We study the occurrence of chaos in the Atkinson–Allen model of four competing species, which plays the role as a discrete-time Lotka–Volterra-type model. We show that in this model chaos can be generated by a cascade of quasiperiod-doubling bifurcations starting from a supercritical Neimark–Sacker bifurcation of the unique positive fixed point. The chaotic attractor is contained in a globally attracting invariant manifold of codimension one, known as the carrying simplex. Biologically, our study implies that the invasion attempts by an invader into a trimorphic population under Atkinson–Allen dynamics can lead to chaos.

中文翻译：

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Atkinson-Allen模型中四个竞争物种的混沌吸引子。

我们研究了四个竞争物种的Atkinson-Allen模型中混沌的发生，该模型起着离散时间Lotka-Volterra型模型的作用。我们表明，在该模型中，可以通过从唯一正定点的超临界Neimark-Sacker分叉开始的拟拟加倍分叉的级联生成混沌。混沌吸引子包含在一个整体吸引不变的二维维数中，称为携带单纯形。从生物学上讲，我们的研究表明，入侵者在阿特金森-艾伦动力学条件下入侵三态种群的企图会导致混乱。