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Plant growth model without environmental forcingThe model without environmental forcing closely follows the original description of the Thornley transport resistance (TTR) model29. A summary of the model parameters is provided in Supplementary Table 2. The shoot and root mass pools (MS and MR, in kg structural dry matter) change as a function of growth and loss (equations (1) and (2)). The litter (kL) and maintenance respiration (r) loss rates (in kg kg−1 d−1) are treated as constants. In the original model description29 r = 0. The parameter KM (units kg) describes how loss varies with mass (MS or MR). Growth (Gs and Gr, in kg d−1) varies as a function of the carbon and nitrogen concentrations (equations (3) and (4)). CS, CR, NS and NR are the amounts (kg) of carbon and nitrogen in the roots and shoots. These assumptions yield the following equations for shoot and root dry matter,$${mathrm{MS}}[t+1]={mathrm{MS}}[t]+{G}_{{mathrm{S}}}[t]-frac{({k}_{{mathrm{L}}}+r){mathrm{MS}}[t]}{1+frac{{K}_{{{M}}}}{{mathrm{MS}}[t]}},$$
(1)
$${mathrm{MR}}[t+1]={mathrm{MR}}[t]+{G}_{{mathrm{R}}}[t]-frac{({k}_{{mathrm{L}}}+r){mathrm{MR}}[t]}{1+frac{{K}_{{{M}}}}{{mathrm{MR}}[t]}},$$
(2)
where GS and GR are$${G}_{{mathrm{S}}}=gfrac{{mathrm{CS}}times {mathrm{NS}}}{{mathrm{MS}}},$$
(3)
$${G}_{{mathrm{R}}}=gfrac{{mathrm{CR}}times {mathrm{NR}}}{{mathrm{MR}}},$$
(4)
and g is the growth coefficient (in kg kg−1 d−1).Carbon uptake UC is determined by the net photosynthetic rate (a, in kg kg−1 d−1) and the shoot mass (equation (5)). Similarly, nitrogen uptake (UN) is determined by the nitrogen uptake rate (b, in kg kg−1 d−1) and the root mass. The parameter KA (units kg) forces both photosynthesis and nitrogen uptake to be asymptotic with mass. The second terms in the denominators of equations (5) and (6) model product inhibitions of carbon and nitrogen uptake, respectively; that is, the parameters JC and JN (in kg kg−1) mimic the inhibition of source activity when substrate concentrations are high,$${U}_{{mathrm{C}}}=frac{a{mathrm{MS}}}{left(1+frac{{mathrm{MS}}}{{K}_{{mathrm{A}}}}right)left(1+frac{{mathrm{CS}}}{{mathrm{MS}}times {J}_{{mathrm{C}}}}right)},$$
(5)
$${U}_{{mathrm{N}}}=frac{b{mathrm{MR}}}{left(1+frac{{mathrm{MR}}}{{K}_{{mathrm{A}}}}right)left(1+frac{{mathrm{NR}}}{{mathrm{MR}}times {J}_{{mathrm{N}}}}right)}.$$
(6)
The substrate transport fluxes of C and N (τC and τN, in kg d−1) between roots and shoots are determined by the concentration gradients between root and shoot and by the resistances. In the original model description29, these resistances are defined flexibly, but we simplify and assume that they scale linearly with plant mass,$${tau }_{{mathrm{C}}}=frac{{mathrm{MS}}times {mathrm{MR}}}{{mathrm{MS}}+{mathrm{MR}}}left(frac{{mathrm{CS}}}{{mathrm{MS}}}-frac{{mathrm{CR}}}{{mathrm{MR}}}right)$$
(7)
$${tau }_{{mathrm{N}}}=frac{{mathrm{MS}}times {mathrm{MR}}}{{mathrm{MS}}+{mathrm{MR}}}left(frac{{mathrm{NR}}}{{mathrm{MR}}}-frac{{mathrm{NS}}}{{mathrm{MS}}}right)$$
(8)
The changes in mass of carbon and nitrogen in the roots and shoots are then$${mathrm{CS}}[t+1]={mathrm{CS}}[t]+{U}_{{mathrm{C}}}[t]-{f}_{{mathrm{C}}}{G}_{{mathrm{s}}}[t]-{tau }_{{mathrm{C}}}[t]$$
(9)
$${mathrm{CR}}[t+1]={mathrm{CR}}[t]+{tau }_{{mathrm{C}}}[t]-{f}_{{mathrm{C}}}{G}_{{mathrm{r}}}[t]$$
(10)
$${mathrm{NS}}[t+1]={mathrm{NS}}[t]+{tau }_{{mathrm{N}}}[t]-{f}_{{mathrm{N}}}{G}_{{mathrm{s}}}[t]$$
(11)
$${mathrm{NR}}[t+1]={mathrm{NR}}[t]+{U}_{{mathrm{N}}}[t]-{f}_{{mathrm{N}}}{G}_{{mathrm{r}}}[t]-{tau }_{{mathrm{N}}}[t]$$
(12)
where fC and fN (in kg kg−1) are the fractions of structural carbon and nitrogen in dry matter.Adding environmental forcing to the plant growth modelIn this section, we describe how the net photosynthetic rate (a), the nitrogen uptake rate (b), the growth rate (g) and the respiration rate (r) are influenced by environmental-forcing factors. These environmental-forcing effects are described in equations (13)–(17) and summarized graphically in Extended Data Fig. 1. All other model parameters are treated as constants. Previous work that implemented the TTR model as a species distribution model30 is used as a starting point for adding environmental forcing. As in this previous work30, we assume that parameters a, b and g are co-limited by environmental factors in a manner analogous to Liebig’s law of the minimum, which is a crude but pragmatic abstraction. The implementation here differs in some details.Unlike previous work30, we use the Farquhar model of photosynthesis47,48 to represent how solar radiation, atmospheric CO2 concentration and air temperature co-limit photosynthesis35. We assume that the Farquhar model parameters are universal and that all vegetation in our study uses the C3 photosynthetic pathway. The Farquhar model photosynthetic rates are rescaled to [0,amax] to yield afqr. The effects of soil moisture (Msoil) on photosynthesis are represented as an increasing step function ({{{{S}}}}(M_{mathrm{soil}},{beta }_{1},{beta }_{2})=max left{min left(frac{M_{mathrm{soil}}-{beta }_{1}}{{beta }_{2}-{beta }_{1}},1right),0right}). This allows us to redefine a as,$$a={a}_{{mathrm{fqr}}} {{{{S}}}}(M_{mathrm{soil}},{beta }_{1},{beta }_{2})$$
