This paper aims to study and analyze the effect of the flow rate and amplitude of walling waves on the peristaltic flow of Jeffrey’s fluid through an irregular channel. In addition, it analyses streamlined patterns and their local and global bifurcation flow. The theory of dynamical systems is used to explore the position of critical points and their bifurcation. Numerical and analytical methods are presented to display effective changes in amplitude and rate of the flow on the transport of fluid. Three types of streamline patterns occur backward, trapping and augmented flow in the channel through movement due to changes in the value rate parameters.