  We are interested in modeling Darwinian evolution resulting from the
interplay of phenotypic variation and natural selection through ecological
interactions. The population is modeled as a stochastic point process whose
generator captures the probabilistic dynamics over continuous time of birth,
mutation, and death, as influenced by each individual's trait values, and
interactions between individuals. An offspring usually inherits the trait
values of her progenitor, except when a random mutation causes the offspring to
take an instantaneous mutation step at birth to new trait values. In the case
we are interested in, the probability distribution of mutations has a heavy
tail and belongs to the domain of attraction of a stable law. We investigate
the large-population limit with allometric demographies: larger populations
made up of smaller individuals which reproduce and die faster, as is typical
for micro-organisms. We show that depending on the allometry coefficient the
limit behavior of the population process can be approximated by nonlinear
L\'evy flights of different nature: either deterministic, in the form of
nonlocal fractional reaction-diffusion equations, or stochastic, as nonlinear
super-processes with the underlying reaction and a fractional diffusion
operator. These approximation results demonstrate the existence of such
nontrivial fractional objects; their uniqueness is also proved.
