Integrability and periodic orbits in the generalized quasispecies model

Abstract

This paper studies a parametric family of systems of differential equations, which is obtained from the quasispecies model assuming arbitrary parameters without biological constraints. We study equilibria, invariant manifolds and the integrability of the family. It is well known that, by restricting the parameters to the biological domain, the quasispecies model is integrable, and here, we show that it is also the case for general parameters. Moreover, we consider a 4-dimensional realization of the model under the influence of a periodic perturbation. After restricting the system to an invariant manifold and relying on the first-order averaging technique, we demonstrate the existence of unstable periodic orbits in a neighborhood of the equilibrium located at the origin. It shows that periodic orbits emanate from the Zero-Hopf bifurcation that the mentioned equilibrium undergoes when the small parameter equals zero.