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,à 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 α 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 θ, quantities characterizing dynamic ranges of populations A and α, and the ecological force F α . 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 conservative ecology. *

Universal ideal behavior and macroscopic work relation of linear irreversible stochastic thermodynamics

“…The present paper is a study of OUP in terms of the latter perspective, in which we have identified the unbalanced circulation as a conservative dynamics, a hallmark of the generalized underdamped thermodynamics [33]. In terms of this conservative dynamics, Boltzmannʼs entropy function naturally enters stochastic thermodynamics, and we discover a relation between the Helmholtz theorem [60] and the various work relations.…”

Bridging discrete and continuous formalisms for biodiversity quantification

“…The result of this procedure, in which the Langevin equation is replaced with yet another set of deterministic ordinary differential equations for X via suitable closure assumptions, is sometimes called a "phenomenological equation". It is rarely the case that an exact, closedform macroscopic equation can be obtained -this is certainly the case for linear processes with natural boundaries, but not for nonlinear processes; although see e.g., Ma and Qian [28]. The phenomenological equations are thus deterministic population models typified by the Lotka-Volterra type equations in ecology, the SIR models in epidemiology, the Michaelis-Menten reaction kinetics, amongst others (see, for example, Murray [29]).…”

Thermodynamics of the general diffusion process: Equilibrium supercurrent and nonequilibrium driven circulation with dissipation

“…In general, x(t) is not ergodic on the entire level set. However, realizing that j(x) is only a deterministic representation of the stochastic circulation, one expects a time-scale separation between the intra-ϕ-level-set motion and motion across level sets [51]. To represent and characterize an entire ϕ level set, Boltzmann's idea was to quantify it using some geometric quantities.…”

“…with parameter α, which has a conserved quantity ϕ(x, α) [51]. Note that due to its definition, the vector field j(x, α) is structurally stable within the framework of the general diffusion process (1).…”