Ulam type stability for von Bertalanffy growth model with Allee effect.

Abstract

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.