In this article, we prove that in connected metric spaces k - cycles are not globally attracting (where k>2). We apply this result to a two species discrete-time Lotka-Volterra competion model with stocking. In particular, we show that an k-cycle cannot be the ultimate life-history of evolution of all population sizes. This solves Yakubu's conjecture but the question on the structure of the boundary of the basins of attraction of the locally stable n-cycles is still open.
Taylor & Francis
Elaydi, S., & Yakubu, A.-A. (2002). Global stability of cycles: Lotka-Volterra competition model with stocking. Journal of Difference Equations and Applications, 8(6), 537-549. doi:10.1080/10236190290027666
Journal of Difference Equations and Applications