Darwin illustrated his theory about emergence and evolution of biological species with a
diagram. It shows how species exist, evolve, appear and disappear. The goal of this work
is to give a mathematical interpretation of this diagram and to show how it can be
reproduced in mathematical models. It appears that conventional models in population
dynamics are not sufficient, and we introduce a number of new models which take into
account local, nonlocal and global consumption of resources, and models with space and
time dependent coefficients.