Applied Sciences and Mathematics, Arizona State University, Mesa, AZ 85212, USA.
J Biol Dyn. 2012;6:358-76. doi: 10.1080/17513758.2011.586064. Epub 2011 Jun 9.
In this article, we study the global dynamics of a discrete two-dimensional competition model. We give sufficient conditions on the persistence of one species and the existence of local asymptotically stable interior period-2 orbit for this system. Moreover, we show that for a certain parameter range, there exists a compact interior attractor that attracts all interior points except Lebesgue measure zero set. This result gives a weaker form of coexistence which is referred to as relative permanence. This new concept of coexistence combined with numerical simulations strongly suggests that the basin of attraction of the locally asymptotically stable interior period-2 orbit is an infinite union of connected components. This idea may apply to many other ecological models. Finally, we discuss the generic dynamical structure that gives relative permanence.
在本文中，我们研究了一个离散二维竞争模型的全局动力学。我们给出了该系统中一个物种持续存在和存在局部渐近稳定内周期 2 轨道的充分条件。此外，我们表明，对于一定的参数范围，存在一个紧致的内吸引子，它吸引除勒贝格测度零集以外的所有内点。这个结果给出了一种较弱的共存形式，称为相对持久性。这种新的共存概念结合数值模拟强烈表明，局部渐近稳定内周期 2 轨道的吸引域是连通分支的无限并集。这个想法可能适用于许多其他生态模型。最后，我们讨论了产生相对持久性的一般动力学结构。
