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.