Here, we briefly explain (i) the failure of the traditional population dynamics, (ii) the state-of-the-art in the mathematical formulation of unified metabolic theory of ecology, and (iii) a problem with the state-of-the-art.
(i) Failure of the traditional population dynamics
Population dynamics is of utmost socio-economic importance because it is used in stock assessments and, more generally, bioresource management (e.g. forestry and fishing). For the purpose of traditional population modeling, the necessary assumptions on growth and reproduction of individuals are obtained by statistically fitting an appropriate length-at-age or fecundity-at-age curves to the available data. In this way, however, the modeled organism becomes completely unresponsive to the changing environment, which contradicts the evidence. For example, captive Pacific bluefin tuna not only grow faster than their wild relatives, but they also mature faster (5 vs. 3 years in the wild and captivity, respectively).
(ii) The state-of-the-art
To compensate for the explained failure of the traditional population dynamics, the state-of-the-art approaches replace statistical data fitting with a model founded on physiological energetics. When such an individual-level model is coupled with a partial differential equation (PDE) that describes the population density, we have the basic dynamical system of unified metabolic theory of ecology.
(iii) Problem with the state-of-the-art
The described dynamical system of unified metabolic theory of ecology is deterministic. However, stochasticity plays an important role because, for example, food availability can randomly fluctuate in the environment. A particularly notorious example, relevant for many economically valued species (e.g. bluefin tuna), is that life history can exhibit extreme stochasticity, especially due to mortality in embryo and larval stages. In this case, however, PDE that describes the population density becomes stochastic. Based on existing research [Bjornstad et al., J Anim Ecol 73: 1157–1167], one can expect rich dynamical phenomena from an individual-level model coupled with a stochastic PDE. Thus far, such a dynamical system has not been analyzed in the literature.