The underlying theoretical model for RED is a continuity equation, for each PFT and spatial location, which describes the time evolution of the number density n of plants per unit area per unit mass m: 1∂n∂t+∂∂mng=-γn. Here g is the growth rate, and γ is the mortality rate of a plant of mass m. In general, g and γ could be any reasonable function of tree size. For large-scale applications we make simplifying assumptions for these functions consistent with observed n from forest inventory data . By default we assume that γ is independent of plant mass and that g follows a power law of plant mass as follows: 2g=g0mm0ϕg. Here g0 is the growth rate of a plant with the reference mass, m0. A value of ϕg=0.75 is assumed by default, consistent with the analysis of field-based measurements by . We also follow in assuming the scaling of plant canopy area a with plant mass as follows: 3a=a0mm0ϕa, where ϕa=0.5 by default. Solutions for n can be integrated over mass to derive the total plant number, N=∫0∞ndm, the total growth rate, G=∫0∞gndm, the total biomass, M=∫0∞mndm, and the fractional area covered ν=∫0∞andm.

We wish to produce a model of vegetation demography that can be updated numerically and which explicitly conserves vegetation carbon, providing a constraint on the number of plants moving between mass classes in the discrete form. In order to do this we integrate Eq. () over finite mass ranges: 4∂Ni∂t+Fi-Fi-1=-γNi, where i denotes the ith mass class; Fi is the flux of plants growing out of the ith mass class and into the (i+1)th mass class; Fi-1 is the flux of plants growing out of the (i-1)th mass class and into the ith mass class; and Ni is the number of plants per unit area in the ith mass class. For clarity, Eq. () is deliberately presented as continuous in time at this stage, as the focus in this subsection is on discretization of the mass profile. The fully numerical version of RED, which includes discretization of time, is described in Sects. 2.4 and 2.5. In order to explicitly conserve carbon, the flux Fi must take the form (see Appendix ) 5Fi=Nigi(mi+1-mi), where mi is the mean mass of a plant in the ith mass class, and gi is the growth rate per plant of the ith mass class (kg C yr-1 plant-1).

To solve Eq. () we also require a lower boundary condition, which represents the rate at which seedlings of mass m0 are introduced into the cohort. Here we assume that a fixed fraction, α, of the total assimilate available to a PFT (P), is devoted to producing new seedlings, with the remainder G=(1-α)P being allocated to the growth of existing plants. Spreading is homogeneous across the entirety of the grid box, but only seedlings established within “unoccupied” space will survive to join the plant cohort. The net incoming flux of seedlings of mass m0 is therefore 6F0=αPm0s=α(1-α)Gm0s, where s is the fractional gap area available for seedlings. The definition of s is assumed to differ by PFT to reflect an underlying tree–shrub–grass dominance hierarchy, as shown schematically in Fig. 1. Therefore, the rate of recruitment F0 is the ratio of a fraction of the carbon assimilate allocated to reproduction, αP, and m0, multiplied by the gap area s.

The space available to the seedlings of the kth PFT is calculated from the area fractions of the PFTs to which it is subdominant as follows: 7sk=1-∑lcklνl, where νl is the area fraction of the lth PFT, and ckl is the competition coefficient for the impact of PFT l on PFT k. If PFT l is within the same plant functional group (trees, shrubs or grasses) as PFT k or dominant over it, ckl=1. If PFT k is dominant over PFT l, ckl=0 (Fig. ). This “gap” boundary condition results in there being no equilibrium solution where the amount of coverage exceeds 1. Doing so would halt the recruitment flux such that mortality processes would bring the fractional coverage back below unity. This is a similar competition regime to the Lotka-inspired TRIFFID model and allows for the coexistence between inter-functional groups (trees, shrubs, and grasses) of PFTs. For instance, a PFT such as broadleaf deciduous tree can coexist with a deciduous shrub and C3 grass. The hierarchy also enables the simulation of succession during regrowth. Faster-growing species of grasses will not be able to expand into space occupied by trees and shrubs, unless there is space created by disturbance. A summary of the competition coefficients is given in Table .

RED updates plant size distributions, biomass, and fractional areal coverage for an arbitrary number of PFTs at each spatial location and can be driven by variables provided by a land carbon cycle model, an Earth system model, or observations (see Fig. ). For each PFT, the minimum required input is a time series of net carbon assimilate (P), defined as the difference between net primary productivity (ΠN) and local litter production due to turnover of leaves, stems, and roots (Λl): 8P=ΠN-Λl. We apply the m3/4 scaling to P. We therefore implicitly assume the same scaling for both gross primary productivity and plant respiration. This is consistent with observations suggesting that plant production also scales approximately as m3/4 . Where available, additional mortality due to disturbance events such as droughts, fires, and anthropogenic deforestation (γd) can be added to the baseline mortality rates (γb), for each PFT as follows: 9γ=γb+γd. Disturbance rates γd can in principle be both PFT dependent and mass dependent (e.g. to capture forestry practices).

