Kondo Masumi, Onitsuka Masakazu
Department of Applied Mathematics, Okayama University of Science, Okayama 700-0005, Japan.
Math Biosci Eng. 2024 Feb 29;21(3):4698-4723. doi: 10.3934/mbe.2024206.
In many studies dealing with mathematical models, the subject is examining the fitting between actual data and the solution of the mathematical model by applying statistical processing. However, if there is a solution that fluctuates greatly due to a small perturbation, it is expected that there will be a large difference between the actual phenomenon and the solution of the mathematical model, even in a short time span. In this study, we address this concern by considering Ulam stability, which is a concept that guarantees that a solution to an unperturbed equation exists near the solution to an equation with bounded perturbations. Although it is known that Ulam stability is guaranteed for the standard von Bertalanffy growth model, it remains unsolved for a model containing the Allee effect. This paper investigates the Ulam stability of a von Bertalanffy growth model with the Allee effect. In a sense, we obtain results that correspond to conditions of the Allee effect being very small or very large. In particular, a more preferable Ulam constant than the existing result for the standard von Bertalanffy growth model, is obtained as the Allee effect approaches zero. In other words, this paper even improves the proof of the result in the absence of the Allee effect. By guaranteeing the Ulam stability of the von Bertalanffy growth model with Allee effect, the stability of the model itself is guaranteed, and, even if a small perturbation is added, it becomes clear that even a small perturbation does not have a large effect on the solutions. Several examples and numerical simulations are presented to illustrate the obtained results.
在许多涉及数学模型的研究中，研究对象是通过应用统计处理来检验实际数据与数学模型解之间的拟合度。然而，如果存在一个解，其会因微小扰动而大幅波动，那么即使在短时间内，实际现象与数学模型的解之间也可能会有很大差异。在本研究中，我们通过考虑乌拉姆稳定性来解决这一问题，乌拉姆稳定性是一个概念，它保证在有界扰动方程的解附近存在未扰动方程的解。虽然已知标准的冯·贝塔朗菲生长模型具有乌拉姆稳定性，但对于包含阿利效应的模型，该问题仍未解决。本文研究了具有阿利效应的冯·贝塔朗菲生长模型的乌拉姆稳定性。从某种意义上说，我们得到了与阿利效应非常小或非常大的条件相对应的结果。特别地，当阿利效应趋近于零时，我们得到了一个比标准冯·贝塔朗菲生长模型现有结果更优的乌拉姆常数。换句话说，本文甚至改进了在不存在阿利效应情况下结果的证明。通过保证具有阿利效应的冯·贝塔朗菲生长模型的乌拉姆稳定性，模型本身的稳定性得到了保证，并且即使添加了微小扰动，也很明显微小扰动对解不会有很大影响。本文给出了几个例子和数值模拟来说明所得到的结果。
