Mathematical modelling that characterizes the qualitative behavior of the interactions of aphids and ladybugs as insectivorous animals

The significance of the agricultural industry is primarily driven by the growth of the human population and the imperative for sustainable food production. Within this sector, the presence of insect pests, such as aphids, demands particular attention, as they exert a detrimental influence on agricultural practices. To devise effective approaches, it is essential to possess a thorough comprehension of aphid control. Thus, this study presents a mathematical model that captures the qualitative dynamics of interactions between aphids and ladybugs. The model is composed of a system of non-linear differential equations, encompassing the interplay between the two species, the impact of insecticides on aphids, and the hibernation stages of ladybugs. The analytical approach is employed to ascertain the criteria for the existence and stability of the equilibrium derived from the constructed model. The study focused on stability and sensitivity analysis to understand the behavior of the model and develop effective strategies for managing aphid populations. Both the findings from the analytical and numerical investigations offer valuable insight that augmenting insectivorous populations, such as the ladybug populations, may serve as an alternative method for aphid control, reducing the reliance on insecticides. Nevertheless, the application of insecticides continues to be a viable approach for aphid control. Further investigation into controlling the aphid population involves an in-depth study of optimal control problems. The study examined different scenarios involving the utilization of insecticides. Our numerical experiments show a potential of insecticide to control the population of aphids effectively. The change in the population of aphids do not give a big disturbance in ladybug population as long as there is an alternative resource for ladybugs. From the cost analysis, we find that a larger intervention needed when the initial population of aphids relatively small. Therefore, insecticide interventions need to consider various factors, not only the biological behavior of ladybugs and aphids but also the initial population of ladybugs and aphids.