An Ecology and Economy Coupling Model. A global stationary state model for a sustainable economy in the Hamiltonian formalism

The severity of the two deeply correlated crises, the environmental and the economic ones, needs to be faced also in theoretical terms; thus, the authors propose a model yielding a global “stationary state”, following the idea of a “steady-state economics” by Georgescu-Rogen and Herman Daly, by constructing only one dynamical system of ecological and economic coupled variables. This is possible resorting to the generalized Volterra model, that, translated in the Hamiltonian formalism and its Hamilton equations, makes possible to “conjugate” every pair of variables, one economic, the other one ecological, in describing the behavior in time of a unique dynamical system. Applying the model to two of the most relevant ecological-economic pairs of variables leads to a suggestive geometry in the “phase space” of the model: the trajectories are curves wrapping a “donut”, their set is the “stationary state” we were looking for. Those trajectories are “quasi-periodic motions”, characterized by two frequencies, for whose values a good estimate is provided in the “small oscillations” approximation. A more general, but more abstract, “stationary state” is defined by virtue of the stability of the solutions of the Hamilton equations, just in this article recognized. The global character of the model is assured when world data of variables are used. A very interesting feature of the model is that the path to a scenario of sustainability is given in terms analogous to the Newtonian Dynamics.