Examples of transient dynamics with single or double saturation rates.

<p>(A) On the intermediate timescale, N-limited plant biomass (solid curve) approaches its quasi equilibrium (dotted line) at the rate <i>µ_N′</i> (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 (<i>m_P</i>(1−<i>κ_δP</i>) + <i>φ_P ≈ φ_P</i>) and the P weathering rate (<i>ψ</i>, 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 <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042045#pone.0042045.s001" target="_blank">Figure S1</a>.