In this paper, we study a drive-response discrete-time dynamical system which has been coupled using convex functions and we introduce a synchronization threshold which is crucial for the synchronizing procedure. We provide one application of this type of coupling in synchronized cycles of a generalized Nicholson-Bailey model. This model demonstrates a rich cascade of complex dynamics from stable fixed point to periodic orbits, quasi periodic orbits and chaos. We explain how this way of coupling makes these two chaotic systems starting from very different initial conditions, quickly get synchronized. We investigate the qualitative behavior of GNB model and its synchronized model using time series analysis and its long time dynamics by the help of bifurcation diagram.
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