Piecewise Fractional Analysis of the Migration Effect in Plant-Pathogen Interactions

Abstract

This study introduces several updated results for the piecewise plant-pathogen-herbivore interactions model with singular-type and nonsingular fractional-order derivatives. A piecewise fractional model has developed to describe the interactions between plants, disease, (insect) herbivores, and their natural enemies. We derive essential findings for the aforementioned problem, specifically regarding the existence and uniqueness of the solution, as well as various forms of Ulam Hyers (U-H) type stability. The necessary results were obtained by utilizing fixed-point theorems established by Schauder and Banach. Additionally, the U-H stabilities were determined based on fundamental principles of nonlinear analysis. To implement the model as an approximate piecewise solution, the Newton Polynomial approximate solution technique is employed. The applicability of the model was validated through numerical simulations both in fractional as well as piecewise fractional format. The motivation of our article is that we have converted the integer order problem to a global piecewise and fractional order model in the sense of Caputo and Atangana-Baleanu operators and investigate it for existence, uniqueness of solution, Stability of solution and approximate solution along with numerical simulation for the validity of our obtained scheme.