Phase Space Formulation of Population Dynamics in Ecology

TL;DR

This paper introduces a phase space framework for population dynamics in ecology, defining M-systems with a conserved M-function analogous to Hamiltonian mechanics, and applies it to key ecological models.

Contribution

It develops a novel phase space formulation for ecological systems using M-functions and brackets, unifying various models under a common geometric approach.

Findings

M-systems include Lotka-Volterra, self-feeding, and logistic models.

Derived equations of motion from a variational principle.

Established a geometric framework analogous to Hamiltonian mechanics.

Abstract

A phase space theory for population dynamics in Ecology is presented. This theory applies for a certain class of dynamical systems, that will be called M-systems, for which a conserved quantity, the M-function, can be defined in phase space. This M-function is the generator of time displacements and contains all the dynamical information of the system. In this sense the M-function plays the role of the hamiltonian function for mechanical systems. In analogy with Hamilton theory we derive equations of motion as derivatives over the resource function in phase space. A M-bracket is defined which allows one to perform a geometrical approach in analogy to Poisson bracket of hamiltonian systems. We show that the equations of motion can be derived from a variational principle over a functional J of the trajectories. This functional plays for M-systems the same role than the action S for hamiltonian systems. Finally, three important systems in population dynamics, namely, Lotka-Volterra, self-feeding and logistic evolution, are shown to be M-systems.