A NOTE ON A CLASS OF DISCRETE LARGE- AND SUB-CENTRE SYSTEMS

For different mappings, there are various methods and techniques to obtain the normal forms and calculate the center manifolds of a system, with the computations being highly complicated. Unfortunately, all the results obtained are merely of a local nature. Recently, a discrete logistic-lottery system with large- and sub-centres was studied. The authors established a flip bifurcation theorem and provided some numerical simulations. Thus, they demonstrated that the lottery competition is indeed driven by the logistic growth of the dominant species. In this note, we have once again considered that system and theoretically proved that all its stable periodic solutions are induced by the main equation. In particular, the method is remarkably straightforward, and the logistic model can be generalized to other recursive equations, such as those describing the weak Allee effect, the strong Allee effect, the Ricker model, or the Caspari-Watson function.