THE PERIOD FUNCTION IN A CLASS OF QUADRATIC KOLMOGOROFF SYSTEMS

In this paper we consider the family of quadratic Kolmogoroff 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 integrability 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. 98.015.0111.0 198 Marco Uribe and Myrna Wallace

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 integrability 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.

Introduction
A Kolmogoroff system is given by a planar differential system of the form This kind of system appears usually in predator-prey models and so only dynamics on the first quadrant is considered.In this paper we consider the quadratic Kolmogoroff system: where 1 < a < ∞.
A system of this class has a family of periodic solutions nested around a center at 1 1 + a , 1 1 + a and filling out the triangle formed by the saddle points (0, 0), (1, 0) and (0, 1).The family of periodic solutions forms a center period annulus The function which associates to any closed curve its period, is called the period function.We are interested in obtaining the global description of the period function defined on the center period annulus.
In section 2 , we analyze the period function and we prove that it is monotone increasing for a = 3.
In section 3 we prove, by means of a linear change of coordinates that system (1.1), in the Bautin's form is of type B 2 for all a > 1 Bautin's systems and their period functions were defined in [1] by C. Chicone and M. Jacob where they conjectured that for B 2 systems the period function is globally monotone increasing.
By calculating the first three periodic coefficients for a > 1, we realize that they are alternatively positive and negative.This result suggests us that there would exists a value of the parameter a such that the corresponding system has a period function with critical points.
In general, aside from their intrinsic interest, monotonicity properties of the period function are important in the question of existence and uniqueness of autonomous boundary value problems [2], in the study of subharmonic bifurcation of periodic oscillations, and in the analysis of the problem of linearization.

The Period Function
System (1.1) has a first integral In fact Since H(x, y) = 0 for (x, y) on the invariant triangle x = 0, y = 0, a+1) .The period of this orbit is given then by the formula We are interested in the study of T as a function of the energy E. We observe that for E going from 0 to (a − 1) a−1 (a + 1) −(a+1) , the closed orbit goes from the polycycle to the center so that a negative derivative dT dE , indicates an increasing period function.
In order to calculate dT dE , we introduce the change of variables on a periodic orbit γ(E) : u = Ev 1−a and γ(E) can be given by (x + (v), y − (v)) and (x − (v), y + (v)) where 2) the integral T (E) given by (2.1) can be written in the new coordinates u and v. Since u and v are related to each other on γ(E), T (E) can be rewritten as an integral in v alone.Using the simmetry of γ(E) , the integral T (E) becomes Now, we consider new a change of variables.Define where Since g has a quadratic maximum at v 0 , h is a smooth function.Moreover, it is easy to see that h (v) > 0, for all v min < v < v max , so h monotone increasing.Setting r = h(v), the integral T (E) can be simplified to This form of the integral suggests the substitution r = √ w 0 − 4E sin θ.We obtain (2.7) In this form the integral can be derivated to obtain sin θdθ (2.8) Now, a straightforward calculation proves the Theorem : The period function is monotone increasing in a neighborhood of a = 3 Proof : If a = 3 in (2.8) , we obtain In fact , because h is monotone increasing.
Remark : Since T (E) is a continuous function on the parameter a, we can assert that T (E) is monotone increasing for a in a neighborhood of a = 3.

The Period Function near the center
In this section we compute the first period constants of system (1.1), using the results of [3] , [4] .
First, using the change of variables we carry the center to the origin Bringing the linear part of (3.1) to the Jordan form we obtain with a rescaling of time t = √ a+1 √ a−1 T , the system is transformed into