Numerical Solutions of Fractional Order Ratio-Dependent Food Chain Model with Caputo Derivative

Year 2025, Volume: 13 Issue: 4, 1544 - 1555, 30.10.2025

Zafer Öztürk

https://doi.org/10.29130/dubited.1600292

Abstract

The study of life within an ecosystem reveals a complex system. Ecosystems are characterised by the presence of all the elements that give rise to chaotic dynamics. Although chaos is often predicted by mathematical models, there is currently only limited evidence of its existence in nature, with the proof of its occurrence remaining scarce and uncertain. Despite the apparent simplicity of food chains, they exhibit highly complex dynamics. Models created several years ago have confirmed that food chains have complex dynamics. In this study, a fractional order ratio-dependent food chain model is considered. This model consists of three compartments: prey population density (𝑋), predator population density (𝑌) and top predator density (𝑍). The fractional derivative is employed in accordance with the Caputo sense. A mathematical analysis of the fractional order ratio-dependent food chain model is conducted. Numerical results are obtained with the aid of the Euler method and the graphs are interpreted.

Keywords

Fractional Order Food Chain Model , Mathematical Modeling , Euler Method , Caputo Derivative

Ethical Statement

This study does not involve human or animal participants. All procedures followed scientific and ethical principles, and all referenced studies are appropriately cited.

Kesir Mertebeden Orana Bağlı Besin Zinciri Modelinin Caputo Türevi ile Sayısal Çözümleri

Abstract

The study of life within an ecosystem reveals a complex system. Ecosystems are characterised by the presence of all the elements that give rise to chaotic dynamics. Although chaos is often predicted by mathematical models, there is currently only limited evidence of its existence in nature, with the proof of its occurrence remaining scarce and uncertain. Despite the apparent simplicity of food chains, they exhibit highly complex dynamics. Models created several years ago have confirmed that food chains have complex dynamics. In this study, a fractional order rate-dependent food chain model is considered. This model consists of three compartments: prey population density (X), predator population density (Y) and top predator density (Z). The fractional derivative is employed in accordance with the Caputo sense. A mathematical analysis of the fractional order rate-dependent food chain model is conducted. Numerical results are obtained with the aid of the Euler method and the graphs are interpreted.

Keywords

Fractional Order Food Chain Model , Mathematical Modeling , Euler Method , Caputo Derivative

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