Applications of the generalized gamma function to a fractional-order biological system

In this work, variety of complex dynamics are found in a fractional-order antimicrobial resistance (AMR) model based on the generalized Gamma function. Firstly, the extended left and right Caputo fractional differential operators, respectively, ELCFDO and ERCFDO are introduced. The basic features of the ELCFDO are outlined. The ELCFDO is shown to have a new fractional parameter that affects the occurrence of the complex dynamics in the fractional AMR system. Secondly, discretization of the ELCFDO is studied using piecewise constant arguments. Then, complex dynamics of the discretized version of the fractional AMR system involving the ELCFDO are investigated such as the existence of Neimark–Sacker (NS) and flip bifurcations, the existence of closed invariant curves (CIC), the existence of strange attractors with fractal or multi-fractal structures, and chaotic attractors. Finally, an extension of the fractal-fractional operator (FFO) that combines fractal and fractional differentiation is carried out based on the generalized Gamma function. The extended FFO (EFFO) is applied to the proposed AMR system, which also generates similar complex dynamics.