Abstract
This paper is devoted to study the zero-Hopf bifurcation of the three dimensional Lotka-Volterra systems. The explicit conditions for the existence of two first integrals for the system and a line of singularities with zero eigenvalue are given. We characteristic the parameters for which a zero-Hopf equilibrium point takes place at any points on the line. We prove that there are three 3-parameter families exhibiting such equilibria. The averaging theory of the first order is also applied but any information about the possible periodic orbits bifurcating from the zero-Hopf equilibria is not provided by this theorem.
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