2013/04/08 by Jesus Martinez-Linares, Martinez-Linares, Jesus
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #FOS: Biological sciences #FOS: Physical sciences #Mathematical Physics (math-ph) #Populations and Evolution (q-bio.PE) #math-ph #math.MP #q-bio.PE
paper · pdf · doi:10.48550/arxiv.1304.2324
4 pages, 1 figure
arxiv created 2013/04/08 · arxiv updated 2013/04/09
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.