A delay non-autonomous model for the combined effects of fear, prey refuge and additional food for predator

In this paper, we investigate the combined effects of fear, prey refuge and additional food for predator in a predator-prey system with Beddington type functional response. We observe oscillatory behaviour of the system in the absence of fear, refuge and additional food whereas the system shows stable dynamics if anyone of these three factors is introduced. After analysing the behaviour of system with fear, refuge and additional food, we find that the system destabilizes due to fear factor whereas refuge and additional food stabilize the system by killing persistent oscillations. We extend our model by considering the fact that after sensing the chemical/vocal cue, prey takes some time for assessing the predation risk. The delayed system shows chaotic dynamics through multiple stability switches for increasing values of time delay. Moreover, we see the impact of seasonal change in the level of fear on the delayed as well as non-delayed system.