Future Work
Flexibility of transition functions. Our assumptions about the
functional forms of the transition functions are easily replaced with
alternatives. For the extinction process, Eq. 31 can be easily modified
to describe a minimum viable population size greater than 2. Alternative
forms of the ontogenic growth equation (e.g., Makareiva et al., 2004)
can be substituted for Eq. 29. Density dependence can be described by a
–d1n 2 term. Moreover, a wide
range of options are possible for modeling the dependence of speciation
rates on n , \(\varepsilon,\) and the state variables. A
modification of the birth rate function can improve the realism of the
transition functions when applied to forest census data that are limited
to trees with some minimum threshold dbh, as for example with the
Smithsonian Tropical Forest census data. Entry into the data set arises
not from birth but from ontogenic growth into the smallest censused size
cohort.
When the SAD and MRDI are not measured. If empirical values of
the state variables and their time derivatives are available, they can
be plugged directly into Eqs. 14-18. Initial investigation suggests that
if the unperturbed values of the transition function rate constants are
known, then the range of possible perturbed values of those constants
that permit solutions for the Lagrange multipliers is fairly tightly
constrained by the constraints imposed by the measured time derivatives
of the state variables. Thus the causes of disturbance could be
identified without requiring knowledge of the abundance and metabolic
rate distributions.
Space. To include space explicitly in DynaMETE, one could
modify the assembly (or colonization) rule (Harte, 2011) that generates
the species-level spatial distributions in METE (Conlisk et al. 2007,
Brush & Harte 2020) by combining the assembly rule with explicit
demographic perturbations incorporated in the transition functionf (n , \(\varepsilon)\).
Traits. DynaMETE currently incorporates individual metabolic
rate (or body size) and abundance. Other traits are implicitly assumed
to be distributed across species and individuals in such a way as to
result in equal fitness opportunities for each taxon. Under disturbance,
however, some trait values may confer more or less fitness and result in
preferential changes in growth and abundance. Letting the transition
rates, f , h , and q , depend upon additional trait
values, as well as upon n and \(\varepsilon\), could allow
investigation of disturbances that affect particular groups of species
or populations more than others, enhancing understanding of how traits
mediate the responses of macroecological metrics to disturbance.
Higher taxonomic categories. The scope and realism of static
METE was enhanced by including an additional state variable
corresponding to the number of families in the community (Harte et al.,
2015). DynaMETE can also accommodate additional state variables
corresponding to the richness of higher taxonomic categories. Applying
such a modified theory at sufficiently long time scales could perhaps
provide insight into the mechanisms of diversification at higher
taxonomic levels, and improve DynaMETE’s realism at short time scales..
Some broader issues. Both anthropogenic stresses and natural
disturbances can cause state variables to rapidly change, and thus
systems experiencing either type of disturbance can fall within
DynaMETE’s criterion for a dynamic ecosystem. Although evidence reviewed
in the Introduction suggests that systems undergoing either type of
disturbance exhibit macroecological patterns that deviate from METE, it
is unclear whether DynaMETE will be applicable both to ecosystems that
are disturbed anthropogenically, and to ecosystems that are adapted to
natural disturbance regimes. Given the importance of finding early
warning signals that distinguish human impact on ecosystems from the
effects of natural disturbance regimes, this is a high priority.
In a provocative article (Goldenfeld and Woese, 2011) suggest that while
physics makes a clean separation between the state of a physical system
and the equations that govern the time-evolution of the system,
successful biological theory will inevitably be self-referential or
recursive in the sense that the state of the system will strongly enter
into the equations that govern dynamics. We observe that this is true of
DynaMETE (Eqs. 14-23; 35-37); the state variables appear explicitly in
the transition functions, which in turn govern state dynamics.
Many academic fields seek to unify complex micro- and macro-level
dynamics, in a speculative vein we suggest that the proposed iterative
procedure at the core of DynaMETE could be of possible application in,
for example, economics (Golan, 2018) and in statistical physics (Jaynes,
1957; 1982), where in both of these fields, static equilibrial patterns
can be captured by MaxEnt but non-equilibrial dynamics has remained
elusive.