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.