(13)
The processes influencing nitrogen availability are complex, and global data products on plant available nitrogen are uncertain. We therefore assume that nitrogen uptake will vary with soil temperature and soil moisture. That is, the nitrogen uptake rate b is assumed to have a maximum rate (bmax) that is co-limited by soil temperature Tsoil and soil moisture Msoil,$$b={b}_{{mathrm{max}}} {{{{S}}}}({T}_{soil},{beta }_{3},{beta }_{4}) {{{{Z}}}}(M_{mathrm{soil}},{beta }_{5},{beta }_{6},{beta }_{7},{beta }_{8}).$$
(14)
In equation (14), we have assumed that the nitrogen uptake rate is a simple increasing and saturating function of temperature. We have also assumed that the nitrogen uptake rate is a trapezoidal function of soil moisture with low uptake rates in dry soils, higher uptake rates at intermediate moisture levels and lower rates once soils are so moist as to be waterlogged. The trapezoidal function is ({{{{Z}}}}(M_{mathrm{soil}},{beta }_{5},{beta }_{6},{beta }_{7},{beta }_{8})=max left{min left(frac{M_{mathrm{soil}}-{{{{{beta }}}}}_{5}}{{{{{{beta }}}}}_{6}-{{{{{beta }}}}}_{5}},1,frac{{{{{{beta }}}}}_{8}-M_{mathrm{soil}}}{{beta }_{8}-{beta }_{7}}right),0right}).The previous sections describe how the assimilation of carbon and nitrogen by a plant are influenced by environmental factors. The TTR model describes how these assimilate concentrations influence growth (equations (3) and (4)). In our implementation, we additionally allow the growth rate to be co-limited by temperature (soil temperature, Tsoil) and soil moisture (Msoil),$$g={g}_{{mathrm{max}}} {{{{Z}}}}({T}_{{mathrm{soil}}},{beta }_{9},{beta }_{10},{beta }_{11},{beta }_{12}) {{{{S}}}}(M_{mathrm{soil}},{beta }_{13},{beta }_{14}).$$
(15)
We use Tsoil since we assume that growth is more closely linked to soil temperature, which varies slower than air temperature. The respiration rate (r, equations (1) and (2)) increases as a function of air temperature (Tair) to a maximum rmax,$$r={r}_{{mathrm{max}}}{{{{S}}}}({T}_{{mathrm{air}}},{beta }_{15},{beta }_{16}).$$
(16)
The parameter r is best interpreted as a maintenance respiration. Growth respiration is not explicitly considered; it is implicitly incorporated in the growth rate parameter (g, equation (15)), and any temperature dependence in growth respiration is therefore assumed to be accommodated by equation (15).Fire can reduce the structural shoot mass MS as follows,$${mathrm{MS}}[t+1]={mathrm{MS}}[t](1-{{{{S}}}}(F,{beta }_{17},{beta }_{18})).$$
(17)
where F is an indicator of fire severity at a point in time (for example, burnt area) and the function S(F, β17, β18) allows MS to decrease when the fire severity indicator F is high. If F = 0, this process plays no role. This fire impact equation was used in preliminary analyses, but the data on fire activity did not provide sufficient information to estimate β17 and β18; we therefore excluded this process from the final analyses.We further estimate two additional β parameters (βa and βb) so that each site can have unique maximum carbon and nitrogen uptake rates. Specifically, we redefine a as ({a}^{{prime} }={beta }_{a} a) and b as ({b}^{{prime} }={beta }_{b} b).Data sources and preparationTo describe vegetation activity, we use the GIMMS 3g v.1 NDVI data26,27 and the MODIS EVI28 data. The GIMMS data product has been derived from the AVHRR satellite programme and controls for atmospheric contamination, calibration loss, orbital drift and volcanic eruptions26,27. The data provide 24 NDVI raster grids for each year, starting in July 1981 and ending in December 2015. The spatial resolution is 1/12° (~9 × 9 km). The EVI data used are from the MODIS programme’s Terra satellite; it is a 1 km data product provided at a 16-day interval. We use data from the start of the record (February 2000) to December 2019. The MODIS data product (MOD13A2) uses a temporal compositing algorithm to produce