Ecology

Our method is based on the assumption that the information structure of a system captures every effective interaction among its agents and thereby reflects the condition of the system. The abstract nature of information theory allows us to analyse systems independently of their specific instantiation. We only rely on the availability of longitudinal data reflecting the strength of the system’s individual components in a very broad sense. Hence, in general, our method can be applied to any complex system. The only condition is that for a given period of time and for every component of the system, a time series of quantitative data reflecting the outcome of interactions exists. Such time series could exemplarily be biomass of a plant species, number of individuals of an animal species, or sales of a company. The data type can differ among the components of the system, i.e. be heterogeneous.
In a first step, networks of information transfer are inferred via pairwise estimation of transfer entropy9 among all agents. Considering these networks and, in particular, their development over time, offers insights into functional interactions.
In the second step, potential, connectedness and resilience are computed solely using the networks of information transfer (see Supplementary A for a review of the adaptive cycle and its defining variables). Here, we utilize capacity and ascendency as being defined by Ulanowicz in the context of ascendency theory11. Note that Ulanowicz also used information theory to define capacity as an entropy of flows and ascendency as mutual information between inflow and outflow. While the first one is a measure of the average indeterminacy in the fluxes of the network, the latter quantifies the efficiency the system has in making use of its capacity12. However, being rooted in systems ecology, Ulanowicz always considered flows of physical quantities, such as energy or resource fluxes. In contrast, we will derive the quantities from networks of information transfer, abstracting from the physical representation of the interaction. Thus, potential is the capacity of the network of information transfer, and connectedness the corresponding ascendency.
The challenging part of our approach is to find an appropriate measure of resilience. There exist various conceptions and following definitions of resilience13. For our purposes, Holling’s definition of resilience fits best, namely to define resilience as ”the magnitude of disturbance that can be absorbed before the system changes the variables and processes that control behavior” (Ref.1, p. 28). There have been various approaches to make this notion measurable, however, all of them either depending on the specific system under observation14,15,16,17 or requiring deep knowledge of the system dynamics18. Resilience has also been studied from a network perspective (see e.g.19,20). Since we are modeling complex systems as networks of information transfer, our definition is inspired by a common concept in spectral graph theory. We use the so-called graph Laplace operator, which captures vulnerability of a network with respect to perturbations of its topological structure.
Taken together, the development of these three variables displays the system’s course through the adaptive cycle, helping to better understand system maturation and to evaluate its current condition. We will now provide a detailed description of our method, its implementation, and its application in the three case studies presented in this paper.
Step 1: estimation of networks of information transfer
Let (mathscr {V}) be a collection of variables, quantifying the state of agents defining a system. Let (I = (i_1, dots , i_N)) and (J = (j_1, dots , j_N)) be two sets of samples of states for the components I and J, say. For example, I and J can be identified with abundances of two interacting species at time points (1, dots , N). We consider the time series I and J as realisations of two approximately stationary discrete Markov processes. This allows us to compute Schreiber’s transfer entropy9, serving as a measure of their effective interaction. Transfer entropy from J to I is defined as

$$begin{aligned} T_{J rightarrow I} = sum _{n = 1}^{N-1} p left( i_{n+1},i_{n}, j_{n} right) cdot log left( frac{p left( i_{n+1}|i_{n}, j_{n} right) }{p left( i_{n+1} | i_{n} right) } right) . end{aligned}$$

(T_{J rightarrow I}) quantifies the average reduction in uncertainty about the future of I given the past of J. In other words, how much additional information do we gain about the next state of I, knowing not only the past of I itself, but the past of J as well. In the literature, a multitude of studies on the interpretation of transfer entropy in general and in specific contexts can be found21,22,23.
As the probabilities occurring in the definition of transfer entropy are in general not known, we have to estimate transfer entropy on the basis of given realizations of the random variables, i.e. the data given as samples of the time series. Typically, we do not use all available samples to estimate transfer entropy at time t but samples falling within a certain window of time preceding time t. The size (w_t) of this windows can either be fixed, or depend on the time t, e.g. (w_t = t). In the first case, the window is “shifted” going along with t to guarantee transfer entropy always being estimated on the same number of samples. In the second case, the window starts at the beginning of the time series and is extended with increasing t. In this case, the full history of the time series is considered for estimating transfer entropy. The choice of the window size depends on the system under consideration. In any case, it should be at least as large as the assumed order of the underlying Markov process. We then compute the information transfer from J to I at time t estimating transfer entropy over the period (t-w_t+1,dots ,t). To be precise,

$$begin{aligned} T_{J rightarrow I}^t = sum _{n = t-w_t+1}^{t} p left( i_{n+1},i_{n}, j_{n} right) cdot log left( frac{p left( i_{n+1}|i_{n}, j_{n} right) }{p left( i_{n+1} | i_{n} right) } right) . end{aligned}$$

Depending on the size of (w_t) and the data being available, it can be useful to increase the number of data points falling within every window by interpolation. For our calculations, we used the Matlab function pchip. Interpolation stabilizes the estimation in case of small window sizes. At the same time, interpolating too many points reduces stochasticity in the time series due to the deterministic component being introduced by the interpolation model. Thus, there is a trade-off between stochasticity and stability which has to be taken into account.
We estimated (T_{J rightarrow I}^t) using the Kraskov-Stögbauer-Grassberger (KSG) estimator TransferEntropyCalculatorKraskov as being provided with the JIDT toolkit24. For all our calculations, the function has been called using the data ((j_{t-w_t+1},dots ,j_t)) and ((i_{t-w_t+1},dots ,i_t)) in the mode computeAverageLocalOfObservations. For all other parameters of the estimation procedure, we used the default values (k=k_{tau}=l=l_{tau}=delay=1). Other choices of these parameters can be reasonable depending on the specific system to be analysed. To distinguish actual interactions from random noise, we tested all estimates via hundred-fold bootstrapping using the function computeSignificance(100) incorporated in the toolkit. Tests passing below a given significance level have been accepted and thus lead to an edge between the corresponding components with the estimated transfer entropy defining the corresponding weight. Estimating and testing for all pairs of components at fixed time t, we finally obtained a weighted, directed graph

