  In this paper, we study the number of limit cycles which bifurcate from the
periodic orbits of cubic polynomial vector fields of Lotka-Volterra type having
a rational first integral of degree 2, under polynomial perturbations of degree
$n$. The analysis is carried out by estimating the number of zeros of the
corresponding Abelian integrals. Moreover, using \emph{Chebyshev criterion}, we
show that the sharp upper bound for the number of zeros of the Abelian
integrals defined on each period annulus is 3 for $n=3$. The simultaneous
bifurcation and distribution of limit cycles for the system with two period
annuli under cubic polynomial perturbations are considered. All configurations
$(u,v)$ with $0\leq u, v\leq 3, u+v\leq5$ are realizable.
