A Thermodynamic Theory of Ecology: Helmholtz Theorem for Lotka-Volterra Equation, Extended Conservation Law, and Stochastic Predator-Prey Dynamics

摘要

We carry out mathematical analyses, {\em \`{a} la} Helmholtz's andBoltzmann's 1884 studies of monocyclic Newtonian dynamics, for theLotka-Volterra (LV) equation exhibiting predator-prey oscillations. In doing soa novel "thermodynamic theory" of ecology is introduced. An important feature,absent in the classical mechanics, of ecological systems is a naturalstochastic population dynamic formulation of which the deterministic equation(e.g., the LV equation studied) is the infinite population limit. Invariantdensity for the stochastic dynamics plays a central role in the deterministicLV dynamics. We show how the conservation law along a single trajectory extendsto incorporate both variations in a model parameter $\alpha$ and in initialconditions: Helmholtz's theorem establishes a broadly valid conservation law ina class of ecological dynamics. We analyze the relationships among meanecological activeness $\theta$, quantities characterizing dynamic ranges ofpopulations $\mathcal{A}$ and $\alpha$, and the ecological force $F_{\alpha}$.The analyses identify an entire orbit as a stationary ecology, and establishthe notion of "equation of ecological states". Studies of the stochasticdynamics with finite populations show the LV equation as the robust, fastcyclic underlying behavior. The mathematical narrative provides a novel way ofcapturing long-term dynamical behaviors with an emergent {\em conservativeecology}.