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This paper presents the temporal and spatiotemporal dynamics of a delayed prey–predator system with a variable carrying capacity. Prey and predator interact via a Holling type-II functional response. A detailed dynamical analysis, including well-posedness and the possibility of coexistence equilibria, has been performed for the temporal system. Local and global stability behavior of the co-existence equilibrium is discussed. Bistability behavior between two coexistence equilibria is demonstrated. The system undergoes a Hopf bifurcation with respect to the parameter β , which affects the carrying capacity of the prey species. The delayed system exhibits chaotic behavior. A maximal Lyapunov exponent and sensitivity analysis are done to confirm the chaotic dynamics. In the spatiotemporal system, the conditions for Turing instability are derived. Furthermore, we analyzed the Turing pattern formation for different diffusivity coefficients for a two-dimensional spatial domain. Moreover, we investigated the spatiotemporal dynamics incorporating two discrete delays. The effect of the delay parameters in the transition of the Turing patterns is depicted. Various Turing patterns, such as hot-spot, coldspot, patchy, and labyrinth, are obtained in the case of a two-dimensional spatial domain. This study shows that the parameter β and the delay parameters significantly instigate the intriguing system dynamics and provide new insights into population dynamics. Furthermore, extensive numerical simulations are carried out to validate the analytical findings. The findings in this article may help evaluate the biological revelations obtained from research on interactions between the species. [ABSTRACT FROM AUTHOR]

In our ecological system, prey species can defend themselves by casting strong and effective defenses against predators, which can slow down the growth rate of prey. Predator has more at stake when pursuing a deadly prey than just the chance of missing the meal. Prey have to "trade off" between reproduction rate and safety and whereas, predator have to "trade off" between food and safety. In this article, we investigate the trade-off dynamics of both predator and prey when the predator attacks a dangerous prey. We propose a two-dimensional prey and predator model considering the logistic growth rate of prey and Holling type-2 functional response to reflect predator's successful attacks. We examine the cost of fear to reflect the trade-off dynamics of prey, and we modify the predator's mortality rate by introducing a new function that reflects the potential loss of predator as a result of an encounter with dangerous prey. We demonstrated that our model displays bi-stability and undergoes transcritical bifurcation, saddle node bifurcation, Hopf bifurcation, and Bogdanov–Taken bifurcations. To explore the intriguing trade-off dynamics of both prey and predator population, we investigate the effects of our critical parameters on both population and observed that either each population vanishes simultaneously or the predator vanishes depending on the value of the handling time of the predator. We determined the handling time threshold upon which dynamics shift, demonstrating the illustration of how predators risk their own health from hazardous prey for food. We have conducted a sensitivity analysis with regard to each parameter. We further enhanced our model by including fear response delay and gestation delay. Our delay differential equation system is chaotic in terms of fear response delay, which is evidenced by the positivity of maximum Lyapunov exponent. We have used numerical analysis to verify our theoretical conclusions, which include the influence of vital parameters on our model through bifurcation analysis. In addition, we used numerical simulations to showcase the bistability between co-existence equilibrium and prey only equilibrium with their basins of attraction. The results that are reported in this article might be useful in interpreting the biological insights gained from studying the interactions between prey and predator. [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]

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]