$$begin{aligned} G^t = left( mathscr {V},{T_{J rightarrow I}^t|(J,I) in mathscr {V} times mathscr {V} } right) end{aligned}$$

as being our inferred model of interaction at time t. Given time series of abundances of length N for each component, this results in a sequence of interaction networks for time points (w_1,dots ,N).
Summarizing, the first step infers models of interaction among the given variables in form of a series of networks capturing the interaction patterns and their strengths. These network models can then be used in the second step to actually determine the position of the system within the adaptive cycle.
Step 2: determining potential, connectedness, and resilience
As mentioned before, our definitions of potential and connectedness are based on Ulanowicz’s notions of capacity and ascendency11. Ulanowicz provides further information on the theoretical background of these measures. Let

$$begin{aligned} T^t = sum _{(J,I) in mathscr {V} times mathscr {V}} T_{J rightarrow I}^t end{aligned}$$

be the total transfer of the system at time t. We further introduce the following shorthand notation

$$begin{aligned} T_{J}^{text {out},t} = sum _{I in mathscr {V}} T_{J rightarrow I}^t qquad text{ and } qquad T_{I}^{text {in},t} = sum _{J in mathscr {V}} T_{J rightarrow I}^t. end{aligned}$$

Define

$$begin{aligned} P^t = – sum _{(J,I) in mathscr {V} times mathscr {V}} T_{J rightarrow I}^t cdot log left( frac{T_{J rightarrow I}^t}{T^t} right) qquad hbox {as the system’s} ,potential ,hbox {at time} ,t end{aligned}$$

and

$$begin{aligned} C^t = sum _{(J,I) in mathscr {V} times mathscr {V}} T_{J rightarrow I}^t cdot log left( frac{T_{J rightarrow I}^tT^t}{T_{J}^{text {out,t}}T_{I}^{text {in,t}}} right) qquad ,hbox {as its} ,connectedness ,hbox {at time}, t. end{aligned}$$

Being essentially a sum over the indeterminacy in each transfer within the system, potential can be interpreted as a measure of the system’s power for evolution and its ability to develop. Recall that development of the system as a whole necessarily relies on communication, i.e., transfer of information among its components. In contrast, connectedness measures the degree of internal coherence of the system by contrasting information leaving one component with information arriving at another component.
In order to define resilience, we need to capture vulnerability of the system with respect to unforeseen perturbation. In terms of graph theory, this can be achieved by studying the eigenvalues of a certain matrix, being associated with the graph. Indeed, the smallest non-trivial eigenvalue of the so-called graph Laplacian of an undirected graph quantifies the vulnerability of the graph with respect to disturbance of the topology of the graph25,26. In our case, we need to transfer this idea to the case of the directed graphs (G^t).
Thus, given (G^t=left( mathscr {V},T^t_{J rightarrow I}|(J,I) in mathscr {V} times mathscr {V}right)) be a non-empty, weighted, directed graph with vertex set (mathscr {V}). Let further (c > 0) be a constant. Let (D_{out}) and (D_{in}) be the diagonal matrix of out-degrees and in-degrees, respectively, and A the weighted adjacency matrix. We then define the following Laplace type operators of (G^t):

$$begin{aligned} L_{out} = c cdot D^{-frac{1}{2}}_{out} left( D_{out}- A right) D^{-frac{1}{2}}_{out}, quad hbox { and }quad L_{in} = c cdot D^{-frac{1}{2}}_{in} left( D_{in} – A right) D^{-frac{1}{2}}_{in}, end{aligned}$$

following the convention that (D^{-frac{1}{2}}_*(u,u) = 0) for (D_*(u,u) = 0). Note that, for the sake of readability, we omitted the superscript t in these definitions. For all case studies presented in this paper, we used

$$begin{aligned} c = frac{1}{max { T^t_{J rightarrow I}|(J,I) in mathscr {V} times mathscr {V} }} end{aligned}$$

as standardization constant.
Since A is no longer symmetric, the spectrum of (L_{out}) and (L_{in}) is complex in general. Nevertheless, the distance of the spectrum to the imaginary axis in the complex plane still determines the stability of the graph. Therefore, we define resilience of the graph G as the smallest, non-trivial absolute value of the real parts of all eigenvalues of its two Laplacian matrices, i.e.

$$begin{aligned} R^t = min left{ |mathfrak {R}sigma | :sigma in {{,mathrm{Spec},}}(L_{out}) cup {{,mathrm{Spec},}}(L_{in}), sigma ne 0right} . end{aligned}$$

See Supplementary E for a more detailed explanation motivating this definition as well as for an alternative definition of the involved Laplacian matrices.
Our definitions of the three systemic variables are summarized in Table 1. In addition, Table 2 displays basic information concerning the data sets and parameters of our case studies. Note that, for visualization purposes, Figs. 3, 4, and 5 show a smoothed version of the estimated variables as being obtained by applying the R functions smooth.spline and splinefun.
Table 1 Summary of the definitions of the three systemic variables.
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Table 2 Data and parameters of the presented case studies.
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Figure 2

Schematic representation of our quantification method. In the first step, time series of abundance data (a) are transferred into networks of information transfer (b). In the second step, the three systemic variables (c) are computed on basis of the networks. The figure depicts the window shifting method.

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We created the R package QtAC in order to enable a straightforward application of our method27. The package comprises all functions required to compute a system’s course through the adaptive cycle and to visualize the results.
Figure 2 illustrates the key idea of our approach. Figure 2a shows randomly generated abundances of five components (A,B,C,D,E). To estimate the position of this small system within the adaptive cycle at time t and (t+1), we estimate transfer entropy for all pairs of components based on the samples within the window ((t-w+1, dots , t)) and ((t-w+2, dots , t+1)), respectively, and test for significance. This results in two inferred interaction networks shown in Fig. 2b. Using these networks, we can compute potential, connectedness and resilience at these two points in time. Figure 2c depicts the shift the system has made in the coordinate system spanned by the three characteristic variables.
The decrease in resilience from t to (t+1) mainly follows from loosing the edge (Drightarrow C) at (t+1). With component D being connected with the rest of the system by one edge, only, the system becomes more vulnerable, since perturbation of the edge (Drightarrow E) would fully decouple D from the rest of the system. Similarly, the loss of this edge also leads to a decrease of potential. Heuristically, the more edges a system has, the more potential there is to change from one state to another. Note that the even distribution of weights also added to the system’s potential, as for example edges (Arightarrow E) and (Arightarrow C) both loose weight. The moderate decline of connectedness follows from the loss of the edge (Drightarrow C) as well as from the smaller capacity of the edges (Arightarrow C) and (Arightarrow E), decreasing the overall total edge weight. More