an estimate every 16 days that filters out atmospheric contamination. The EVI is designed to reduce the effects of atmospheric, bare-ground and surface water on the vegetation index28.For environmental forcing, we use the ERA5-Land data31,32 (European Centre for Medium-Range Weather Forecasts Reanalysis v. 5; hereafter, ERA5). The ERA5 products are global reanalysis products based on hourly estimates of atmospheric variables and extend from present back to 1979. The data products are supplied at a variety of spatial and temporal resolutions. We used the monthly averages from 1981 to 2019 at a 0.1° spatial resolution (~11 km). The ERA5 data provide air temperature (2 m surface air temperature), soil temperature (0–7 cm soil depth), surface solar radiation and volumetric soil water (0–7 cm soil depth). Fire data were taken from the European Space Agency Fire Disturbance Climate Change Initiative’s AVHRR Long-Term Data Record Grid v.1.0 product49. This product provides gridded (0.25° resolution) data of monthly global (from 1982 to 2017) burned area derived from the AVHRR satellite programme. As mentioned, the fire data did not enrich our analysis, and the analyses we present here therefore exclude further consideration of the fire data.All data were resampled to the GIMMS grid. The mean pixel EVI was then calculated for each GIMMS cell for each time point in the MODIS EVI data. We used linear interpolation on the NDVI, EVI and ERA5 environmental-forcing data to estimate each variable on a weekly time step. This served to set the time step of the TTR difference equations to one week and to synchronize the different time series.Site selectionThe GIMMS grid cells define the spatial resolution of our sample points. GIMMS grid cells are large (1/12°, ~9 km), meaning that most grid cells contain multiple land-cover types. We focused on wilderness landscapes, and our aim was to find multiple grid cells for the major ecosystems of the world. We used the following classification of ecosystem types to guide the stratification of our grid-cell selection: tropical evergreen forest (RF), boreal forest (BF), temperate evergreen and temperate deciduous forest (TF), savannah (SA), shrubland (SH), grassland (GR), tundra (TU) and Mediterranean-type ecosystems (MT).We used the following criteria to select grid cells. (1) Selected grid cells should contain relatively homogeneous vegetation. Small-scale heterogeneity was allowed (for example, catenas, drainage lines, peatlands) as long as many of these elements are repeated in the scene (for example, rolling hills were accepted, but elevation gradients were rejected). (2) Sites should have no signs of transformative human activity (for example, tree harvesting, crop cultivation, paved surfaces). We used the Time Tool in Google Earth Pro, which provides annual satellite imagery of the Earth from 1984 onwards, to ensure that no such activity occurred during the observation period (note that the GIMMS record starts in July 1981; however, it is likely that evidence of transformative activity between July 1981 and 1984 would be visible in 1984). Grid cells with extensive livestock holding on non-improved pasture were included. In some cases, a small agricultural field or pasture was present, and such grid cells were used as long as the field or pasture was small and remained constant in size. (3) Grid cells should not include large water bodies, but small drainage lines or ponds were accepted as long as they did not violate the first criterion. (4) Grid cells should be independent (neighbouring grid cells were not selected) and cover the major ecosystems of the world. Using these criteria, we were able to include 100 sites in the study (Extended Data Figs. 2 and 3 and Supplementary Table 4).State-space modelWe used a Bayesian state-space approach. Conceptually, the analysis takes the form,$$M[t]=f(M[t-1],{boldsymbol{beta}},{boldsymbol{theta}}_{t-1},{epsilon }_{t-1})$$
(18)
$${mathrm{VI}}[t]=m M[t]+eta .$$
(19)