The input values of net assimilate for each PFT (P) define the total structural growth rate, G=(1-α)P, and the seedling flux F0 (via Eq. ), using PFT-specific values of the parameter α (see Table ). The definition of the total structural growth rate at a given time step is 10G=∑iNigi, which can be combined with the growth scaling given by Eq. () to derive the reference growth rate, g0, from the net assimilate, P, which is a driving input: 11g0=(1-α)P∑iNimim0ϕg. This in turn enables the growth rate of each mass class to be calculated using Eq. (). For each PFT, the number of plants in a mass class (Ni) is updated using a discretized form of Eq. (): 12Ni(j+1)=Ni(j)+ΔtFi-1(j)-Fi(j)-γ(j)Ni(j), where Δt is the RED time step (typically 1 month), and the superscript (j) denotes the jth time step. Our results are robust to changes in model time step so long as the time step remains small compared to the characteristic timescales associated with regrowth (m0/g0∼ 4 years) and plant mortality (1/γ∼ 20 years). The lower-boundary seedling flux is calculated from Eq. () using Eq. (). We impose a zero-flux condition out of the upper mass class, under the assumption that there will be enough mass classes to ensure that this flux is negligible. However, to ensure carbon conservation on the land we add any plants that grow out of the upper mass class into a demographic litterfall term for each PFT, which is a RED output. This demographic litterfall term, Λd, keeps track of the carbon lost from the vegetation due to competition, mortality, and the carbon in any such plants that grow out of the largest resolved mass class (class I) as follows: 13Λd=αP(1-s)+∑iγiMi+gINI. The first term on the righthand side of this equation represents carbon loss due to the shading of seedlings; the second term represents mortality of the resolved mass classes (which may include disturbance events); and the third term, which is normally very small, is the loss of vegetation carbon due to plants growing beyond the modelled mass classes. In order to initiate regrowth from bare soil, RED also assumes a minimum effective fractional area of each PFT. Where the net assimilate would be sufficiently negative to take the vegetation fraction below this minimum, the minimum value is maintained by subtraction from the demographic litter. The demographic litterfall term therefore represents the net addition litter production consistent with the prescribed net assimilate flux, the disturbance rate, and the change in vegetation carbon modelled by RED. When coupling to an ESM or land carbon model, the demographic litterfall term (Λd) should be added to the input local litterfall (Λl) (as used in Eq. ) to calculate the total litterfall flux into the soil and/or litter system.

The steady state of the continuum model defined by Eqs. () and () can be solved analytically for each PFT . The continuum analytical solutions for the equilibrium mass distribution neq(m), the total plant number (Neq), biomass (Meq), growth rate (Geq), and fractional area (νeq) are summarized in Appendix . The shape of the mass distribution and each of these parameters depend on the ratio of plant mortality to growth, which we choose to define for the reference mass class m0 as follows: 14μ0=γm0g0. In order to initialize the numerical RED model in a drift-free initial state, we also derive the steady state of the discrete model (of Eq. ), which will differ slightly from the continuum model for a finite number of mass classes. The equilibrium solution of Eq. () is derived in Appendix , based on the balance between seedling recruitment and total cohort mortality that defines the equilibrium state. The discretized version of RED thus yields formulae for the coverage (Eq. ) and biomass densities (Eq. ) which depend on the lowest mass class through the value of μ0. Similarly, analytical expressions can be derived for total plant number and total growth rate of each PFT at equilibrium.

The total equilibrium stand density, Neq:15Neq=N0XN.

Here we use the analytical forms for the equilibrium state (Sect. ) and observations of global vegetation cover to diagnose the corresponding map of PFT-specific mortality rates. These mortality rates are therefore consistent with the current observed vegetation state and rates of net assimilation (P) provided from UKESM . The UKESM simulation provides net primary productivity (NPP) and local litterfall per unit area of each PFT. We multiply by PFT fraction to get the grid-box mean values required to drive RED (using ESA land-cover data, as explained below). The observed maps of PFTs are provided by the ESA LC_CCI dataset for 2008–2012 , projected onto the nine JULES PFTs (Fig. ). Maps of the prescribed annual mean values of the rate of net assimilation (P) are shown in Fig. .

We use the procedure outlined in Sect.  to estimate spatially varying values of μ0 for each PFT, using Eq. () and then Eq. () to estimate N0. This method successfully reproduces the ESA map of dominant PFT to good accuracy, as shown in Fig.  and Table .

