  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.
