This topic presents a study on a host–parasitoid model with a Holling type III functional
response. In population dynamics, when host density rises, the parasitoid response initially
accelerates due to the parasitoid's improved searching efficiency. However, above
a certain density threshold, the parasitoid response will reach a saturation level due to
the influence of reducing the handling time. Thus, we incorporated a Holling type III
functional response into the model to characterize such a phenomenon. The dynamics
of the current model are discussed in this paper. We first obtained the existence
and local stability conditions of the positive fixed point of the model. Furthermore, we
investigated the bifurcation behaviors at the positive fixed point. More specifically, we
used bifurcation theory and the center manifold theorem to prove that the model possess
both period doubling and Neimark–Sacker bifurcations. Then, the chaotic behavior
of the model, in the sense of Marotto, is proven. Furthermore, we apply a state-delayed
feedback control strategy to control the complex dynamics of the present model.
Finally, numerical examples are provided to support our analytic results.