This paper develops a method for studying bifurcations that occur in a neighborhood of the extinction equilibrium in nonlinear semelparous Leslie matrix models. The method uses a Lotka-Volterra equation with cyclic symmetry to detect the existence and to evaluate the stability of bifurcating equilibria and cycles. An application of the method provides sharp stability conditions for both a single-class cycle and a positive equilibrium bifurcating from the extinction equilibrium. The stability condition for a bifurcating single-class cycle confirms that the periodicity observed in periodical insects occurs if competition is more severe between than within age-classes. The developed method is also used to investigate two examples of nonlinear semelparous Leslie matrix models incorporating predator satiation. The investigation shows that a single-class cycle, which is associated with the periodicity in periodical insects, is a unique stable cycle in a neighborhood of the extinction equilibrium if the density effects in survival probabilities are identical among age-classes.

中文翻译：

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非线性对称Leslie矩阵模型中的循环分支。

本文开发了一种方法，用于研究非线性半对称莱斯利矩阵模型中灭绝平衡附近发生的分叉。该方法使用具有循环对称性的Lotka-Volterra方程来检测存在性并评估分叉平衡和周期的稳定性。该方法的应用为单类循环和从灭绝平衡分叉的正平衡提供了尖锐的稳定性条件。分叉单类周期的稳定性条件证实，如果相互竞争比同年龄组内的竞争更为激烈，则周期性昆虫中观察到的周期性就会发生。所开发的方法还用于研究包含捕食者饱和的非线性同质莱斯利矩阵模型的两个示例。