ABSTRACT A class of autonomous Kolmogorov systems that are dissipative and competitive with the origin as a repellor are considered when each nullcline surface is either concave or convex. Geometric method is developed by using the relative positions of the upper and lower planes of the nullcline surfaces for global asymptotic stability of an interior or a boundary equilibrium point. Criteria are also established for global repulsion of an interior or a boundary equilibrium point on the carrying simplex. This method and the theorems can be viewed as a natural extension of those results for Lotka-Volterra systems in the literature.

中文翻译：

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Kolmogorov 系统全局稳定性和斥力的几何方法

摘要 当每个零斜面是凹面或凸面时，考虑了一类耗散且与原点竞争的自主 Kolmogorov 系统作为排斥面。几何方法是利用零斜面上下平面的相对位置发展起来的，用于内部或边界平衡点的全局渐近稳定性。还为承载单纯形上的内部或边界平衡点的全局排斥建立了标准。这种方法和定理可以看作是文献中 Lotka-Volterra 系统结果的自然延伸。