THE PERIOD FUNCTION IN A CLASS OF QUADRATIC KOLMOGOROFF SYSTEMS ∗

In this paper we consider the family of quadratic Kolmogo-roff systems with a center in the real quadrant: {. x = x (1 − x − ay) y = y (−1 + ax+ y), where 1 < a < ∞.This system has three invariant lines (the coordinate axes and the line x+ y − 1 = 0) and a family of periodic solutions nested around a center and filling out the triangle determined by the three invariant lines. Using integra-bility of this system we reduce the abelian integral representing the period function and its derivative. The main result is that the corresponding period function is monotone increasing for values of the parameter near a = 3. ∗Partially supported by a Grant of Universidad de Concepción P.I. 98.015.011