Distributed Delay Model and Von Foerster's equation: Different points of view to describe insects' life cycles with chronological age and physiological time

Mathematical models are a powerful tool when used to describe ectotherms' life cycles, above all for their suitability in being utilised for decision support systems. In particular, two models continue to arouse the interest of the scientific community and inspire new developments: the Manetsch-VanSickle Distributed Delay Model and the Von Foerster equation. Even though these models have been widely studied, discussed and applied, some aspects relating to their different points of view in representations of the same life cycle are yet to be explored. One of the main issues open for ongoing investigation is the different modes of division in preimaginal stages, which leads to different interpretations of the concept of age between the two models. The Distributed Delay Model considers a subdivision in h chained preimaginal stages with the same size, based on the concept of physiological time, in which the development of the species is related to the daily average temperature. On the other hand, the Von Foerster equation considers chronological age, defined commonly as a time with a different scale. This work highlights the analogies between the two models and shows, using the case study of L. botrana, how to obtain the number of the h stages considered by the Distributed Delay Model, from the number of observed preimaginal stages of the Von Foerster equation. To make the models comparable, the upwind scheme has been applied to the Von Foerster equation, leading to a system of ordinary differential equations that is similar to the Distributed Delay Model.