Energy intake functions and energy budgets of ectotherms and endotherms derived from their ontogenetic growth in body mass and timing of sexual maturation

Ectothermic and endothermic vertebrates differ not only in their source of body temperature (environment vs. metabolism), but also in growth patterns, in timing of sexual maturation within life, and energy intake functions. Here, we present a mathematical model applicable to ectothermic and endothermic vertebrates. It is designed to test whether differences in the timing of sexual maturation within an animal's life (age at which sexual maturity is reached vs. longevity) together with its ontogenetic gain in body mass (growth curve) can predict the energy intake throughout the animal's life (food intake curve) and can explain differences in energy partitioning (between growth, reproduction, heat production and maintenance, with the latter subsuming any other additional task requiring energy) between ectothermic and endothermic vertebrates. With our model we calculated from the growth curves and ages at which species reached sexual maturity energy intake functions and energy partitioning for five ectothermic and seven endothermic vertebrate species. We show that our model produces energy intake patterns and distributions as observed in ectothermic and endothermic species. Our results comply consistently with some empirical studies that in endothermic species, like birds and mammals, energy is used for heat production instead of growth, and with a hypothesis on the evolution of endothermy in amniotes published by us before. Our model offers an explanation on known differences in absolute energy intake between ectothermic fish and reptiles and endothermic birds and mammals. From a mathematical perspective, the model comes in two equivalent formulations, a differential and an integral one. It is derived from a discrete level approach, and it is shown to be well-posed and to attain a unique solution for (almost) every parameter set. Numerically, the integral formulation of the model is considered as an inverse problem with unknown parameters that are estimated using a series of empirical data.