Trade-off and chaotic dynamics of prey-predator system with two discrete delays.

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.