Investigating the Dynamic Behavior of Integer and Noninteger Order System of Predation with Holling’s Response

Kolade M. Owolabi (), Sonal Jain and Edson Pindza
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Kolade M. Owolabi: Department of Mathematical Sciences, Federal University of Technology, Akure PMB 704, Ondo State, Nigeria
Sonal Jain: School of Technology, Woxsen University, Hyderabad 502345, Telangana, India
Edson Pindza: Department of Decision Sciences, College of Economic and Management Sciences, University of South Africa (UNISA), Pretoria 0002, South Africa

Mathematics, 2024, vol. 12, issue 10, 1-25

Abstract: The paper’s primary objective is to examine the dynamic behavior of an integer and noninteger predator–prey system with a Holling type IV functional response in the Caputo sense. Our focus is on understanding how harvesting influences the stability, equilibria, bifurcations, and limit cycles within this system. We employ qualitative and quantitative analysis methods rooted in bifurcation theory, dynamical theory, and numerical simulation. We also delve into studying the boundedness of solutions and investigating the stability and existence of equilibrium points within the system. Leveraging Sotomayor’s theorem, we establish the presence of both the saddle-node and transcritical bifurcations. The analysis of the Hopf bifurcation is carried out using the normal form theorem. The model under consideration is extended to the fractional reaction–diffusion model which captures non-local and long-range effects more accurately than integer-order derivatives. This makes fractional reaction–diffusion systems suitable for modeling phenomena with anomalous diffusion or memory effects, improving the fidelity of simulations in turn. An adaptable numerical technique for solving this class of differential equations is also suggested. Through simulation results, we observe that one of the Lyapunov exponents has a negative value, indicating the potential for the emergence of a stable-limit cycle via bifurcation as well as chaotic and complex spatiotemporal distributions. We supplement our analytical investigations with numerical simulations to provide a comprehensive understanding of the system’s behavior. It was discovered that both the prey and predator populations will continue to coexist and be permanent, regardless of the choice of fractional parameter.

Keywords: fractional reaction–diffusion; Holling type IV response; Hopf bifurcation; local stability analysis; numerical methods and experiments (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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