On the Non-Existence of Periodic Orbits for a Class of Two-Dimensional Kolmogorov Systems

In this paper we introduce an explicit expression of first integral, then we prove the nonexistence of periodic orbits, then consequently the non-existence of limit cycles of two-dimensional Kolmogorov system, where R(x, y), S (x, y), P (x, y), Q(x, y),M (x, y), N (x, y) are homogeneous polynomials of degrees m, a, n, n, b, b, respectively. We introduce concrete example exhibiting the applicability of our result.