Stable and periodic solutions of an exergy-based model of population dynamics

The long-term qualitative behavior of most real ecological populations can be mathematically classified as either asymptotic equilibrium or periodic. The reasons underlying these behaviors have been, and are still, debated. Aim of this paper is to show that it is possible to characterize both of these behavious through a novel (in general non-autonomous) population dynamics model, proposed and developed in previous work by the authors, in which the resource consumption is quantified solely in terms of exergy flows. A limited but significant number of reasonable assumptions on the phenomenological characteristics of the interacting species result in different evolutionary scenarios, corresponding respectively to asymptotic stability or periodic attractors. In the case of asymptotic periodic motion, the system is described by a set of non-linear, non-autonomous differential equations and an analytical investigation becomes arduous even in the simplest cases. Explicit results, both analytical and numerical, are discussed in this paper. The theoretical description is compared with the behavior of real, well-known biological systems: the snowshoe hare-lynx predator-prey system for the periodic case and the reindeer herds of the Pribilof Islands for the case of asymptotic stability. The assumptions that must be posited to obtain a periodic behavior in ecological populations appear to have been never recognized before and the possibility to adopt them as an universal mechanism of population cycles is discussed in the conclusions.