A thermodynamic theory of ecology: Helmholtz theorem for Lotka-Volterra equation, extended conservation law, and stochastic predator-prey dynamics

Abstract

We carry out mathematical analyses la Helmholtz's and Boltzmann's 1884 studies of monocyclic Newtonian dynamics, for the LotkäCVolterra (LV) equation exhibiting predator-prey oscillations. In doing so, a novel thermodynamic theory' of ecology is introduced. An important feature, absent in the classical mechanics, of ecological systems is a natural stochastic population dynamic formulation of which the deterministic equation (e.g. the LV equation studied) is the infinite population limit. Invariant density for the stochastic dynamics plays a central role in the deterministic LV dynamics. We show how the conservation law along a single trajectory extends to incorporate both variations in a model parameter α and in initial conditions: Helmholtz's theorem establishes a broadly valid conservation law in a class of ecological dynamics. We analyse the relationships among mean ecological activeness È, quantities characterizing dynamic ranges of populations A and α, and the ecological force Fα. The analyses identify an entire orbit as a stationary ecology, and establish the notion of an ¡® equation of ecological states'. Studies of the stochastic dynamics with finite populations show the LV equation as the robust, fast cyclic underlying behaviour. The mathematical narrative provides a novel way of capturing long-term dynamical behaviours with an emergent conservative ecology.