Reagents and equipment
Para-nitrophenol, analytical grade; triethanolamine, analytical grade; calcium acetate (Ca(CO2CH3)2), analytical grade, China National Pharmaceutical Group Co., Ltd. Potassium chromate (K2CrO4), analytical grade; hydrochloric acid (HCl), analytical grade, Beijing Chemical Works. Calcium chloride (CaCl2·2H2O), analytical grade, Shanghai Macklin Biochemical Co., Ltd. Sodium hydroxide (NaOH), analytical grade, Xilong Chemical Industry Incorporated Co., Ltd. pH meter, Thermo Fisher Scientific, USA. ICP-OES 730, Agilent. X ray diffractometer (XRD), Bruker D8, Germany.
Soil conditioner: Developed by Tianjin Cement Industry Design and Research Institute Co., Ltd. Specific preparation method: The raw materials were K-feldspar (produced in Inner Mongolia, with ({mathrm{KAlSi}}_{3}{mathrm{O}}_{8}), ({mathrm{NaAlSi}}_{3}{mathrm{O}}_{8}), and SiO2 being the main components), CaCO3, and CaMg(CO3)2, which were crushed and ball-milled in order to get the sizes that could pass through an 80 μm sieve, before being mixed in appropriate ratios. Next, these were sintered by using an alumina crucible placed in a box furnace at 1270 °C for 60 min, then naturally cooled inside, and finally ground to approximately 80 μm to obtain the MSCs.
Soils samples
14 typical acid soils were listed in Table 1, which are from three provinces in China: Hunan (No. 1–5), Sichuan (No. 6–10), and Guangdong (No. 11–14). Soils were sampled in 0–20 cm, with all vegetation residues in the surface layer removed, then were placed indoors, naturally air-dried, and passed through a 2 mm sieve. A portion of each sample was used for testing and analysis, and the remained was used for the culture test.
Table 1 Information of the 14 types of acid soil samples in China.
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The evaluation of relationship between LR from schematics of ΔpH and soil acidity by using SMP-DB method
Mclean’s improved SMP-DB method was used to calculate LR. The calculation principle for the double buffer method is shown in Fig. 1, while the specific operating method and calculation principles of the experiment are as follows.
Figure 1

Computation of soil LR from the double buffer schematics and the relationships of the resulting similar triangles. (Notes: d = acidity of the soil neutralized by the buffer solution when the soil–buffer solution was at the ideal pH (6.5); d1 = acidity of the soil neutralized by the buffer solution when pH of the soil–buffer solution reduced from 7.5 to 1; d2 = acidity of the soil neutralized by the buffer solution when pH of the soil–buffer solution reduced from 6.0 to 2; pH1 = pH of the soil–buffer solution after addition of the SMP buffer solution; and pH2 = pH of the soil–buffer solution after addition of HCl.).

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Based on the schematics of the double buffer method and in accordance with the isosceles triangle principle, the proportional relationship was established as shown in Eq. (1):

$$frac{d-{d}_{2}}{{d}_{1}-{d}_{2}}=frac{6.5-{pH}_{2}}{{pH}_{1}-{pH}_{2}}$$
(1)

Equation (2) was obtained after conversion of Eq. (1):

$$d={d}_{2}+left({d}_{1}-{d}_{2}right)frac{6.5-{pH}_{2}}{{pH}_{1}-{pH}_{2}}$$
(2)

According to Fig. 1, Eqs. (3) and (4) were obtained by making (Delta {pH}_{1}=7.5-{pH}_{1}) and (Delta {pH}_{2}=6.0-{pH}_{2}), respectively. ({left(frac{Delta x}{Delta y}right)}^{^circ }) is the milligram equivalent (meq) of H+ that must be depleted to increase the pH of the buffer solution by one unit. It is determined based on the standard curve of the buffer solution.

$$frac{{d}_{1}}{Delta {pH}_{1}}={left(frac{Delta x}{Delta y}right)}^{^circ }$$
(3)

$$frac{{d}_{2}}{Delta {pH}_{2}}={left(frac{Delta x}{Delta y}right)}^{^circ }$$
(4)

Equations (5) and (6) were obtained after conversion of Eqs. (3) and (4):

$${d}_{1}=Delta {pH}_{1}times {left(frac{Delta x}{Delta y}right)}^{^circ }$$
(5)

$${d}_{2}=Delta {pH}_{2}times {left(frac{Delta x}{Delta y}right)}^{^circ }$$
(6)

These were substituted into Eq. (2) and then integrated to obtain Eq. (7):

$$mathrm{d}=Delta {pH}_{2}times {left(frac{Delta x}{Delta y}right)}^{^circ }+left(Delta {pH}_{1}-Delta {pH}_{2}right)times {left(frac{Delta x}{Delta y}right)}^{^circ }times frac{6.5-{pH}_{2}}{{pH}_{1}-{pH}_{2}}$$
(7)

Equation (7), intended for theoretical calculations, was derived from the mathematical relationships among the various parameters. (d) is the equivalent acidity of the soil neutralized by the buffer solution when the soil–buffer solution was at the ideal pH (6.5). The LR required to neutralize 5 g acid soil to pH 6.5 could then be extrapolated based on the measured data and the aforementioned equation.
Since the molecular weight of 1 mol of CaCO3 is 100, there is a clear conversion relationship between lime and CaCO3. For calculation convenience, LR is commonly expressed as the mass of CaCO3 needed to deplete the H+ present in 100 g of soil. In other words, ({mathrm{L}}_{R}= {text{meq CaCO}}_{3}/100 ,mathrm{g}=20mathrm{d}). After comparing the results obtained via the SMP-DB method and the Ca(OH)2-titrated acidity method, Mclean found that Eq. (8), a revision of the earlier equation, had a better correlation with the actual situation.

