We study a competition model of two competing species in population biology having exponential and rational growth functions described by Alexander et al. [1992]. They observed that, for some choice of parameters, the competition model has a chaotic attractor [Formula: see text] for which the basin of attraction is riddled. Here, we give a detailed analysis to illustrate what happens when the normal parameter in this model changes. In fact, by varying the normal parameter, we discuss how the geometry of the basin of attraction of [Formula: see text], the region of coexistence or extinction, changes and illustrate the transitions between the set [Formula: see text] being an asymptotically stable attractor (extinction of rational species), a locally riddled basin attractor and a normally repelling chaotic saddle (extinction of exponential species). Additionally, we show that the riddling and the blowout bifurcation occur. Numerical simulations are presented graphically to confirm the validity of our results. In particular, we verify the occurrence of synchronization for some values of parameters. Finally, we apply the uncertainty exponent and the stability index to quantify the degree of riddling basin. Our observation indicates that the stability index is positive for Lebesgue for almost all points whenever the riddling occurs.

中文翻译：

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两物种竞争模型中的共存和灭绝分析

我们研究了种群生物学中两种竞争物种的竞争模型，该模型具有 Alexander 等人描述的指数和理性增长函数。[1992]。他们观察到，对于某些参数的选择，竞争模型有一个混沌吸引子 [公式：见正文]，其中的吸引盆是千疮百孔。在这里，我们给出一个详细的分析来说明当这个模型中的正常参数发生变化时会发生什么。事实上，通过改变法线参数，我们讨论了[公式：见文本]的吸引力盆地的几何形状，即共存或灭绝的区域，如何变化并说明集合[公式：见文本]之间的过渡。渐近稳定的吸引子（理性物种的灭绝），局部分布的盆地吸引子和通常排斥的混沌鞍（指数物种的灭绝）。此外，我们还表明发生了谜语和井喷分叉。数值模拟以图形方式呈现，以确认我们结果的有效性。特别是，我们验证了某些参数值的同步发生。最后，我们应用不确定性指数和稳定性指数来量化迷彩盆地的程度。我们的观察表明，每当出现谜语时，几乎所有点的 Lebesgue 稳定性指数都是正的。