Contents lists available at ScienceDirect Ecological Informatics journal homepage: www.elsevier.com/locate/ecolinf Distributed Delay Model and Von Foerster's equation: Different points of view to describe insects' life cycles with chronological age and physiological time Luca Rossini ⁎ , Stefano Speranza, Mario Contarini Università degli Studi della Tuscia, Dipartimento di Scienze Agrarie e Forestali (DAFNE), Via San Camillo de Lellis snc 01100, Viterbo, Italy ARTICLE INFO Keywords: Decision support systems Integrated pest management Insect pests Age-structured models Heat units Development rates ABSTRACT 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. 1. Introduction Nowadays, the availability of reliable mathematical models de- scribing the life cycle of ectotherms is broad, as are their applications in decision support systems. Among this variety of choice, two models in particular continue to arouse the interest of the scientific community, though they were introduced several years ago: the Von Foerster equation (VFE) (Von Foerster, 1959) and the Distributed Delay Model (DDM) (Manetsch, 1976; Vansickle, 1977). Their distinguishing feature is their ability to represent a cohort of individuals developing over time and through their life stages, making them suitable to describe popu- lations of ectotherms. After their first introduction, in fact, these models were widely ap- plied in several case studies, ranging from plants and insects to fish populations, and they were an important milestone for the development of more recent physiologically-based models (DeAngelis and Huston, 1987; Rossini et al., 2019a; Sinko and Streifer, 1967). The VFE considers age as a time with a different scale, while the DDM connects the mean development time of the population with a series of sub- models, which describe the relationship between the species and en- vironmental parameters. Therefore, the DDM has the added value of considering age not as a time with a different scale, but as a “physio- logical time” (Gutierrez et al., 1975; Mirhosseini et al., 2017), essen- tially defined as the units of degree days (Gutierrez et al., 1985; Plant and Wilson, 1986). The units of degree days are represented in different ways, ranging from the simplest definition of “thermal summation” (Nance et al., 2018; Severini and Gilioli, 2002), to more accurate de- velopment rate functions (Ikemoto and Kiritani, 2019). With this precondition, VFE and DDM seem at first sight to be two different models and thus not correlated; however, the work of Plant and Wilson (1986) showed that it was possible to obtain the DDM from the VFE. Their work suggests that these two models only differ from each other in having different point of view in representing the same life cycle, suggesting a connection between chronological age and https://doi.org/10.1016/j.ecoinf.2020.101117 Received 31 March 2020; Received in revised form 14 April 2020; Accepted 26 April 2020 ⁎ Corresponding author. E-mail address: luca.rossini@unitus.it (L. Rossini). Ecological Informatics 59 (2020) 101117 Available online 12 June 2020 1574-9541/ © 2020 Elsevier B.V. All rights reserved. T