  Various field and laboratory experiments show that prey refuge plays a
significant role in the stability of prey-predator dynamics. On the other hand,
theoretical studies show that delayed system exhibits a much more realistic
dynamics than its non-delayed counterpart. In this paper, we study a
multi-delayed prey-predator model with prey refuge. We consider modified
Holling Type II response function that incorporates the effect of prey refuge
and then introduce two discrete delays in the model system. A negative feedback
delay is considered in the logistic prey growth rate to represent density
dependent feedback mechanism and a positive feedback delay is considered to
represent the gestation time of the predator. Our study reveals that the system
exhibits different dynamical behaviors, viz., stable coexistence, periodic
coexistence or chaos depending on the values of the delay parameters and degree
of prey refuge. The interplay between two delays for a fixed value of prey
refuge has also been determined. It is noticed that these delays work in a
complementary fashion. In addition, using the normal form theory and center
manifold argument, we derive the explicit formulae for determining the
direction of the bifurcation, the stability and other properties of the
bifurcating periodic solutions.
