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In the natural world, there are many insect species whose individual membershave a life history that takes them through two stages, immature and mature.Moreover, the rates of survival, development, and reproduction almost alwaysdepend 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 fornatural enemies is investigated. Using characteristic equations, a set ofsufficient conditions for local asymptotic stability of all the feasibleequilibria is obtained. Moreover, using approach as in (Beretta and Kuang,2002), the possibility of the existence of a Hopf bifurcation for the interiorequilibrium with respect to maturation delay is explored, which shows that thematuration delay plays an important role in the dynamical behavior of threespecies system. Also obtain some threshold values of maturation delay for thestability-switching of the particular system. In succession, using the normalform theory and center manifold argument, we derive the explicit formulas whichdetermine the stability and direction of bifurcating periodic solutions.Finally, a numerical simulation for supporting the theoretical analysis isgiven.

Theoretical Study of Pest Control Using Stage Structured Natural Enemies with Maturation Delay: A Crop-Pest-Natural Enemy Model