EXPLICIT FORM FOR THE FIRST INTEGRAL AND LIMIT CYCLES OF A CLASS OF PLANAR KOLMOGOROV SYSTEMS

In this paper we characterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form

{x′=x(R(x,y)exp⁡(A(x,y)B(x,y))+P(x,y)exp⁡(C(x,y)D(x,y))),y′=y(R(x,y)exp⁡(A(x,y)B(x,y))+Q(x,y)exp⁡(V(x,y)W(x,y))),

where A(x,y), B(x,y), C(x,y), D(x,y), P(x,y), Q(x,y), R(x,y), V(x,y), W(x,y) are homogeneous polynomials of degree a, a, b, b, n, n, m, c, c, respectively. Concrete example exhibiting the applicability of our result is introduced.