A discrete logistic model with conditional Hyers–Ulam stability

Douglas R. Anderson; Masakazu Onitsuka; Douglas R. Anderson; Masakazu Onitsuka

doi:10.3934/math.2025298

AIMS Mathematics

2025, Volume 10, Issue 3: 6512-6545. doi: 10.3934/math.2025298

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A discrete logistic model with conditional Hyers–Ulam stability

Douglas R. Anderson ¹,
Masakazu Onitsuka ^{2
,
,}

1.
Department of Mathematics, Concordia College, Moorhead, MN 56562, USA
2.
Department of Applied Mathematics, Okayama University of Science, Okayama 700-0005, Japan

Received: 16 December 2024 Revised: 17 March 2025 Accepted: 19 March 2025 Published: 24 March 2025
MSC : 39A30, 39A60

This study investigates the conditional Hyers–Ulam stability of a first-order nonlinear $ h $-difference equation, specifically a discrete logistic model. Identifying bounds on both the relative size of the perturbation and the initial population size is an important issue for nonlinear Hyers–Ulam stability analysis. Utilizing a novel approach, we derive explicit expressions for the optimal lower bound of the initial value region and the upper bound of the perturbation amplitude, surpassing the precision of previous research. Furthermore, we obtain a sharper Hyers–Ulam stability constant, which quantifies the error between true and approximate solutions, thereby demonstrating enhanced stability. The Hyers–Ulam stability constant is proven to be in terms of the step-size $ h $ and the growth rate but independent of the carrying capacity. Detailed examples are provided illustrating the applicability and sharpness of our results on conditional stability. In addition, a sensitivity analysis of the parameters appearing in the model is also performed.
- $ h $-difference equation,
- Hyers–Ulam stability,
- Hyers–Ulam stability constant,
- perturbation,
- conditional stability,
- logistic growth
Citation: Douglas R. Anderson, Masakazu Onitsuka. A discrete logistic model with conditional Hyers–Ulam stability[J]. AIMS Mathematics, 2025, 10(3): 6512-6545. doi: 10.3934/math.2025298

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Abstract

This study investigates the conditional Hyers–Ulam stability of a first-order nonlinear $ h $-difference equation, specifically a discrete logistic model. Identifying bounds on both the relative size of the perturbation and the initial population size is an important issue for nonlinear Hyers–Ulam stability analysis. Utilizing a novel approach, we derive explicit expressions for the optimal lower bound of the initial value region and the upper bound of the perturbation amplitude, surpassing the precision of previous research. Furthermore, we obtain a sharper Hyers–Ulam stability constant, which quantifies the error between true and approximate solutions, thereby demonstrating enhanced stability. The Hyers–Ulam stability constant is proven to be in terms of the step-size $ h $ and the growth rate but independent of the carrying capacity. Detailed examples are provided illustrating the applicability and sharpness of our results on conditional stability. In addition, a sensitivity analysis of the parameters appearing in the model is also performed.

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