$${mathrm{L}}_{R}= {text{meq CaCO}}_{3}/100 ,mathrm{g}=1.69left(20dright)-0.86$$
(8)

Under normal circumstances, the weight of a 20 cm thick layer of ploughed soil would be 2250 t per hectare. For one hectare of soil, the LR of CaCO3 could be calculated using Eq. (9), with the unit being tons per hectare. This amount is expressed in meq CaCO3/100 g soil; to obtain approximate rates in metric tons per hectare (0–20 cm), it can be multiplied by 1.125.

$${L}_{R}= {text{meq CaCO}}_{3}/100 ,mathrm{g}times 1.125=left[1.69left(20dright)-0.86right]times 1.125=38.03d-0.97$$
(9)

SMP-DB buffer performance
Preparation of buffer solution27
800 mL distilled water was poured into a 1 L beaker, and then 1.8 g of para-nitrophenol, 2.5 mL of triethanolamine, 3.0 g of K2CrO4, 2.0 g of Ca(CO2CH3)2, and 53.1 g of CaCl2·2H2O were added into the beaker; the mixture was then stirred and mixed. NaOH 40% (w/w) or HCl 50% (v/v) was used to adjust the pH to 7.5 before the buffer solution was transferred to a 1 L volumetric flask. The beaker was rinsed for 3 times with 50 mL of distilled water, and the rinses were transferred to the volumetric flask. The eventual constant volume was 1 L.
Test method for buffer standard curve titration
1 mL of 0.05 M HCl was added into a 50 mL beaker with 10 mL of buffer solution in it and the pH of the solution was measured after stirring for 1 min. This operation was repeated 8 times and the corresponding pH was recorded to plot a titration curve.
The standard curve of the prepared buffer solution is shown in Fig. 2. It has a pH of 5–8 and its standard curve is linear, which could be fitted using a proportional function. The fitting yielded the linear equation y = −7.19x + 8.05, r2 = 0.998, which was highly significant. The buffering performance of the buffer solution was calculated based on the fitting equation, and the specific calculation process is stated below.
Figure 2

XRD pattern of the MSCs.

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Two points (x1, y1), (x2, y2) were selected and substituted into the fitting equation to obtain Eqs. (10) and (11):

$${y}_{1}=-7.19{x}_{1}+8.05$$
(10)

$${y}_{2}=-7.19{x}_{2}+8.05$$
(11)

The two equations were subtracted to obtain Eq. (12):

$${mathrm{y}}_{1}-{mathrm{y}}_{2}=-7.19left({x}_{1}-{x}_{2}right)$$
(12)

Let (Delta y={y}_{1}-{y}_{2}, Delta x=-left({x}_{1}-{x}_{2}right)), then Eq. (13) would be established.

$$frac{Delta y}{Delta x}=frac{{y}_{1}-{y}_{2}}{-left({x}_{1}-{x}_{2}right)}=7.19$$
(13)

When the pH increased by one unit, ∆ y = 1 was substituted into Eq. (13) to obtain Eq. (14):

$${left(frac{Delta x}{Delta y}right)}^{^circ }=frac{-left({x}_{1}-{x}_{2}right)}{{y}_{1}-{y}_{2}}=frac{1}{7.19}=0.139$$
(14)

In other words, 0.139 meq of H+ must be depleted per unit increase in the pH of the buffer solution.
During the process of the buffer titration test, the pH of the buffer was adjusted from 7.5 to 6.0. It was shaken again to determine pH2. Titration was performed using 0.05 M HCl (1 mL of 0.05 M HCl = 0.05 meq HCl). When the pH of the buffer solution was adjusted from 7.5 to 6.0, the reduction of 1.5 units required 0.139 × 1.5 = 0.2085 meq HCl, which converted to 4.2 mL of 0.05 M HCl.
Test procedure for buffer titration28
Soil pH was measured using a glass electrode pH meter. 5.00 g soil sample was weighed and placed in a 50 mL beaker. Deionized water was added at a 1:1 water-to-soil ratio, and the beaker was shaken for 10 min at 250 r min−1. After standing for 30 min, the pH (suspension) was measured. 10.00 mL of the SMP buffer solution was added and the mixture was shaken again for 10 min and then allowed to stand for 30 min. The suspension’s pH was measured to obtain pH1.
After measuring the pH and pH1, 4.2 mL of 0.05 M HCl was added to the suspension. This was the equivalent amount needed to adjust the buffer solution’s pH from 7.5 to 6.0 and was calculated according to the buffer solution’s standard curve. The mixture was shaken again for 10 min, and stand for 30 min before the pH of the soil suspension (pH2) was measured. The steps were repeated for 3 times.
ICP-OES measurement
The MSC main elemental contents were determined using ICP-OES. The operating parameters are presented in Table 2.
Table 2 Operating parameters of the ICP-OES spectrometer.
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0.200 g of each MSC was weighed and put into a 30 mL platinum crucible, and 1.500 g of molten agent was added (the mass ratio of sodium carbonate to sodium tetraborate was 2:1). After the molten agent and samples were mixed, the crucible was placed in a muffle furnace and its temperature was raised to 950 °C for 60 min to melt the contents. The crucible was taken out of the furnace after cooling and the sample inside was leached using 70 mL of HCl (3 + 7) to reach a constant volume of 100 mL. This solution was directly used to determine the Ca, Mg, Ba, Ti, and Mn content. Next, the solution was diluted 10 times to determine the high concentrations of K, Al, Si, Na, and Fe content.
XRD measurement
The MSC samples were ground to 0.045 mm (300 mesh) using an agate mortar and uniformly distributed inside sample frames. These were then pressed, flattened, and compacted using glass slides before being placed on the sample stage of the XRD sample chamber for analysis. The powder XRD patterns were obtained using a Bruker D8 Advance powder diffractometer working at 40 kV and 40 mA, using monochromatized Cu Kα radiation (λ = 0.154056 nm). The measurement was performed in the range angle 2θ = 15°–70°. Before the XRD test, all the samples were ground to 80 μm. The MDI Jade 5.0 software package (USA Materials Data Inc.) was used for qualitative analysis of the XRD spectra being tested.
Soil culture experiment
The soil samples were mixed with the MSCs and cultured for 30 days. Changes in the soil pH values were used to calculate the MSCs’ pH adjustment capacity and MSCR. The test treatments involved the addition of 0, 0.2%, 0.4%, 0.8%, 1.2%, or 1.6% of MSCs (total six levels) to the 14 soil samples and have 3 replications. The specific operating steps were as follows: 50 g of each soil sample and MSCs were mixed in each plastic cup uniformly, and water was added to 60% of the field moisture capacity. The cups were then sealed with plastic film to prevent excessive evaporation. The soil pH was measured after 30 days (1:1 water-to-soil ratio, measured after 10 min of shaking)29. The measurements were repeated twice. The measured pH and actual MSCR were subjected to regression analysis, and the regression equation was used to determine the MSCR required to neutralize the soil pH to 6.5 (the ideal pH for this study). More

