Behavior of limit cycle bifurcations for a class of quartic Kolmogorov models in a symmetrical vector field

Abstract

In this study, we consider the limit cycle bifurcation problem for a class of quartic Kolmogorov models with five positive singular points, i.e., (1,1), (1,2), (2,1), (1,3), and (3,1), which lie in a symmetrical vector field relative to the line y=x. We classify these singular points. We show that points (1,2) and (2,1) can bifurcate into three small limit cycles by simultaneous Hopf bifurcation, and that points (1,3) and (3,1) can bifurcate into three small limit cycles by simultaneous Hopf bifurcation. In addition, we construct limit cycles for this model and we show that four positive singular points, i.e., (1,1), (1,2), (2,1), and (1,3), can bifurcate into eight limit cycles in total, among which six cycles may be stable. Few previous studies have considered a symmetrical Kolmogorov model with several positive singular points. Our results are good in terms of the Hilbert number for the Kolmogorov model.