A new method for the explicit integration of Lotka-Volterra equations

In this work we study some first order nonlinear ordinary differential equations describing the time evolution (or "motion") of those hamiltonian systems provided with a first integral linking implicitly both variables to a motion constant. An application has been performed on the Lotka-Volterra predator-prey system, turning to a strongly nonlinear- differential equation in the phase variables. Our method grasps all the capabilities of modern computer algebra in order to solve (algebraic approximation) some equations of third and fourth degree with intricate forcing terms, obtaining symbolic explicit expressions osculating the solution in a neighborhood of the initial conditions. Another approach is also developed managing a Taylor truncated series and inverting it (asymptotic approximation). After having evaluated how both approximations differ from the traditional numerical techniques, finally we accomplish the much more probatory control of the approximants' accuracy referred, through the motion constant, to the first integral of the equation itself.