  In the natural world, there are many insect species whose individual members
have a life history that takes them through two stages, immature and mature.
Moreover, the rates of survival, development, and reproduction almost always
depend on age, size, or development stage. Keeping this in mind, in this paper,
a three species crop-pest-natural enemy food chain model with two stages for
natural enemies is investigated. Using characteristic equations, a set of
sufficient conditions for local asymptotic stability of all the feasible
equilibria is obtained. Moreover, using approach as in (Beretta and Kuang,
2002), the possibility of the existence of a Hopf bifurcation for the interior
equilibrium with respect to maturation delay is explored, which shows that the
maturation delay plays an important role in the dynamical behavior of three
species system. Also obtain some threshold values of maturation delay for the
stability-switching of the particular system. In succession, using the normal
form theory and center manifold argument, we derive the explicit formulas which
determine the stability and direction of bifurcating periodic solutions.
Finally, a numerical simulation for supporting the theoretical analysis is
given.
