Abstract

We consider a mathematical model describing the production of the components of some living system under the influence of positive and negative feedback. The model is presented in the form of the Cauchy problem for a nonlinear delay integro-differential equation. A theorem of the existence, uniqueness, and nonnegativity of the solutions to the model on the half-axis is proved for nonnegative initial data. The questions of the asymptotic behavior of the solutions and the stability of the equilibria of the model are investigated. Sufficient conditions for the asymptotic stability are obtained for nontrivial equilibria and the boundaries of their attraction domains are estimated. Examples illustrating the application of the obtained theoretical results are given.

中文翻译：

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生命系统模型中出现的延迟积分微分方程解的渐近行为

摘要

我们考虑一个数学模型，描述在正反馈和负反馈的影响下某些生命系统组件的生产。该模型以非线性延迟积分微分方程的柯西问题的形式呈现。对于非负的初始数据，证明了半轴上模型解的存在性、唯一性和非负性定理。研究了解的渐近行为和模型平衡的稳定性问题。对于非平凡平衡，获得了渐近稳定性的充分条件，并估计了它们的吸引力域的边界。给出了说明所获得的理论结果的应用的例子。