Study sites and experimental design
The study sites are part of the NutNet experiment (Supplementary Data 1; http://nutnet.org/)27. Plots at each site are 5 × 5 m separated by at least 1 m. All sites included in the analyses presented here included unmanipulated plots and fertilized plots with nitrogen (N), phosphorus (P), and potassium and micronutrients (K) added in combination (NPK+). N, P, and K were applied annually before the beginning of the growing season at rates of 10 gm−2 y−1. N was supplied as time-release urea ((NH2)2CO) or ammonium nitrate (NH4NO3). P was supplied as triple super phosphate (Ca(H2PO4)2), and K as potassium sulfate (K2SO4). In addition, a micronutrient mix (Fe, S, Mg, Mn, Cu, Zn, B, and Mo) was applied at 100 gm−2 y−1 to the K-addition plots, once at the start of the experiment, but not in subsequent years to avoid toxicity. Treatments were randomly assigned to the 25 m2 plots and were replicated in three blocks at most sites (some sites had fewer/more blocks or were fully randomized). Sampling was done in 1 m2 subplots and followed a standardized protocol at all sites27.
Site selection
Data were retrieved on 1 May 2020. To keep a constant number of communities per site and treatment, we used three blocks per site, excluding additional blocks from sites that had more than three (Supplementary Data 1). Sites spanned a broad envelope of seasonal variation in precipitation and temperature (Supplementary Fig. 1), and represent a wide range of grassland types, including alpine, desert and semiarid grasslands, prairies, old fields, pastures, savanna, tundra, and shrub-steppe (Supplementary Data 1).
Stability and asynchrony measurements are sensitive to taxonomic inconsistencies. We adjusted the taxonomy to ensure consistent naming over time within sites. This was usually done by aggregating taxa at the genus level when individuals were not identified to species in all years. Taxa are however referred to as “species”.
We selected sites that had a minimum of 4 years, and up to 9 years of posttreatment data. Treatment application started at most sites in 2008, but some sites started later resulting in a lower number of sites with increasing duration of the study, from 42 sites with 4 years of posttreatment duration to 15 sites with 9 years of duration (Supplementary Data 1). Longer time series currently exist, but for a limited number of sites within our selection criteria.
Primary productivity and cover
We used aboveground live biomass as a measure of primary productivity, which is an effective estimator of aboveground net primary production in herbaceous vegetation36. Primary productivity was estimated annually by clipping at ground level all aboveground live biomass from two 0.1 m2 (10 × 100 cm) quadrats per subplot. For shrubs and subshrubs, leaves and current year’s woody growth were collected. Biomass was dried to constant mass at 60 °C and weighed to the nearest 0.01 g. Areal percent cover of each species was measured concurrently with primary productivity in one 1 × 1 m subplot, in which no destructive sampling occurred. Cover was visually estimated annually to the nearest percent independently for each species, so that total summed cover can exceed 100% for multilayer canopies. Cover and primary productivity were estimated twice during the year at some sites with strongly seasonal communities. This allowed to assemble a complete list of species and to follow management procedures typical of those sites. For those sites, the maximum cover of each species and total biomass were used in the analyses.
Diversity, asynchrony, and stability across spatial scales
We quantified local scale and larger-scale diversity indices across the three replicated 1-m2 subplots for each site, treatment and duration period using cover data37,38. In our analysis, we treated each subplot as a “community” and the collective subplots as the “larger scale” sensu Whittaker28. Local scale diversity indices (species richness, species evenness, Shannon, and Simpson) were measured for each community, and averaged across the three communities for each treatment at each site resulting in one single value per treatment and site. Species richness is the average number of plant species. Shannon is the average of Shannon–Weaver indices39. Species evenness is the average of the ratio of the Shannon–Weaver index and the natural logarithm of average species richness (i.e., Pielou’s evenness40). Simpson is the average of inverse Simpson indices41. Due to strong correlation between species richness and other common local diversity indices (Shannon: r = 0.90 (95% confidence intervals (CIs) = 0.87–0.92), Simpson: r = 0.88 (0.86–0.91), Pielou’s evenness: r = 0.62 (0.55–0.68), with d.f. = 324 for each), we used species richness as a single, general proxy for those variables in our models. Results using these diversity indices did not differ quantitatively from those presented in the main text using species richness (Supplementary Fig. 5), suggesting that fertilization modulate diversity effects largely through species richness. Following theoretical models15,16, we quantified abundance-based gamma diversity as the inverse Simpson index over the three subplots for each treatment at each site and abundance-based beta diversity, as the multiplicative partitioning of abundance-based gamma diversity: abundance-based beta equals the abundance-based gamma over Simpson28,42, resulting in one single beta diversity value per treatment and site. We used abundance-based beta diversity index because it is directly linked to ecosystem stability in theoretical models15,16, and thus directly comparable to theories. We used the R functions “diversity”, “specnumber”, and “vegdist” from the vegan package43 to calculate Shannon–Weaver, Simpson, and species richness indices within and across replicated plots.
Stability at multiple scales was determined both without detrending and after detrending data. For each species within communities, we detrended by using species-level linear models of percent cover over years. We used the residuals from each regression as detrended standard deviations to calculate detrended stability17. Results using detrended stability did not differ quantitatively from those presented in the main text without detrending. Stability was defined by the temporal invariability of biomass (for alpha and gamma stability) or cover (for species stability and species asynchrony), calculated as the ratio of temporal mean to standard deviation14,17. Gamma stability represents the temporal invariability of the total biomass of three plots with the same treatment, alpha stability represents the temporal invariability of community biomass averaged across three plots per treatment and per site, and species stability represents the temporal invariability of species cover averaged across all species and the three plots per treatment14. The mathematical formula are:

