Fractional dynamics of populations

Nature often presents complex dynamics, which cannot be explained by means of ordinary models. In this paper, we establish an approach to certain fractional dynamic systems using only deterministic arguments. The behavior of the trajectories of fractional non-linear autonomous systems around the corresponding critical points in the phase space is studied. In this work we arrive to several interesting conclusions; for example, we conclude that the order of fractional derivation is an excellent controller of the velocity how the mentioned trajectories approach to (or away from) the critical point. Such property could contribute to faithfully represent the anomalous reality of the competition among some species (in cellular populations as Cancer or HIV). We use classical models, which describe dynamics of certain populations in competition, to give a justification of the possible interest of the corresponding fractional models in biological areas of research. Fractional calculus emerged from the interest in generalizing the ordinary integrals and derivatives. Different derivatives and integrals of arbitrary (fractional, real or complex) order have been studied. For example, in [8,5] the authors show that the fractional calculus constitutes a meeting place of multiple disciplines: stochastic processes, probability, integro-differential equations, integral transforms, special functions, numerical analysis, etc. Here, we pay attention to fractional differential equations and systems of equations, that is, equations with derivatives of real or complex order ([1,5,6]):