Mathematical analysis of the new α - difference operator with an application to prey-predator model with harvesting. Quadratic invariant

Abstract

In this research paper, we introduce a novel mathematical operator known as the alpha-difference operator (α-DO) and its corresponding integral. We establish the foundational theorems related to this operator and demonstrate its applications in both linear and nonlinear dynamical equations. A key focus of our study is the application of α-DO in the context of the prey-predator model with harvesting. In the linear scenario, we derive exact solutions for the model. For the nonlinear case, we develop an iterative scheme to obtain approximate solutions. We also prove a theorem that guarantees the convergence of this scheme. We conduct a thorough investigation of the dynamical behavior of the system as the parameter varies. This is visualized through graphical representations. Our findings reveal that the system exhibits local memory, which significantly influences the evolution of the system. We observe that the α-DO is particularly effective in describing dynamical systems that undergo a change in behavior at a specific characteristic time. This is especially relevant to the system under consideration. A prime example of such a system is the Exposed-Infected-Recovery System (EIRS). Lastly, we construct the Hamiltonian function using a quadratic invariant. This provides further insights into the energy conservation and stability properties of the system. Our research opens up new insight for the application of the α-DO in various fields of science and engineering.