Global dynamics of an ecological model in presences of fear and group defense in prey and Allee effect in predator

In this article, we have studied dynamical behavior of a predator-prey system with fear and group defense in prey and Allee effect in predator. The existence and non-existence of equilibria are shown under some sufficient conditions via a thorough theoretical analysis, followed by an investigation of their stability. The system experiences several bifurcations of co-dimension one such as saddle-node, Hopf, homoclinic, coalescence of periodic orbits, and bifurcations of co-dimension two such as Bautin and Bogdanov-Takens. These bifurcations are used to illustrate the complex dynamical structure of the model. Along with the analysis of the one-parameter bifurcation, bi-parametric plane of Allee and fear parameters is separated into various regions through the selective change of these vital parameters together, enabling a deeper comprehension of the dynamics of the system within each region. The system exhibits multi-stability (bi-stability and tri-stability) and global stability in different regions of the bi-parametric plane which indicates that the survival of predator species depends on their initial population. It is noticed that degree of fear and Allee strength play a vital role in survival and extinction of the predator. The occurrence of all bifurcations has been ensured by verifying their transversality and genericity conditions and further validated by numerical examples and graphical illustrations.