Reformulation of the Distributed Delay Model to describe insect pest populations using count variables

Among the models used to describe insect pest populations, the Distributed Delay Model has been applied in several case studies in recent years. Its success is due mainly to its simplicity, and its versatility to be easily included in software to calculate numerical solutions. In its original formulation, the Distributed Delay Model provides, as a solution, the distribution of the insects’ maturation flow; then, this is compared with monitoring in field applications. A different form of the model can be obtained, with the same assumptions, to describe the distribution of the number of individuals which are in a specific life stage at time t. The first aim of this work was to show the mathematical details in order to obtain the second form of the Distributed Delay Model, and to calculate its analytical solutions. The second aim was to analyse the model's behaviour in describing insect pest's population in varying environmental conditions, specifically in terms of temperature. To pursue this second aim, two case studies of noteworthy relevance in agriculture were considered: the pepper weevil, Anthonomus eugenii and the European grapevine moth, Lobesia botrana. For each case study, field populations were simulated with both the Distributed Delay Model versions, and the results were compared to determine the most appropriate model for application in the case of insect pest populations. Both the case studies highlighted that the novel formulation presented in this work significantly improves simulation, providing a more reliable representation of field data.