Limit Cycles Bifurcations for a Class of Kolmogorov Model in Symmetrical Vector Field

The problem of limit cycles for Kolmogorov model is interesting and significant both in theory and applications. Our work is concerned with limit cycles bifurcations problem for a class of quartic Kolmogorov model with two positive singular points (i.e. (1, 2) and (2, 1)). The investigated model is symmetrical with regard to y = x. We show that each one of points (1, 2) and (2, 1) can bifurcate five small limit cycles at the same step under a certain condition. Hence, the two positive singular points can bifurcate ten limit cycles in sum, in which six cycles can be stable. In terms of symmetrical Kolmogorov model, published references are less. In terms of the Hilbert Number of Kolmogorov model, our results are new.