$${mathrm{Species}},{mathrm{stability}} = frac{{sum _{i,k}m_{i,k}}}{{sum _{i,k}sqrt {w_{ii,kk}} }},$$
(1)

$${mathrm{Alpha}},{mathrm{stability}} = frac{{sum _kmu _k}}{{sum _ksqrt {v_{kk}} }},$$
(2)

$${mathrm{Gamma}},{mathrm{stability}} = frac{{sum _kmu _k}}{{sqrt {sum _{k,l}nu _{kl}} }},$$
(3)

where mi,k and wii,kk denote the temporal mean and variance of the cover of species i in subplot k; μk and vkk denote the temporal mean and variance of community biomass in subplot k, and vkl denotes the covariance in community biomass between subplot k and l. We then define species asynchrony as the variance-weighted correlation across species, and spatial asynchrony as the variance-weighted correlation across plots:

$${mathrm{Species}},{mathrm{asynchrony}} = frac{{sum _{i,k}sqrt {w_{ii,kk}} }}{{sum _ksqrt {sum _{ij,kl}w_{ij,kl}} }},$$
(4)

$${mathrm{Spatial}},{mathrm{asynchrony}} = frac{{sum _ksqrt {v_{kk}} }}{{sqrt {sum _{k,l}nu _{kl}} }},$$
(5)

where wij,kl denotes the covariance in species cover between species i in subplot k and species j in subplot l.
These two asynchrony indices quantify the incoherence in the temporal dynamics of species cover and community biomass, respectively, which serve as scaling factors to link stability metrics across scales14 (Fig. 1). To improve normality, stability, and asynchrony measures were logarithm transformed before analyses. We used the R function “var.partition” to calculate asynchrony and stability across spatial scales14.
Climate data
Precipitation and temperature seasonality were estimated for each site, using the long-term coefficient of variation of precipitation (MAP_VAR) and temperature (MAT_VAR), respectively, derived from the WorldClim Global Climate database (version 1.4; http://www.worldclim.org/)44.
Analyses
All analyses were conducted in R 4.0.2 (ref. 45) with N = 42 for each analysis unless specified. First, we used analysis of variance to determine the effect of fertilization, and period of experimental duration on biodiversity and stability at the two scales investigated. Models including an autocorrelation structure with a first-order autoregressive model (AR(1)), where observations are expected to be correlated from 1 year to the next, gave substantial improvement in model fit when compared with models lacking autocorrelation structure. Second, we used bivariate analyses and linear models to test the effect of fertilization and period of experimental duration on biodiversity–stability relationships at the two scales investigated. Again, models including an autocorrelation structure gave substantial improvement in model fit (Supplementary Table 1)46,47,48. We ran similar models based on nutrient-induced changes in diversity, stability, and asynchrony. For each site, relative changes in biodiversity, stability, and asynchrony at the two scales considered were calculated, as the natural logarithm of the ratio between the variable in the fertilized and unmanipulated plots (Supplementary Fig. 9). Because plant diversity, asynchronous dynamics, and temporal stability may be jointly controlled by interannual climate variability22, we ran similar analyses on the residuals of models that included the coefficient of variation among years for each of temperature and precipitation. Results of our analyses controlling for interannual climate variability did not differ qualitatively from the results presented in the text (Supplementary Fig. 4). In addition, to test for temporal trends in stability and diversity responses to fertilization, we used data on overlapping intervals of four consecutive years. Results of our analyses using temporal trends did not differ qualitatively from the results presented in the text (Supplementary Fig. 6). Inference was based on 95% CIs.
Second, we used SEM29 with linear models, to evaluate multiple hypothesis related to key predictions from theories (Table 1). The path model shown in Fig. 1e was evaluated for each treatment (control and fertilized), and we ran separate SEMs for each period of experimental duration (from 4 to 9 years of duration). We generated a summary SEM by performing a meta-analysis of the standardized coefficients across all durations for each treatment. We then tested whether the path coefficients for each model differed by treatment by testing for a model-wide interaction with the “treatment” factor. A positive interaction for a given path implied that effects of one variable on the other are significantly different between fertilized and unfertilized treatments. We used the R functions “psem” to fit separate piecewise SEMs49 for each duration and combined the path coefficients from those models, using the “metagen” function50.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article. More

