Rich dynamics of a delay-induced stage-structure prey–predator model with cooperative behaviour in both species and the impact of prey refuge

The prey–predator interaction between organisms living in an ecosystem is greatly affected by the cooperation among the species as well as immature prey refuge in presence of time delay. This study has developed and analysed a stage-structured prey–predator model that includes refuge behaviour for prey and two types of cooperative behaviours, namely cooperative behaviour among prey as well as cooperative behaviour among predators, in addition to gestational delay. The presence of Saddle–node, Transcritical, and Hopf-bifurcations is explained by analytically and numerically. We examine the model parameter’s sensitivity to determine the species’ favourable and unfavourable impacts and to identify the most significant parameters. The dynamics of delay and non-delay systems are compared, and we recognise that a consistent increase in time delay causes chaotic oscillations with period-doubling fluctuations which can be regulated by the refuge for immature prey. However, we demonstrate that the immature prey refuge destabilises the system in absence of time delay while it acts as a control parameter in presence of delay. Finally, we wrap up the paper by highlighting the key biological implications of the numerical and analytical findings.