Figure 1.
Model ecosystem described in equations 1–7.
Living plant biomass (B) contains carbon, nitrogen (N), and phosphorus (P) in a fixed stoichiometric ratio. When plant biomass turns over it becomes litter. The model keeps track of litter N (LN) and P (LP) separately. Some litter is decomposed into plant-unavailable soil organic matter (SOM; DN, DP), some is mineralized to plant-available soil nutrients (AN, AP), and some is lost from the system. Some SOM is mineralized into plant-available nutrients, and some is lost. Plant-available nutrients come from mineralization and external inputs, are taken up by plants, and are lost from the ecosystem. Symbiotic N fixation enters directly into the plant pool. We use “mineralization” as shorthand for “mineralization and depolymerization” to indicate the conversion of nutrient to plant-available form, which can include some organic forms.
Table 1.
Variables, functions, and parameters.
Table 2.
Parameter assumptions for analytical simplifications.
Figure 2.
Fit of timescale approximation to full numerical integration.
Parameters are as in Table 1, with no symbiotic N fixation. Both the approximation (solid red line) and the numerical integration (dashed blue line) began with biomass = 300 kg C ha−1, litter N and P = 0.1 kg N ha−1 and 0.05 kg P ha−1, soil organic matter (SOM) N and P = 300 kg N ha−1 and 40 kg P ha−1, and plant-available N and P = 1 kg N ha−1 and 0.003 kg P ha−1. Black dotted (N limited) and dashed (P limited) lines are quasi equilibrium and equilibrium values, displayed only for the relevant timescales. Vertical black lines indicate the timescale breaks: we used short timescale approximations to the left of the first black vertical line, intermediate timescale approximations between the two lines, and long timescale approximations to the right of the second line. Panels show (A) plant biomass, (B) which nutrient is limiting, (C) litter N, (D) litter P, (E) SOM N, (F) SOM P, (G) plant-available N, and (H) plant-available P. In this case net primary production begins P limited because there is an abundance of plant-available N, as from a small N fertilization, but becomes N limited within a couple weeks due to the large P:N ratio of inputs relative to plant demand and preferential P recycling. Plants remain N limited through succession and for thousands of years due to the preferential recycling of P and concomitant high loss ratio of dissolved organic N:P. At the long timescale plants become P limited because weathering inputs of P are negligible and dust P inputs are small. The time at which P limitation appears is controlled by the rock P weathering rate. See Fig. 4 for a different development trajectory. Note the logarithmic time axis.
Figure 3.
Examples of transient dynamics with single or double saturation rates.
(A) On the intermediate timescale, N-limited plant biomass (solid curve) approaches its quasi equilibrium (dotted line) at the rate µN′ (equation 9, which is the exponent in equation 11). With a logarithmic time axis (as here) this appears sigmoidal, but with a linear time axis it would be a saturating curve (similar to the Michaelis-Menten, or Type II, curve). P-limited plant biomass would have the same shape. (B) When NPP is P limited on the long timescale, SOM P has two controlling rates, the SOM P loss rate (mP(1−κδP) + φP ≈ φP) and the P weathering rate (ψ, equation 17). When these are sufficiently different, SOM P (solid line) first approaches an intermediate quasi equilibrium (dashed line) at the SOM P loss rate, then proceeds to its equilibrium (dotted line) at the P weathering rate. Here the rates are different enough to yield an overshoot, but not sufficiently different to yield full quasi equilibration at the intermediate point. Other details of timescale dynamics are given in Figure S1.
Figure 4.
Timescale approximation illustrating the effects of obligate N fixation and high continual P inputs.
Parameters and starting conditions are as in Fig. 2 except that F = 0.07 kg N kg C−1 y−1 for the short and intermediate timescales and α = 0.2 kg P ha−1 y−1 throughout. At the end of the intermediate timescale F is set to 0 to simulate the exclusion of N fixers. (A) Plant biomass, (B) which nutrient limits net primary production (NPP), (C) plant-available N, and (D) plant-available P are shown. Breaks in timescales, quasi equilibria, and equilibria are shown as in Fig. 2. In this case obligate N fixers fix enough to overcome N limitation, hence P limits NPP through the short and intermediate timescales. Although P weathering depletes rock P as in Fig. 2, relatively high dust deposition and high losses of plant-unavailable N relative to P combine to make N limit NPP at the terminal steady state.
Figure 5.
Approximate determination of N versus P limitation at different timescales.
