The classic Lotka-Volterra model belongs to a family of differential equations known as “Generalized Lotka-Volterra”, which is part of a classification of four models of quadratic fields with center. These models have been studied to address the Hilbert infinitesimal problem, which consists in determine the number of limit cycles of a perturbed hamiltonian system with center. In this work, we first present an alternative proof of the existence of centers in Lotka-Volterra predator-prey models. This new approach is based in algebraic equations given by Kapteyn, which arose to answer Poincaré’s problem for quadratic fields. In addition, using Hopf Bifurcation theorem, we proof that more realistic models, obtained by a non-linear perturbation of a classic Lotka-Volterra model, also possess limit cycles.

Las ecuaciones diferenciales ordinarias surgen juntamente con la aparición del cálculo, en la célebre polémica de Newton y Leibniz, a finales del siglo XVII. Doscientos años después, Van der Pool y Appleton obtuvieron ecuaciones diferenciales relacionadas con los circuitos eléctricos. La teoría cualitativa de las ecuaciones diferenciales nació´ de los trabajos de Poincare´ y Liapunov a finales del siglo XIX y en los inicios del siglo XX. En el problema infinitesimal de Hilbert, se observa que bajo pequeñas perturbaciones de un campo se puede obtener ciclos límites en el campo perturbado. Este problema esta´ relacionado con la existencia de ciclos límites, por ello, es importante saber cuándo un campo posee un centro, el cual es llamado el problema del centro. Este problema fue´ resuelto por Poincare´ para campos polinomiales. Posteriormente Liapunov generalizó´ este r...

The classic Lotka-Volterra model belongs to a family of differential equations known as “Generalized Lotka-Volterra”, which is part of a classification of four models of quadratic fields with center. These models have been studied to address the Hilbert infinitesimal problem, which consists in determine the number of limit cycles of a perturbed hamiltonian system with center. In this work, we first present an alternative proof of the existence of centers in Lotka-Volterra predator-prey models. This new approach is based in algebraic equations given by Kapteyn, which arose to answer Poincaré’s problem for quadratic fields. In addition, using Hopf Bifurcation theorem, we proof that more realistic models, obtained by a non-linear perturbation of a classic Lotka-Volterra model, also possess limit cycles.