HOPF BIFURCATION PROBLEM FOR A CLASS OF KOLMOGOROV MODEL WITH A POSITIVE NILPOTENT CRITICAL POINT

In this paper, We discuss the Hopf bifurcation problem of a three-order positive nilpotent critical point (1, 1) of a class of Kolmogorov model. By using the method offered by [12], we obtain the expressions of quasi-Lyapunov constants with the help of computer algebra system-MATHEMATICA. By analyzing the structure of these quasi-lyapunov constants, we divide them into two kinds of cases and study their bifurcation behavior separately. For case 1, the nilpotent critical point (1, 1) can bifurcate 5 small amplitude limit cycles. For case 2, 6 small amplitude limit cycles can bifurcate from the three-order nilpotent critical point (1, 1). In addition, We also give the integrability conditions (i.e., center condition) for each case. In terms of limit cycle bifurcation for Kolmogorov model with nilpotent positive critical points, our result is new.