In holometabolous insects, adult organs develop as imaginal discs within the growing larvae. Once larvae pass the critical size, juvenile hormone (JH) titers decline and imaginal discs begin to grow rapidly. The subsequent rise in ecdysteroid titers causes the larva to stop feeding and wander away from its food to find a place to metamorphose. The period between attainment of the critical size and ecdysteroid secretion is called the terminal growth period (TGP). Although all holometabolous insects are thought to have the critical size for metamorphosis, the mechanisms for nutritional regulation of the critical size and the TGP differ between species. For example, in Manduca sexta, low nutrition reduces the critical size and has no effect on the duration of the TGP, whereas in Drosophila melanogaster, low nutrition lengthens the duration of the TGP and has no effect on the critical size.
To study the ecological causes of the diversity in the mechanisms for nutritional regulation, we develop a resource allocation model of a holometabolous insect and analyze it with the Pontryagin's maximum principle with free-final-time. A typical optimal growth schedule is biphasic (the so-called bang-bang control): all nutrition is devoted to growth of a larval body until some switching time, after which point all nutrition is devoted to growth of an imaginal disc until the end of development. Regarding the larval body size at the switching time and the second phase as the "critical size" and the "TGP", we investigate how these values depend on nutrition for different fitness functions.
Considering that the fitness function is proportional to the size of an imaginal disc at the end of development, the combination of a constant TGP and a nutrition-dependent critical size is optimal. On the other hand, the combination of a nutrition-dependent TGP and a constant critical size is optimal if the time to obtain a mature imaginal disc (fixed size) is minimized. These optimal strategies are in accord with the pattern observed for Manduca sexta and Drosophila melanogaster, respectively.