On the period function in a class of generalized Lotka–Volterra systems

摘要

In this note, motivated by the recent results of Wang et al. [Wang et al., Local bifurcations of critical periods in a generalized 2D LV system, Appl. Math. Comput. 214 (2009) 17–25], we study the behaviour of the period function of the center at the point (1,1) of the planar differential systemu′=up(1-vq),v′=μvq(up-1),where p,q,μ∈R with pq>0 and μ>0. Our aim is twofold. Firstly, we determine regions in the parameter space for which the corresponding system has a center with a monotonic period function. Secondly, by taking advantage of the results of Wang et al., we show some properties of the bifurcation diagram of the period function and we make some comments for further research. The differential system under consideration is a generalization proposed by Farkas and Noszticzius of the Lotka–Volterra model.