Perturbed Abundance and Metabolic Rate Distributions in DynaMETE.
Here we examine, in a first order approximation to a fully iterated solution, how different types of disturbance give rise to characteristic departures of the SAD, \(\phi\left(n\right),\) and the MRDI,\(\psi\left(\varepsilon\right),\) from their steady state form. In particular, we truncate the iteration procedure, stopping with step 6 in Table 1. A single iteration at a one-year time step, however, results in changes in the structure function that are too small to show interesting deviations from steady state, so to generate a discernible effect for a single iteration we use a time step of 25 years. Specifically, we assume the static structure function, with Lagrange multipliers “frozen” at their static numerical values as prescribed in Eqs. 21-23, perturb the transition functions by changing one or more rates constants, and then derive from Eqs. 21-23 a set of time-differential equations for the state variables. These equations differ from Eqs. 38-40 because the latter were derived by updating the Lagrange multipliers at each time step. We then ran these equations out to t = 25 and take the values of the {X (25)} and the {dX (25)/dt} as constraints in Eqs. 14-18 to calculate using Max Ent a perturbed structure function. That function will be of the form of Eq. 19 and from it we derive perturbed forms for the species abundance distribution (SAD) and the metabolic rate distribution over individuals (MRDI) using the same summations as performed to derive Eqs. 8 and 9.
The results of that calculation are shown in Fig. 5. The five derived Lagrange multipliers are given in Table 3. Setting the immigration rate constant, m 0, to zero only slightly alters the SAD and the MRDI in this first iteration of the full structure function (Fig. 5a,b), even S decreases significantly. A 25% increase in the death rate constant, d 0, shifts the SAD toward a lognormal shape as indicated by the curved rank-log(abundance) graph at intermediate abundances (Fig. 5c). The rank-log(metabolism) graph shifts in a more complex manner, weaving around the METE prediction, and predicting more of the very smallest trees (\(\varepsilon=1)\), fewer individuals with low (\(\varepsilon=2-100)\ \)metabolism, and more trees with relatively high (\(\varepsilon=100-100\)000) metabolism, and a reduction in the sizes of the very largest individuals (Fig. 5d). A 5% decrease in the growth rate of individuals, w 0, generates a roughly mirror-image shift in the SAD relative to that from an increase in the death rate; the resulting SAD is approximately described by either an exponential distribution or an inverse power function with exponent > 1 (Fig. 5e). Similarly, the shift in the MRDI generated by a decrease in growth rate is roughly the mirror image of the shift induced by an increase in the death rate (Fig. 5f).
Figs. 5g,h show the effect of the same combination of changes in the rate constants used to generate Fig. 3d. We do not attempt here a detailed comparison to real data because of the first order approximation used to obtain these theoretical curves, but we do note their rough similarity to the empirical SAD and MRDI at BCI (see insets in Figs. 5g,h).
We emphasize that a full iterative solution of Eqs. 14-23 in, say, 25 one-year time steps, over a period of 25 years could result in output that differs considerably from the truncated solutions in Fig. 5. Subsequent work will explore this.