Abstract

Bistability, the presence of alternative stable states, is an important feature of population models as it indicates that long-term predictions are dependent on the current population density. Two distinct kinds of bistability re-occur in population modelling studies, Allee Bistability and Positive Bistability. In this article, we show that a novel codimension-3 bifurcation, the cusp-transcritical interaction, can act as an organising centre for ordinary differential equations that exhibit both Allee Bistability and Positive Bistability. We first show how a normal form for cusp-transcritical interactions emerges from the unfolding of a particular one-dimensional degeneracy. We then illustrate the ecological relevance of the cusp-transcritical interaction. Finally, we provide a comprehensive example of normal-form analysis of an existing population model that demonstrates the occurrence of the codimension-3 bifurcation. We note that Allee Bistability and Positive Bistability may manifest unexpectedly in complex, ecological models, and therefore, this bifurcation-focused approach can provide valuable insight into the behaviour of newly developed ecosystem models.

中文翻译：

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尖峰-跨临界相互作用的正态分析：在人口动态中的应用

摘要

双稳态（存在其他稳定状态）是人口模型的重要特征，因为它表明长期预测取决于当前的人口密度。人口建模研究中再次出现两种不同的双稳态：Allee双稳态和正双稳态。在本文中，我们显示了一种新颖的codimension-3分叉，即尖顶跨临界相互作用，可以充当展示Allee双稳态和正双稳态的常微分方程的组织中心。我们首先显示尖峰-跨临界相互作用的正常形式是如何从特定一维简并的展开中出现的。然后，我们说明了尖顶跨临界相互作用的生态相关性。最后，我们提供了一个现有人口模型的范式分析的综合示例，该示例演示了codimension-3分叉的发生。我们注意到，Allee双稳态和正双稳态可能会在复杂的生态模型中意外出现，因此，这种以分叉为重点的方法可以为新近开发的生态系统模型的行为提供有价值的见解。