Dynamic extension of the maximum entropy theory of ecology (METE) using a master equation approach
Natural and anthropogenic disturbances can alter patterns in the abundance, distribution, and energetics of species. While existing theories have had success predicting the functional forms of macroecological patterns in static systems, a dynamic theory of the time dependence of the shape of these patterns under succession or disturbance is lacking. One approach to predicting static patterns combines the concept of ecological state variables with the MaxEnt inference procedure, using total species richness, total abundance, and total metabolic rate of a community as state variables to predict species-area relationships, the metabolism-abundance relationship, and species abundance distributions. In that type of theory, distributions of microstate variables such as metabolic rates of individuals and abundances of species, are contingent upon the values of state variables. To extend this approach to the dynamic regime, we adopt a master equation approach. In contrast to neutral theory models, in which a master equation incorporates demographic rates for a random species and thereby generates an abundance distribution across species, we construct and solve a master equation for the time-dependent probability distribution for the state variables themselves. Using Bayes law, an integral over the state variables of the product of that distribution times the contingent distributions that arise from MaxEnt-based theory then yields the time-dependent distributions of macroecology.
The form of the master equation is too complex to solve analytically, but by taking advantage of the very different time constants characterizing growth of individuals, births and deaths, and diversification, the master equation can be approximated by a set of three weakly coupled equations for the three state variables. Analytic solutions for the static case are obtained, and the Bayesian stochastic macroecological metrics that then result are shown to fairly closely resemble the results from the deterministic theory. We then solve numerically for the time-dependent outcomes in response to either an initial displacement from steady state, or in response to a new patch of bare landscape suddenly becoming colonizable. These dynamic metrics help explain observed patterns of shifting abundance and body-size distributions for arthropods observed across Hawaiian islands of varying age and for plants observed in disturbed meadow ecosystems.