Abelian integrals and limit cycles for a class of cubic polynomial vector fields of Lotka-Volterra type with a rational first integral of degree 2

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.