The delayed logistic equation (also known as Hutchinson’s equation or Wright’s equation) was originally introduced to explain oscillatory phenomena in ecological dynamics. While it motivated the development of a large number of mathematical tools in the study of nonlinear delay differential equations, it also received criticism from modellers because of the lack of a mechanistic biological derivation and interpretation. Here, we propose a new delayed logistic equation, which has clear biological underpinning coming from cell population modelling. This nonlinear differential equation includes terms with discrete and distributed delays. The global dynamics is completely described, and it is proven that all feasible non-trivial solutions converge to the positive equilibrium. The main tools of the proof rely on persistence theory, comparison principles and an \(L^2\)-perturbation technique. Using local invariant manifolds, a unique heteroclinic orbit is constructed that connects the unstable zero and the stable positive equilibrium, and we show that these three complete orbits constitute the global attractor of the system. Despite global attractivity, the dynamics is not trivial as we can observe long-lasting transient oscillatory patterns of various shapes. We also discuss the biological implications of these findings and their relations to other logistic-type models of growth with delays.

中文翻译：

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细胞生物学引起的新型时滞逻辑方程的全局动力学

最初引入时滞逻辑方程（也称为Hutchinson方程或Wright方程）来解释生态动力学中的振荡现象。尽管它激发了非线性延迟微分方程研究的大量数学工具的发展，但由于缺乏机械的生物学推导和解释，它也受到建模者的批评。在这里，我们提出了一个新的时滞逻辑方程，该方程具有明显的生物学基础，来自细胞群体建模。该非线性微分方程包括具有离散和分布式延迟的项。完整描述了全局动力学，并且证明了所有可行的非平凡解都收敛于正平衡。证明的主要工具依赖于持久性理论，\（L ^ 2 \）-摄动技术。使用局部不变流形，构造了一个连接不稳定的零和稳定的正平衡的独特的异质轨道，我们证明了这三个完整的轨道构成了系统的整体吸引子。尽管具有全局吸引力，但动力学并不是一件容易的事，因为我们可以观察到各种形状的持久瞬态振荡模式。我们还讨论了这些发现的生物学含义，以及它们与其他具有延迟的逻辑类型的增长模型之间的关系。