Caputo and Conformable Fractional Order Guava Model for Biological Pest Control: Discretization, Stability and Bifurcation

Two predator-prey model describing the guava borers and natural enemies are studied in this paper. Positivity, existence, and uniqueness of the solution, global and local stability analysis of the fixed points of the first model based on the Caputo fractional operator are studied. By adding piecewise constant functions to the second model including conformable fractional operator allows us to transition discrete dynamical system via discretization process. Applying Schur-Cohn criterion to the discrete system, we hold some regions where the equilibrium points in the discretized model are local asymptotically stable. We prove that discretized model displays supercritical Neimark–Sacker bifurcation at the equilibrium point. Theoretical and numerical results show that the discretized system demonstrates richer dynamic properties such as quasi-periodic solutions, bifurcation, and chaotic dynamics than the fractional order model with Caputo operator. All theoretical results are interpreted biologically and the optimum time interval for the harvesting of the guava fruit is given.