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The present work highlights the reverse side of the same ecological coin by considering the counter-attack of prey on immature predators. We assume that the birth rate of prey is affected by the fear of adult predators and its carry-over effects (COEs). Next, we introduce two discrete delays to show time lag due to COEs and fear-response. We observe that the existence of a positive equilibrium point and the stability of the prey-only state is independent of fear and COEs. Furthermore, the necessary condition for the co-existence of all three species is determined. Our system experiences several local and global bifurcations, like, Hopf, saddle-node, transcritical, and homoclinic bifurcation. The simultaneous variation in the attack rate of prey and predator results in the Bogdanov–Takens bifurcation. Our numerical results explain the paradox of enrichment, chaos, and bi-stability of node-focus and node-cycle types. The system, with and without delay, is analyzed theoretically and numerically. Using the normal form method and center manifold theorem, the conditions for stability and direction of Hopf-bifurcation are also derived. The cascade of predator attacks, prey counter-attacks, and predator defense exhibit intricate dynamics, which sheds light on ecological harmony. [ABSTRACT FROM AUTHOR]

Depending on behavioral differences, reproductive capability and dependency, the life span of a species is divided mainly into two classes, namely immature and mature. In this paper, we have studied the dynamics of a predator–prey system considering stage structure in prey and the effect of predator-induced fear with two discrete time delays: maturation delay and fear response delay. We consider that predators cooperate during hunting of mature prey and also include its impact in fear term. The conditions for existence of different equilibria, their stability analysis are carried out for non-delayed system and bifurcation results are presented extensively. It is observed that the fear parameter has stabilizing effect whereas the cooperative hunting factor having destabilizing effect on the system via occurrence of supercritical Hopf-bifurcation. Further, we observe that the system exhibits backward bifurcation between interior equilibrium and predator free equilibrium and hence the situation of bi-stability occurs in the system. Thereafter, we differentiate the region of stability and instability in bi-parametric space. We have also studied the system's dynamics with respect to maturation and fear response delay and observed that they also play a vital role in the system stability and occurrence of Hopf-bifurcation is shown with respect to both time delays. The system shows stability switching phenomenon and even higher values of fear response delay leads the system to enter in chaotic regime. The role of fear factor in switching phenomenon is discussed. Comprehensive numerical simulation and graphical presentation are carried out using MATLAB and MATCONT. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a system of delay differential equations incorporating prey's refuge, fear, fear-response delay, extra food for predators and their gestation lag. First, we examined the system without delay. The persistence, stability (local and global) and various bifurcations are discussed. We provide detailed analysis for transcritical and Hopf-bifurcation. The existence of positive equilibria and the stability of prey-free equilibrium are interrelated. It is shown that (i) fear can stabilize or destabilize the system, (ii) prey refuge in a specific limit can be advantageous for both species, (iii) at a lower energy level (gained from extra food), the system undergoes a supercritical Hopf-bifurcation and (iv) when the predator gains high energy from extra food, it can survive through a homoclinic bifurcation, and prey may become extinct. The possible occurrence of bi-stability with or without delay is discussed. We observed switching of stability thrice via subcritical Hopf-bifurcation for fear-response delay. On changing some parametric values, the system undergoes a supercritical Hopf-bifurcation for both delay parameters. The delayed system undergoes the Hopf-bifurcation, so we can say that both delay parameters play a vital role in regulating the system's dynamics. The analytical results obtained are verified with the numerical simulation. [ABSTRACT FROM AUTHOR]