First integrals of a class of $n$-dimensional Lotka-Volterra differential systems

Lotka-Volterra model is one of the most popular in biochemistry. It is used to analyze cooperativity, autocatalysis, synchronization at large scale and especially oscillatory behavior in biomolecular interactions. These phenomena are in close relationship with the existence of first integrals in this model. In this paper we determine the independent first integrals of a family of $n$--dimensional Lotka-Volterra systems. We prove that when $n=3$ and $n=4$ the system is completely integrable. When $n\geq6$ is even, there are three independent first integrals, while when $n\geq5$ is odd there exist only two independent first integrals. In each of these mentioned cases we identify in the parameter space the conditions for the existence of Darboux first integrals. We also provide the explicit expressions of these first integrals.