  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.
