Global stability and Hopf-bifurcation of prey-predator system with two discrete delays including habitat complexity and prey refuge.

Highlights • A two-dimensional prey-predator model in presence of habitat complexity and prey refuge with two discrete delays has been proposed and analyzed. • The proposed model system is persistent and globally asymptotically stable if the intrinsic growth rate of prey population is greater than a threshold value, which depends upon the habitat complexity and refuge. • The negative feedback delay and gestation delay both destabilize the behavior of the system. • The numerical experiments conducted shows that the Hopf-bifurcation is supercritical, the bifurcated periodic solution is stable and its period increases. Abstract In this paper, we consider a two-dimensional prey-predator system with two delays. One delay is for negative feedback of the prey population and another is for gestation delay of the predator population. The predator is partially dependent on the prey followed by Holling type-II functional response. Due to habitat complexity and prey refuge, the Holling type-II functional response is modified in this work. We discuss the boundedness, permanence, local and global asymptotic behavior of the non-delayed and delayed models. The existence of periodic solutions via Hopf-bifurcation with respect to both the delays is established. The stability and direction of Hopf-bifurcation is also analyzed by using Normal form theory and Centre manifold theory. Lastly, numerical simulations have been carried out to confirm the analytical findings. The main objective of this work is to balance the prey-predator relationship in the presence of habitat complexity, prey refuge and delays. [ABSTRACT FROM AUTHOR]