Stability and bifurcation analysis of a delayed stage-structured predator–prey model with fear, additional food, and cooperative behavior in both species

The stability of predator–prey interactions in ecosystems is influenced by both inherent species interactions and external factors. For instance, the presence of additional food, as an external factor, may affect the system. To further explore this, a stage-structured predator–prey model is constructed, incorporating the influences of fear and delay on prey-population growth, which provides additional food for immature predators and facilitates cooperative behavior between mature and immature predators. The analysis evaluates the positivity, boundedness, equilibrium points, local stability around each equilibrium point, and certain bifurcations of the system. Additionally, numerical simulations are provided to correspond with the results of the theoretical analysis. It is observed that an appropriate level of fear contributes positively to system stability. While cooperation among predators can benefit immature predators, it also has the potential to harm the overall system. The introduction of additional food complicates the system dynamics, although it benefits predators, it places prey at a disadvantage. Furthermore, we observe a correlation between the level of fear and the effects of additional food, as well as the capacity of additional food to mitigate the influence of delay.