Maize, also known as corn, is one of the most popular food in the world. Managing its Lepidoptera pests is a big deal as the larvae, which cause severe damages, develop inside maize stalks and ears. The most popular means of controlling insect pests remain chemical, but agricultural practices as the management of crop residues have been proven efficient. In this paper, we analyze the effect of the Maize Stalk Borer (MSB) Busseola fusca on maize crops and the efficiency of residue management over several cropping seasons. More precisely, we build two multi-seasonal mathematical models for the dynamics of MSB interacting with maize tissues. The models account for two types of residue management: either plant residues are burned after harvest or, are buried in the soil at the end of the cropping season to reduce pest reservoir for the following cropping seasons. A qualitative analysis of the models is carried out including the computation of a pest-free solution. Moreover, for each model, we compute the ecological threshold, also known as the basic reproduction number, that shapes the stability of the pest-free solution. To characterize the impact of parameter variations on the state variables of the models, we carry out a sensitivity analysis. The sensitivity analysis points out the leading role of the plant infestation rate and the senescence rate of crop residues on model outputs. In addition, numerical simulations are presented to illustrate our theoretical results and support the discussion. Notably, we found that the basic reproduction numbers have a linear relationship with the proportion of immature pests destroyed at the end of cropping seasons. Therefore, advocating management policies that lead to an increase in the proportion of immature pests destroyed at the end of cropping seasons could be helpful to control the dynamics of MSB.