Dynamical behaviour of a stage-structured predator-prey model by incorporating cost and benefits of group defense

In predator-prey theory, the predator can affect on prey population by the killing of the prey and by causing predation fear on the prey population. The prey population also adjusts some behavioral approaches to reduce their predation risk which may influence their long term survival. In the present study, we formulate a predator-prey model dividing the prey population into two stages: juvenile and adult. We assume that when adult preys are sensitive to predation, they adapt group defense as an anti-predator strategy to lower their predation risk. To include group defense in the adult prey population, we consider Holling type IV functional responses for adult prey and predator interaction. But group defense has a negative effect by decreasing their reproduction potential. A parameter predator-taxis sensitivity introduces to interlink benefits of group defense and its costs. Increasing predator-taxis sensitivity also increases the group defense level of adult preys and benefits them by lowering predation risk. But also causes a detrimental effect by decreasing their reproduction rate simultaneously. We study some mathematical properties such as positivity, boundedness, local stability of equilibrium points, and bifurcation behaviors of the model. Our result suggests that the maturation rate can destabilize the system by producing oscillatory coexistence. For higher maturation rate the predator population suddenly extinct from the system, where oscillatory coexistence may disappear and the system becomes stable around the predator-free state. We also observe that predator-taxis sensitivity enhances the destabilizing nature of the system. However, for increasing the level of fear, the destabilization vanishes and the system shows stable behavior. It is also observed that predator- taxis sensitivity can be beneficial for adult prey as their density may increase with increasing the values of predator-taxis sensitivity. We also notice that above a threshold value of predator- taxis sensitivity the system shows bistable behavior. Our fear-induced stage-structured model exhibits interesting and rich dynamical behaviors.