On the dynamics of a class of Kolmogorov systems

known as Kolmogorov system, the derivatives are performed with respect to the time variable and F , G are two functions in the variables x and y. If F and G are linear (LotkaVolterraGause model), then it is well known that there is at most one critical point in the interior of the realistic quadrant (x > 0, y > 0), and there are no limit cycles [10,14,16]. There are many natural phenomena which can be modeled by the Kolmogorov systems such as mathematical ecology and population dynamics [12,17,18] chemical reactions, plasma physics [13], hydrodynamics [5], etc. In the qualitative theory of planar dynamical systems [7–9, 15], one of the most important topics is related to the second part of the unsolved Hilbert 16th problem. There is a huge literature about the limit cycles, most of it deals essentially with their detection, number and stability [1–4, 11]. We recall that in the phase plane, a limit cycle of system (1) is an isolated periodic orbit in the set of all periodic orbits of system (1). System (1) is integrable on an open set Ω of R if there exists a non constant continuously differentiable function H : Ω → R, called a first integral of the system on Ω , which is constant on the trajectories of the system (1) contained in Ω, i.e. if