Every species has an evolutionary history. Without having some knowledge about this history it is very difficult or even impossible to understand many recent biological phenomena. One important evolutionary event in vertebrate evolution was the transition from an ectothermic (“cold-blooded”, e.g. reptiles, fishes) to an endothermic (“warm-blooded”, e.g. birds, mammals) metabolism, which for example we humans have. Several hypothesis on the evolution of endothermy already exist, although none of these is generally accepted. We have developed a hypothesis, the mechanistic hypothesis on the evolution of endothermy (MHEE), which mechanistically describes how this transition might have occurred. Our MHEE gives a physiological explanation for the transition from an ectothermic to an endothermic metabolism. Because of our mechanistic approach, and contrary to many other hypotheses on the evolution of endothermy, the assumptions and predictions of the MHEE are testable, and it is possible to describe and analyze the MHEE mathematically. The mathematical approach enables us to simulate events over an evolutionary time scale and to simulate hypothetical physiological processes, or intermediary evolutionary stages, which cannot be analyzed within real biological systems, or this would be too costly and time expensive. The first objective of our project is to establish a mathematical model, that is sufficiently biological realistic and to validate it by means of real biological data. Our next objective is to put the MHEE into a larger evolutionary context. Therefore we have already developed a mathematical model under the paradigm of Adaptive Dynamics; the mathematical theory that focuses on phenotypic evolution of a species driven by rare mutations and competition, and takes into account the evolution of the environmental conditions. In particular we want to study the evolutionary shift of ectothermic to endothermic metabolism using a parabolic Lotka-Volterra model that includes small mutations.