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

TL;DR

This paper develops a thermodynamic framework for ecological dynamics based on the Lotka-Volterra model, incorporating stochastic effects, conservation laws, and the concept of ecological states to better understand predator-prey oscillations.

Contribution

It introduces a novel thermodynamic theory of ecology, extending classical mechanics concepts to ecological systems with stochastic dynamics and conservation laws.

Findings

Invariant density links stochastic and deterministic dynamics.

Helmholtz's theorem extends conservation laws to ecological parameters.

Lotka-Volterra dynamics are robust under stochastic effects.

Abstract

We carry out mathematical analyses, {\em \`{a} la} Helmholtz's and Boltzmann's 1884 studies of monocyclic Newtonian dynamics, for the Lotka-Volterra (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 $\alpha$ and in initial conditions: Helmholtz's theorem establishes a broadly valid conservation law in a class of ecological dynamics. We analyze the relationships among mean ecological activeness $\theta$, quantities characterizing dynamic ranges of populations $\mathcal{A}$ and $\alpha$, and the ecological force $F_{\alpha}$. The analyses identify an entire orbit as a stationary ecology, and establish the notion of "equation of ecological states". Studies of the stochastic dynamics with finite populations show the LV equation as the robust, fast cyclic underlying behavior. The mathematical narrative provides a novel way of capturing long-term dynamical behaviors with an emergent {\em conservative ecology}.