Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler 2667-419X Eskisehir Technical University 10.20290/estubtdb.1704403 Biological Mathematics Biyolojik Matematik CAPUTO DERIVATIVE FRACTIONAL ORDER PREY-PREDATOR MODEL AND MATHEMATICAL ANALYSIS CAPUTO DERIVATIVE FRACTIONAL ORDER PREY-PREDATOR MODEL AND MATHEMATICAL ANALYSIS https://orcid.org/0000-0001-5662-4670 Öztürk Zafer NEVŞEHİR HACI BEKTAŞ VELİ ÜNİVERSİTESİ 02 25 2026 14 1 20 30 05 22 2025 12 24 2025 Copyright © 2010, Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler 2010 Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler

A foundational process in the field of ecology is the interaction between prey and predator. This interaction refers to the changes in the population density of two species in relation to each other, due to the fact that prey and predator share the same environment and one species preys on the other. The application of differential equations, otherwise referred to as Lotka-Volterra equations, in modeling prey-predator systems is of significant importance in studying the dynamics of interacting biological populations. Prey-predator models are population models that show the relationship between two species sharing the same environment. The present study involves the analysis of stability and the numerical solutions of a discrete-time fractional-order predator-prey mathematical model. This model is divided into two compartments: the prey population (N) and the predator population (P). The Caputo sense is employed to utilize the notion of a fractional derivative. The numerical results obtained for the prey-predator fractional mathematical model were achieved through the implementation of the Euler method, thus yielding graphs for visualization. The results show that the prey population increases steadily over time, while the predator population also increases gradually and steadily.

A foundational process in the field of ecology is the interaction between prey and predator. This interaction refers to the changes in the population density of two species in relation to each other, due to the fact that prey and predator share the same environment and one species preys on the other. The application of differential equations, otherwise referred to as Lotka-Volterra equations, in modeling prey-predator systems is of significant importance in studying the dynamics of interacting biological populations. Prey-predator models are population models that show the relationship between two species sharing the same environment. The present study involves the analysis of stability and the numerical solutions of a discrete-time fractional-order predator-prey mathematical model. This model is divided into two compartments: the prey population (N) and the predator population (P). The Caputo sense is employed to utilize the notion of a fractional derivative. The numerical results obtained for the prey-predator fractional mathematical model were achieved through the implementation of the Euler method, thus yielding graphs for visualization. The results show that the prey population increases steadily over time, while the predator population also increases gradually and steadily.

 Fractional-Order Prey-Predator Population Model Mathematical Modeling Euler Method Caputo Derivative Stability Analysis Fractional-Order Prey-Predator Population Model Mathematical Modeling Euler Method Caputo Derivative Stability Analysis [1] Boshan C, Jiejie C. Bifurcation and chaotic behaviour of a discrete singular biological economic system. Appl. Math and Comp. 2012; 219: 2371-2386. [2] Fournier AK, Gelle, ES. Behavior analysis of companion-animal overpopulation: A conceptualization of the problem and suggestions for intervention. Behavior and Social Issues, 2004; 13(1): 51- 69. [3] Anisiu MC. Lotka, Volterra and their model. Didactica Mathematica, 2014; 32(1): 9-17. [4] Venkataiah K & Ramesh, K. On the stability of a Caputo fractional order predator-prey framework including Holling type-II functional response along with nonlinear harvesting in predator. Partial Differential Equations in Applied Mathematics, 2024; 11: 100777. [5] Podlubny I. Fractional Differential Equations. Academy Press, San Diego CA, 1999. [6] Çelik C. The stability and Hopf Bifurcations for a predator-prey system with time delay. Chaos Solutions & Fractals, 2008; 37: 87-99. [7] Allen LJ. An Introduction to Mathematical Biology, 2007. [8] Hu D, Cao H. Bifurcation and chaos in a discrete-time predator-prey system of Holling and Leslie type, Commun Nonlinear Sci Simulat. 2015; 22: 702-715. [9] Öztürk Z. Fractional order Lorenz Chaos model and numerical application. Journal of Universal Mathematics, 2025; 8(1), 40-51. [10] Brauer F, Van den Driessche P & Allen LJ. Mathematical Epidemiology, 2008; 1945: 3-17. [11] Merdan H, Duman O. On the stability analysis of a general discrete-time population model involving predation and Allee effects. Chaos, Solutions & Fractals, 2009; 40(3): 1169-1175. [12] Kangalgil F. Neimark–Sacker bifurcation and stability analysis of a discrete-time prey–predator model with Allee effect in prey. Advances in Difference Equations, 2019; 2019(1): 92. [13] Zhou SR, Liu YF, Wang G. The stability of predator-prey system subject to the Alle Effects. Theoretical Population Biology 2005; 67(1): 23-31. [14] Öztürk Z, Bilgil H & Sorgun S. A new application of fractional glucose-insulin model and numerical solutions. Sigma Journal of Engineering and Natural Sciences, 2024; 42(2): 407-413. [15] Yaro D, Omari-Sasu SK, Harvim P, Saviour AW & Obeng, BA. Generalized Euler method for modeling measles with fractional differential equations. Int. J. Innovative Research and Development, 2015; 4(4): 358-366. [16] Singh H, Dhar J & Bhatti HS. Discrete time bifurcation behavior of a prey-predator system with generalized predator. Advances in Difference Equations, 2015; 2015(1): 206. [17] Öztürk Z, Bilgil H & Sorgun S. Application of Fractional SIQRV Model for SARS- CoV-2 and Stability Analysis. Symmetry, 2023; 15(5): 1048. [18] Öztürk Z. A New Fractional Order Hand-Foot-Mouth Disease Model and Numerical Solutions. New Mathematics and Natural Computation, 2025; 1-15. [19] Özturk Z, Bilgil H & Sorgun, S. Fractional SAQ alcohol model: stability analysis and Türkiye application. International Journal of Mathematics and Computer in Engineering, 2024; 3(2): 125-136. [20] Kermack WO & McKendrick AG. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London, Series A, Containing Papers of a Mathematical and Physical Character, 1927; 115(772): 700-721. [21] Ara R & SM SR. Complex Dynamics and Chaos Control of Discrete Prey–Predator Model With Caputo Fractional Derivative. Complexity, 2025; 2025(1): 4415022. [22] Ghanbari B. On approximate solutions for a fractional prey–predator model involving the Atangana–Baleanu derivative. Advances in Difference Equations, 2020; 2020(1): 679.