Bohner Martin, Mesquita Jaqueline, Streipert Sabrina
Missouri S & T, Department of Mathematics and Statistics, Rolla, MO 65409-0020, USA.
Universidade de Brasília, Departmento de Matemática, 70910-900 Brasília, DF, Brazil.
Math Biosci Eng. 2022 Aug 15;19(11):11693-11716. doi: 10.3934/mbe.2022544.
In this work, we formulate the Beverton-Holt model on isolated time scales and extend existing results known in the discrete and quantum calculus cases. Applying a recently introduced definition of periodicity for arbitrary isolated time scales, we discuss the effects of periodicity onto a population modeled by a dynamic version of the Beverton-Holt equation. The first main theorem provides conditions for the existence of a unique ω -periodic solution that is globally asymptotically stable, which addresses the first Cushing-Henson conjecture on isolated time scales. The second main theorem concerns the generalization of the second Cushing-Henson conjecture. It investigates the effects of periodicity by deriving an upper bound for the average of the unique periodic solution. The obtained upper bound reveals a dependence on the underlying time structure, not apparent in the classical case. This work also extends existing results for the Beverton-Holt model in the discrete and quantum cases, and it complements existing conclusions on periodic time scales. This work can furthermore guide other applications of the recently introduced periodicity on isolated time scales.
在这项工作中，我们在孤立时间尺度上构建了贝弗顿 - 霍尔特模型，并扩展了离散和量子微积分情形下已知的现有结果。应用最近引入的任意孤立时间尺度的周期性定义，我们讨论了周期性对由贝弗顿 - 霍尔特方程的动态版本建模的种群的影响。第一个主要定理给出了存在唯一的全局渐近稳定的ω周期解的条件，这解决了孤立时间尺度上的第一个库欣 - 亨森猜想。第二个主要定理涉及第二个库欣 - 亨森猜想的推广。它通过推导唯一周期解的平均值的上界来研究周期性的影响。所得到的上界揭示了对基础时间结构的依赖性，这在经典情形中并不明显。这项工作还扩展了离散和量子情形下贝弗顿 - 霍尔特模型的现有结果，并且补充了关于周期时间尺度的现有结论。此外，这项工作可以指导最近引入的孤立时间尺度上的周期性的其他应用。