The fit of the RED equilibrium vegetation coverage to the ESA observations is generally very good (Table ). However, it is imperfect in some areas (e.g. Central Asia, Sahel) where the driving net assimilate from UKESM is zero or negative. Also, areas where the observational dataset indicates coexisting PFTs within the same vegetation class (e.g. broadleaf trees and needleleaf trees) are not well simulated by this first version of RED, which leads to competitive exclusion in the equilibrium state (see Discussion). Since we now have diagnosed values of μ0 and N0, along with prescribed values of P, we can also diagnose the mean plant mortality rate γ, for each location and for each PFT, from Eq. () as follows: 21γ=μ0g0m0, where g0 is given by Eq. () combined with Eqs. () and (). Maps of γ values, derived in this way, are shown in Fig. .

The mortality rate derived is dependent on the assumed areal coverage and the total assimilate. A high coverage with a low growth rate will result in a compensating low diagnosed mortality rate (and vice versa). Furthermore, the choice of α (Eq. ) and m0 also influence the diagnosed value of γ. An analysis of the sensitivity of the inferred value of γ to these factors is presented in Appendix . Assuming ±20 % uncertainty on assimilate, α, and m0 and ±5 % on the coverage gives an uncertainty bound of ±35 % on γ. Under the assumption that high coverages are indicative of the baseline mortality for a given PFT, we take a subsample of the grid boxes that are within the top quartile of nonzero coverages (νeq>0.01) (Table ). The median μ0 value diagnosed from the top quartile of BET-Tr of 0.232-0.007+0.008 (Table ) is very close to the value calculated in our previous paper of approximately 0.235 for all of South America using the RAINFOR sites.

Site-level assessments of the rates of stand mortality within pantropical forests conclude a range of background rates . estimates mortality rates collected across 40 pantropical sites for tree sizes greater than 10–25 cm dbh. Later work by used the demographic data from the RAINFOR dataset of trees ≥ 10 cm dbh. Using these site assessments, we can make a comparison to BET-Tr equilibrium mortality rates by looking at the values of γ in areas where we would expect to see old-growth forests. We use the top 25 % of coverages of the BET-Tr PFT to represent plausible areas of undisturbed forest. Figure  shows that the diagnosed baseline mortality rates are in reasonable agreement with these observational estimates for Amazonia.

There is a need to better understand the influence of mortality arising from disturbance events such as droughts and fire in order to constrain model projections . Here we investigate if the equilibrium mortality rates implicitly capture areas of disturbances, by comparing the mean tree mortality rate to fire and land-use surveys (the mean mortality is defined here by weighting grid-box γ values by grid-box fractional coverages). There are a number of surveys relating stand mortality in regions prone to wildfires . In a broad sense, post-fire mortality rates can range from 0.06 yr-1 to catastrophic rates around 0.8 yr-1 and can vary quite considerably depending on tree species, fire frequency, and drought severity. The drought–fire interaction is responsible for significantly increasing mortality post fire and can be a driving cause of regional dieback . Using the ESA FIRE_CCI dataset we can estimate the burnt vegetation fraction per year. Taking the average burnt vegetation fraction for the months between 2000 and 2010 and converting into annual burn rate, we gain an estimate of fire severity.

Another key issue is anthropogenic land use and land-use change . Fragmentation of natural forests is understood to raise the mortality of the remaining forest and to decrease the overall resilience of the ecosystem . In order to maintain a near-constant agricultural fraction, regular disruption such as grazing is needed to prevent recolonization and secondary succession . We carry out a comparison with land use using the 2000 ESA LC_CCI inferred crop coverages .

In Fig. , we see the derived observations for burn area (panel a) and crop fraction (panel b), along with the derived mean γ for the tree PFTs (panel c). From Fig. d, we see that there are areas of large mortality (γ>0.075yr-1) that do correspond to areas where we see large fire activity (burn rate >0.1yr-1) and increased crop fraction (> 0.25). However, large burn rates are seen to overlap in parts of central Brazil around the Cernado region, southern Africa, and north Western Australia where fires are understood to play a significant part within the ecosystem . There are also some areas of agriculture which correspond to deforestation, such as in the Atlantic forests of Brazil and in Indonesia . Areas of increased disturbances result in grasses and shrubs dominating (Fig. ).

Analysis of the RED equilibrium is an indirect approach to estimating tree mortality based on simple yet mechanistic principles of demography and relying on few inputs (vegetation cover and assimilate). It is however conditional on the assumed estimates of vegetation coverage and net rates of assimilation.