Credit: Tad Theimer

Joseph (Joe) Connell altered both what and how ecologists study. Tree by tree, coral by coral, barnacle by barnacle, he saw patterns and processes across diverse ecosystems. Simply and with incontrovertible evidence, he demonstrated that interactions such as competition and predation could determine where species lived.
Before his classic experiments on Scotland’s rocky shores, field ecology was mainly descriptive, focusing on physical conditions such as temperature or moisture in determining where species lived. Connell, who died last month aged 96, inspired thousands of ecologists to test their hypotheses by manipulating conditions in the field.
Connell established long-term studies of coral reefs at Heron Island in the Great Barrier Reef and of tropical rainforests in Queensland, Australia, that spanned more than three and five decades, respectively. Monitoring revealed the dynamic nature of plant and animal communities that had long been considered stable. He discovered that natural variability in biological interactions and physical factors maintains diversity in these and other endangered ecosystems.
Born in 1923, Connell grew up just outside Pittsburgh, Pennsylvania. When the United States entered the Second World War in 1941, he enlisted in the Army Air Corps and was trained in meteorology. Later, conducting weather surveillance in the Azores — the Portuguese Atlantic archipelago — in support of army operations in Europe, he spent his free time birdwatching and identifying trees. Meeting army recruits who worked as wildlife managers, he realized it was possible to have a career as a biologist. After the war, and a degree in meteorology at the University of Chicago, Illinois, he headed to the University of California, Berkeley, for a master’s in zoology.
Connell produced what he described as a dull, unsatisfying thesis on brush rabbits (Sylvilagus bachmani) in the Berkeley Hills. Discouraged by the difficulties of conducting a population study (he trapped only 40 rabbits in 2 years), he adopted a rule of thumb — never again to study anything bigger than his thumb. As a doctoral student at the University of Glasgow, UK, he gleefully discovered what Charles Darwin had found a century before: that thousands of barnacles could easily be studied on the seashore, no traps required.
Connell realized that he could test his hypotheses about what factors determined where on the shore certain species lived by removing, adding or transplanting barnacles and their snail predators. Classic papers ensued, inspiring other ecologists to rethink distribution patterns, and, importantly, to test their ideas with controlled field experiments.
After a postdoc at the Woods Hole Oceanographic Institution in Massachusetts, Connell joined the faculty at the University of California, Santa Barbara, where he remained for the rest of his career. He was curious about processes that affected distribution and abundance, and those that might keep biodiversity high. Shifting to species that live for hundreds or thousands of years on coral reefs and in rainforests, he set up his Australian long-term monitoring studies in 1962 and 1963. Both recorded the demography and interactions of organisms in permanent plots, tracking community dynamics and the impact of disturbances, ranging from fallen trees to cyclones.
Visiting Connell’s sites with him in the 1970s and 1990s, we were impressed with his foresight and inspired by his insights. On the reef, he explained, physical disturbance by large waves associated with recurring cyclones intermittently reduced the cover of dominant species such as staghorn coral (Acropora aspera). This prompted recolonization by a diverse assemblage of weaker competitors such as encrusting or mound-like species. Connell coined the term ‘intermediate disturbance hypothesis’ to describe this process.
We strolled through the larger of his rainforest plots, avoiding stinging trees, biting flies, ticks and leeches, and relishing the richness — more than 300 tree species and about 100,000 individual plants. Connell outlined another hypothesis, that forests are more diverse when rarer species such as the conifer Sundacarpus amarus are favoured over more common ones such as the flowering tree Planchonella sp. Patterns of seedling establishment, growth or survival depend on that difference in frequency. Because common species grow more densely than rare ones, they are more vulnerable to specialist herbivores or pathogens.
This pattern of density-dependent predation or infection thins out common species, enabling a richer mix to coexist. It is a central component of the Janzen–Connell hypothesis (independently proposed by US ecologist Daniel Janzen in 1970), which predicts that seedlings are more likely to die under the canopies of their parent trees than farther away, ensuring diversity.
Connell was unfailingly kind, generous and devoted to his family. He never lost his profound curiosity about the natural world or his delight in exploring ideas with students and colleagues. He loved to be challenged and, if proven wrong, he gladly moved on to a new hypothesis or question. He sought truth, not fame. Moreover, he empowered everyone around him to think critically by focusing on ideas and evidence, not personalities. Fortunately for the world, his way of exploring science proved powerful, infectious, fun and enduringly productive. More

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Sediment reduction analysis method
This section presents the methods used to calculate sediment reduction caused by the major contributors, i.e., reservoir works, water diversion works, soil and water conservation works, and rainfall.
Sediment reduction by reservoir works
Reservoir works reduce sediment by impounding and retaining the sediment. Recent variations in sediment reduction due to reservoir works were analysed according to the variations in the average annual sediment deposition in the reservoirs of the basin during different periods.
The average annual sediment reduction by various reservoirs can be calculated by dividing the accumulated sediment in each reservoir during a certain period by the number of years:

$$ WS_{r} = sumlimits_{1}^{n} {D_{i} /N} , $$
(1)

where (D_{i}) is the accumulated sediment (100,000,000 t) in a reservoir during a certain period, N is the number of years in the period, and WSr is the average annual sediment reduction (100,000,000 t) in the period by all reservoirs in the basin.
The Hydrological Bureau under the Yellow River Water Conservancy Commission annually measures and calculates the deposition of sediment in all reservoirs in the Yellow River Basin. Two methods can be used, namely, the topographic method and the section method. In the topographic method, the area enclosed by contour lines on the topographic map of the reservoir area is measured to calculate the reservoir volume. The cumulative deposition of sediment during a specific period is the difference between the current and the previous reservoir volume. The topographic method requires closed contour lines on the map. In reality, however, the contour lines cannot be closed due to the presence of farmland, houses, and other artificial structures in the reservoir area, resulting in measurement errors. Therefore, the section method is mainly used at present. Here, M test sections were deployed in the reservoir area, and the test section data were used to calculate the total storage capacity of the reservoir in sections by period, as follows:

$$ V_{i} = sumlimits_{m = 1}^{M – 1} {V_{i,m} } . $$
(2)

The difference in the storage capacity measured twice is the cumulative deposition of sediment in reservoir (D_{i}):

$$ D_{i} = V_{i – 1} – V_{i} , $$
(3)

where Vi is the storage capacity measured at the end of period i and Vi,m is the storage capacity measured in section m – 1.
Sediment reduction by water diversion works
During water diversion, a certain amount of sediment is diverted, along with water, and is deposited in irrigation areas, resulting in a decrease in the volume of the sediment in the river channel. The average annual sediment reduction by water diversion works can be calculated by multiplying the average annual water diversion in different periods in the Jinghe River Basin by the average annual sediment concentration in the water diversion period, as follows:

$$ WS_{d} = sumlimits_{1}^{n} {W_{di} /N times overline{S}} /{1}000, $$
(4)

where (W_{di}) is the cumulative water diversion (100,000,000 m3) in the basin in the water diversion period, N is the number of years in the period, (overline{S}) is the average annual sediment concentration in the period (kg/m3), and (WS_{d}) is the average annual sediment reduction (100,000,000 t) in the basin during the period. Recent variations in sediment discharge caused by water diversion works were analysed according to the variations in the average annual water diversion in the basin in different periods.
Sediment reduction by soil and water conservation works
A commonly used method to compute the sediment reduction by soil and water conservation works is to multiply the area subject to the soil and water conservation works, such as terracing, forestation, grassing, creating enclosures, and constructing silt-arrester dams, by the sediment reduction by each measure per unit area, followed by their summation, as follows:

$$ WS_{SC} = sumlimits_{1}^{n} {F_{i} times S_{j} /10^{8} ,} $$
(5)

