First Integrals and a Zero-Hopf Bifurcation of the Four-Dimensional Lotka-Volterra Systems

In this paper, the integrability and a zero-Hopf bifurcation of the four-dimensional Lotka-Volterra systems are studied. The requirements for this kind of system's integrability and a line of singularities with two zero eigenvalues are provided. We identify the parameters that lead to a zero-Hopf equilibrium point at each point along the line of singularities. We show that there is only one parameter that displays such equilibria. The first-order averaging method is also employed, although this method will not give any information about the bifurcate periodic solutions that bifurcate from the zero-Hopf equilibria.