Summary: The ecological framework gains special interest in the dynamical complexity of a system of interacting species influenced by the adventure of prey refuge along with alternative food source for the predator. The work undertaken provides thorough investigation on several recent findings of long time response to stability-instability phenomena of the system, the dynamical deportment including sensitivity of the system parameters and the occurrence of bifurcations of codimension 1 and 2 as well. The bubbling phenomenon, denoting a change in the amplitudes of successive cycles, can be captured in the current two-dimensional (2D) continuous system by selecting appropriate parameter values. Interestingly, the controlling system parameter of bubbling phenomena appears to be the most sensitive one for the proposed model system. The global-in-time trajectories based on the introduction of the concept of outcomes are examined along with the inclusion of fundamental preliminaries. The notable features of the prediction and identification of bifurcations occurring in the dynamical system arising from the onset of instability to the transition to local and global bifurcations appear to be crucial for both the theoretical and field researchers. Based on the choice of a couple of system parameters of significance at a time, the whole parametric space is partitioned into multiple regions. In addition to these, efforts are made to account the number of ecologically feasible equilibria including their dynamical feedback in each region of concern. Various numerical simulations are ventured upon in order to establish the validity of analytical outcomes in general and the applicability of model system dynamics in terms of several role-playing parameters, in particular.