Degenerate homoclinic and Hopf bifurcation in a stage-structured model with parental care in fearful prey and group defense

Parental care varies widely among animal species and plays a vital role in their reproduction, development, and overall life history. Fear is as important as direct killing as presence of predators influences prey population. As an anti-predator response, prey individuals engage in group defense to reduce the risk of being victimized. This study aims to explore a 3D stage-structure model with the joint impact of parental care, fear and group defense in prey. The interaction between mature fearful prey and predator population is modified with Holling type IV functional response. We analysed the existence of steady state solutions of the system and their stability. Our model undergoes several codimension one and two bifurcations including transcritical, saddle–node, supercritical and subcritical Hopf, supercritical and subcritical homoclinic, saddle–node homoclinic, Bautin, Bogdanov–Takens, saddle–node-transcritical, cusp, and generalized homoclinic bifurcation. The analysis of the parameters such as parental care, prey’s searching efficiency, transition rate, cost of fear, half saturation constant, and hunting rate on mature prey shows the rich and complex dynamics of the model. We found that the lower parental care supports coexistence, whereas higher parental care leads to the catastrophic extinction of the predators. Additionally, fear and half saturation rate play a vital role in the elimination of oscillations. Bistability and tristability behaviours were observed in the system, highlighting the dependence of initial populations for the persistence of the species.