Phase Space Formulation of Population Dynamics in Ecology

A phase space theory for population dynamics in Ecology is presented. This theory applies for a certain class of dynamical systems, that will be calledM-systems, for which a conserved quantity, the M-function, can be defined in phase space. ThisM-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.