A class of autonomous discrete dynamical systems as population models for competing species are considered when each nullcline surface is a hyperplane. Criteria are established for global attraction of an interior or a boundary fixed point by a geometric method utilising the relative position of these nullcline planes only, independent of the growth rate function. These criteria are universal for a broad class of systems, so they can be applied directly to some known models appearing in the literature including Ricker competition models, Leslie-Gower models, Atkinson-Allen models, and generalised Atkinson-Allen models. Then global asymptotic stability is obtained by finding the eigenvalues of the Jacobian matrix at the fixed point. An intriguing question is proposed: Can a globally attracting fixed point induce a homoclinic cycle?

中文翻译：

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离散种群模型全局稳定性的几何方法

当每个零位线表面都是超平面时，将考虑一类作为离散物种的种群模型的自主离散动力系统。通过几何方法，仅利用这些零折线平面的相对位置，而不依赖于增长率函数，来建立内部或边界固定点的全局吸引的标准。这些标准对于广泛的系统类别是通用的，因此可以直接应用于文献中出现的一些已知模型，包括Ricker竞争模型，Leslie-Gower模型，Atkinson-Allen模型和广义的Atkinson-Allen模型。然后，通过在固定点处找到雅可比矩阵的特征值，获得全局渐近稳定性。提出了一个有趣的问题：一个全局吸引的固定点是否会引起同宿循环？