Here M[t] is the plant growth model’s prediction of biomass (M = MS + MR) at time t, and ϵt−1 is the process error associated with the state variables. In the model, each underlying state variable (MS, MR, CS, CR, NS and NR) has an associated process error term. The function f(M[t − 1], β, θt−1, ϵt−1) summarizes that the development of M is influenced by the state variables MS, MR, CS, CR, NS and NR, the environmental-forcing data θt−1 and the β parameters. The observation equation (equation (19)) uses the parameter m to link the VI (vegetation index, either NDVI or EVI) observations to the modelled state M. The parameter η is the observation error. Equation (19) assumes that there is a linear relationship between modelled biomass (M) and VI, which is a simplification of reality50,51,52. The observation error η is structured by our simplification of the data products quality scores (coded Q = 0, 1, 2, with 0 being good and 2 being poor; Supplementary Table 3) to allow the error to increase with each level of the quality score. Specifically, we define η = e0 + e1 × Q.The model was formulated using the R package LaplacesDemon53. All β parameters are given vague uniform priors. The parameter m is given a vague normal prior (truncated to be >0). The process error terms are modelled using normal distributions, and the variances of the error terms are given vague half-Cauchy priors. The ex parameters are given vague normal priors. The model also requires the parameterization of M[0], the initial vegetation biomass; M[0] is given a vague uniform prior. We used the twalk Markov chain Monte Carlo (MCMC) algorithm as implemented in LaplacesDemon53 and its default control parameters to estimate the posterior distributions of the model parameters. We further fitted the model using DEoptim54,55, which is a robust genetic algorithm that is known to perform stably on high-dimensional and multi-modal problems56, to verify that the MCMC algorithm had not missed important regions of the parameter space. The models estimated with MCMC had significantly lower log root-mean-square error than models estimated with DEoptim (paired t-test NDVI analysis: t = –2.9806, degrees of freedom (d.f.) = 99, P = 0.00362; EVI analysis: t = –4.6229, d.f. = 99, P = 1.144 × 10–5), suggesting that the MCMC algorithm performed well compared with the genetic algorithm.Anomaly extraction and trend estimationWe use the ‘seasonal and trend decomposition using Loess’ (STL57) as implemented in the R58 base function stl. STL extracts the seasonal component s of a time series using Loess smoothing. What remains after seasonal extraction (the anomaly or remainder, r) is the sum of any long-term trend and stochastic variation. We estimate the trend in two ways. First, we estimate the trend by fitting a quadratic polynomial (r = a + bx + cx2) to the remainder (r is the remainder, x is time and a, b and c are regression coefficients). The use of polynomials allows the data to specify whether a trend exists, whether the trend is linear, cup or hat shaped and whether the overall trend is increasing or decreasing. As a second method, we estimate the trend by fitting a bent-cable regression to the remainder. Bent-cable regression is a type of piecewise linear regression for estimating the point of transition between two linear phases in a time series59,60. The model takes the form r = b0 + b1x + b2 q(x, τ, γ)60. Here r is the remainder, x is time, b0 is the initial intercept, b1 is the slope in phase 1, the slope in phase 2 is b2 − b1 and q is a function that defines the change point: (q(x,tau ,gamma )=frac{{(x-tau +gamma )}^{2}}{4gamma }I(tau -gamma < tau +gamma )+(x-tau )I(x > tau +gamma )); τ represents the location of the change point and γ the span of the bent cable that joins the two linear phases; I(A) is an indicator function that returns 1 if A is true and 0 if A is false. The bent-cable model allows the data to specify whether a trend exists and whether there has been a switch in the trend, thereby allowing the identification of whether the trend is linear, cup or hat shaped and whether the overall trend is increasing or decreasing. Both the polynomial and bent-cable models were estimated using LaplacesDemon’s53 Adaptive Metropolis MCMC algorithm and vague priors, although for the bent-cable model we constrained τ to be in the middle 70% of the time series and γ to be at most two years.The STL extraction of the seasonal components in the air temperature, soil temperature, soil moisture and solar radiation data (there is no stochasticity or seasonal trend in the CO2 data we used) allows us to simulate detrended time series d of these forcing variables as (d=bar{y}+s+{{{{N}}}}(mu ,sigma )) where N(μ, σ) is a normally distributed random variable with mean and standard deviation estimated from the remainder r (we verified that r was well described by the normal distribution), (bar{y}) is the mean of the data over the time series and s is the seasonal component extracted by STL. For CO2, the detrended time series is simply the average CO2 over the time series. More

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