variability between plant characteristics and soil ECa. We evaluated the correlation between variables using Pearson’s correlation, as well as their spatial association. To control for the spatial non-independence, we derived a corrected Perason’s correlation by using the hypothesis testing procedure propsed by Clifford, Richardson, and Hémon (1989)53. The method uses Moran’s I54 to compute the spatial autocorrelation in the spatial data sets and estimate the correct degrees of freedom, which are then used to assess the significance of the correlation. The method uses the Sturges’ formula to identify the number of distance classes55. More