where Sj is the sediment reduction due to each soil and water conservation measure (t/hm2), published by the soil and water conservation monitoring institutions in each basin based on the analysis of the long-term observation data, Fi is the area subjected to each measure (hm2), and WSSC is the comprehensive sediment concentration for each measure (100,000,000 t). The variations in sediment reduction by soil and water conservation works were analysed based on the variations in the soil and water conservation areas in the basin during different periods.
Analysis of rainfall-induced sediment yield
The deduction method was adopted to analyse the rainfall-induced variations in the sediment yield. Recent variations in sediment reduction attributable to reservoirs, water diversion, and soil and water conservation works were computed and deducted from the measured sediment reduction in recent years (2000–2015):

$$ Delta WS_{p} = Delta WS_{t} – Delta WS_{r} – Delta WS_{d} – Delta WS_{sc} , $$
(6)

where (Delta WS_{t}) is the recently measured sediment reduction (100,000,000 t), (Delta WS_{r}) is the recent variation in the sediment reduction (100,000,000 t) caused by variations in the sediment retention due to reservoir works, (Delta WS_{d}) is the recent variation in sediment reduction (100,000,000 t) caused by variations in water diversion, (Delta {text{WS}}_{{{text{SC}}}}) is the recent variation in sediment reduction (100,000,000 t) caused by variations in the soil and water conservation area, and (Delta WS_{p}) is the recent variation in the rainfall-induced sediment yield caused by variations in rainfall.
Sediment yield calculation method
Figure 6 depicts the computational process for the sediment calculation. First, a reduction calculation of the natural runoff was performed as follows:

$$ W_{0} = W_{m} + W_{cum} + W_{s} + W_{e} + W_{SC} , $$
(7)

where W0 is the natural runoff, Wm is the measured runoff, Wcuw is the industrial water consumption in the basin, Ws is the water retention by reservoirs, We is the water evaporation and seepage losses, Wsc is the water reduction by soil and water conservation, and W0 is the natural water volume in the basin. All these terms are in 100,000,000 m3.
Second, the runoff-sediment relationship in the natural state was established based on the measured runoff and sediment data in periods with negligible human activity, as well as when the underlying surface was in a nearly natural state. Natural sediment discharge was calculated using the relationship between runoff and sediment discharge. According to the observation data from the basin for the past 35 years, runoff was closely related to sediment discharge. Given China’s climatic conditions and economic growth, the basin was nearly in a natural state up to 1960 because human activity had a minor impact on runoff and sediment discharge. Based on the runoff and sediment discharge measurements at Zhangjiashan Station from 1932 to 1960, the relationship between the natural runoff and sediment discharge was established as WS0 = f(W0). Natural sediment discharge in the basin was calculated considering the restored natural runoff.
Third, the natural sediment discharge was calculated using the natural runoff results and the runoff-sediment relationship. Based on the major contributors to sediment reduction in the basin, the future sustainable sediment reduction was calculated as the sum of sediment reduction due to reservoirs, water diversion, and soil and water conservation measures. Sediment reduction caused by variations in rainfall was limited to certain periods. For example, recent reduced heavy rainfall has led to a decreased rainfall-induced sediment yield and consequently a decreased sediment discharge. However, according to forecasts by the Intergovernmental Panel on Climate Change (2014)50, extreme weather and heavy rainfall events are likely to increase in the future. The reduction in sediment due to variations in rainfall was calculated as follows:

$$ WS_{d} = WS_{r} + WS_{d} + W_{SC} , $$
(8)

where WSr is the future sediment reduction caused by reservoir works, i.e., the sum of the sediment retention potential of the remaining capacity of the existing reservoirs and that of planned future reservoirs; WSd is the sediment reduction caused by future water diversion works, which can be obtained by multiplying the water diversion in the basin forecasted according to the social and economic development by the average sediment concentration in the water diversion period; WSsc is the future sediment reduction caused by soil and water conservation, obtained from areas subject to existing and planned soil and water conservation works and the corresponding sediment reduction rates; and WSd is the forecasted value of sediment reduction in the basin. All these terms are in 100,000,000 t.
Fourth, the sustainable sediment reduction in the basin was calculated considering variations in the contributions to sediment reduction in a future period and their effect. Future sediment discharge in the basin is the difference between the natural and future sediment reduction, as follows:

$$ WS_{f} = WS_{0} – WS_{d} , $$
(9)

where WS0 is the natural sediment discharge in the basin, WSd is the forecasted sediment reduction in the basin, and WSf is the forecasted sediment discharge in the basin. All these terms are in 100,000,000 t.
Finally, future river sediment discharge was obtained by subtracting the future sustainable sediment reduction from the natural sediment discharge.
Data acquisition
Hydrological data
A total of 28 hydrometric stations and 190 rainfall stations are located along the main stream and tributaries of the Jinghe River to effectively monitor rainfall, runoff, and sediment in the basin.
Zhangjiashan Station, located at the outlet of the Jinghe River Basin, has a catchment area of 432,160,000 km2, covering 95% of the total area of the basin. Few hydrometric and rainfall stations were operational in this basin before 1956, and hence incomplete data were collected. Analyses in this study were based on data from the Zhangjiashan Station from 1956–2015. At this station, the cross-sections in the main stream and Jinghui Canal (a water diversion canal) were hydrologically measured to determine the discharge, sediment transport rate, and sediment concentration.
Engineering data
Data on sediment reduction due to reservoir works and terraces, forests, grasslands, enclosures, and dams in the basin were based on the results of the National Water Resources Census and official data collated by the Upper and Middle Yellow River Bureau of the Yellow River Conservancy Commission. These data are thus accurate and reliable.
For data collection and erosion–deposition calculations, DL/T 5089–1999 “Specification for Sediment Design of Hydropower and Water Conservancy Projects” provided that “The calculated results of erosion and deposition should be compared with the measured data for several years of operation. If the amount and location of sedimentation are 70% consistent, and the elevation of sedimentation in the reservoir differs by 1 to 2 m, then the calculated results are deemed reliable. For erosion–deposition calculation results, only reliability is considered”.
Relevant data from the stations were systematically verified and collated by the Hydrological Bureau of the Yellow River Conservancy Commission and are therefore accurate and reliable. More

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