Complex dynamics generated by negative and positive feedback delays of a   prey-predator system with prey refuge: Hopf bifurcation to Chaos
  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. 
Debaldev Jana, R. Gopal and M. Lakshmanan
math.DS