(A) The beginning of the intermediate timescale (beginning of succession), (B) end of the intermediate timescale (end of succession), and (C) terminal steady state (end of long-term ecosystem development) are shown with simplifying assumptions from Table 2 and no symbiotic N fixation. (A) At the beginning of succession there is negligible soil organic matter (SOM), so limitation is determined by input fluxes (the axes), plant demand, and litter N versus P mineralization. The dividing line between N and P limitation (condition 13) is plotted with parameters in Table 1 and 10 kg C ha−1 initial plant biomass (solid line) as well as for high (litter N:P = 20, P:N mineralization rates = 1.5; dashed line) and low (litter N:P = 8, P:N mineralization rates = 1; dotted line) parameter values. The circle represents parameters in Table 1. Ecosystems with high rock P inputs would be at the right end of the panel. Ecosystems with high N deposition would be near the top of the panel. (B) At the end of succession, mineralization from SOM far exceeds the balance of abiotic inputs and losses in unpolluted ecosystems, so N versus P limitation is determined by SOM N and P (the axes), plant N:P demand, and the mineralization of litter and SOM N versus P (condition 15). The lines are as in (A), and the circle is the initial SOM N and P in Fig. 2. (C) At the terminal steady state, N versus P limitation is determined by the balance of continual inputs and losses, plant N:P demand, and the mineralization of litter and SOM N versus P (condition 18). Axes are N and P input fluxes. Lines and open circle as in (A), with the closed circle indicating the input fluxes for Fig. 4.
Figure 6.
Effects of preferential P versus N mineralization.
Effects are different for the (A) intermediate and (B) long timescales. Parameters are from Table 1 except for the mineralization rates mN and mP and the fractions of litter N and P decomposition mineralized, εN and εP, which vary to give the ratios along the horizontal axis. Timescale approximations of soil organic matter (SOM) N:P (solid black line), N:P losses of plant-unavailable nutrients from SOM and litter combined (termed DON:DOP in the legend for brevity, but in reality incorporating dissolved and particulate hydrologic losses and erosion losses; red dashed line), N:P mineralization fluxes from SOM and litter combined (blue dashed-dotted line), and plant N:P demand (black dotted line; which here is equivalent to P use efficiency:N use efficiency, litter N:P, and plant uptake) are plotted. The near-1∶1 correspondence between SOM N:P and DON:DOP stems primarily from the equivalence of φN and φP, which is not always true (see Table 2 and [37]). However, any monotonic relationship would yield the same timescale dynamics exhibited here.
Figure 7.
Contrasting effects of inputs versus losses in determining N versus P limitation over long timescales.
Effects are shown for simulations in (A) Fig. 2 and (B) Fig. 4. The N:P ratio of plant-unavailable losses (termed DON:DOP in the legend for brevity, but in reality incorporating dissolved and particulate hydrologic losses and erosion losses; dotted lines) increases over decades due to preferential P mineralization (see Fig. 6), increasing the likelihood of N limitation over the intermediate and the beginning of the long timescales. On millennial and longer timescales the input N:P ratio increases due to the decline in P weathering, increasing the likelihood of P limitation over very long timescales. The transition time from N to P limitation in Fig. 2 corresponds to the crossing of the input and loss N:P ratio lines in (A) (indicated by the vertical line), which is controlled primarily by the weathering rate (see Fig. 8). In Fig. 4 and (B) there is no transition to P limitation over long timescales because the input N:P remains lower than the DON:DOP loss ratio, despite increases in both. The condition for whether limitation switches from one nutrient to the other is given in conditions 14, 15 and 18 (also see Fig. 5). The dip in the DON:DOP loss ratio when the input and loss line cross (when limitation switches) in (A) is an artifact of the timescale separation approximation–it remains constant at its saturation level in the full numerical integration (results not shown)–but all other trends reflect the true dynamics of the system. Note that both axes are logarithmic.
Figure 8.
Transition time to P limitation on the weathering timescale.
The main determinant of the transition time, if it occurs on the weathering timescale, is the weathering rate. Equation 19 is plotted as a function of the weathering rate, with the rest of the parameters as in Table 1 (solid diagonal line), high dust deposition and parent material P (α = 0.05 kg P ha−1 y−1, γ = 1 kg P ha−1; dashed diagonal line) and low dust deposition and parent material (α = 0.001 kg P ha−1 y−1, γ = 0.2 kg P ha−1; dotted diagonal line). The weathering rate and transition time in Table 1 and Fig. 2 are plotted as an open circle. All parameter sets plotted here yield a transition to P limitation, although many possible parameters sets do not (see Figs. 4, 5). Weathering rates from the Hawai’i (solid vertical line) and Franz Josef (dotted vertical line) chronosequences from [51] are also plotted. Note that both axes are logarithmic.
Figure 9.
Model predictions for N vs. P limitation in the youngest Hawai’i and Franz Josef sites.
Axes and equations are as in Fig. 5, but with Hawai’i in a solid line and closed circle and Franz Josef in a dotted line and open circle. At the beginning of primary succession the Hawai’i site would be N limited but fairly close to co-limitation, whereas the Franz Josef site would be strongly N limited without any symbiotic N fixation. Input and nutrient use efficiency parameters come from the 300 year old Hawaiian site (IN = 9.6 kg N ha−1 y−1, IP = 0.63 kg P ha−1 y−1, ωN = 382 g C g N−1, ωP = 6780 g C g P−1 [37]) and the 5–7 year old New Zealand site (IN = 1.5 kg N ha−1 y−1 [78], IP = 2.2 kg P ha−1 y−1–calculated from the decrease in soil inorganic P from the 5 to the 60 year old site–ωN = 45 g C g N−1, ωP = 643 g C g P−1 [5]), with the other parameters as in Fig. 5a.