<p>Due to the residual effect of pesticides, repeated pesticide applications can exert a cumulative lethal effect on pests. In this paper, we first formulate the cumulative lethal rate precisely via a dose-response function that accounts for cumulative doses of pesticides. As a key analytical tool, we derive a spray-period integral of the cumulative lethal rate function under different spraying strategies. As an application, we introduce and analyze a logistic growth model for the pest population incorporating this cumulative lethal rate. Using the derived integral, we analyze the complete dynamics of the model, including the existence, uniqueness and stability of periodic solutions. The results reveal a threshold dynamics: when the spraying period is below a critical value, the extinction equilibrium is globally asymptotically stable; otherwise, the model admits a unique positive periodic solution that is asymptotically stable. Furthermore, based on theoretical findings and numerical simulations, the optimal pesticide application strategy can be determined by comparing the total pesticide dosage required for effective pest suppression and the time of the effective control corresponding to different spraying regimens.</p>