  We introduce a broad class of spatial models to describe how spatially
heterogeneous populations live, die, and reproduce. Individuals are represented
by points of a point measure, whose birth and death rates can depend both on
spatial position and local population density, defined via the convolution of
the point measure with a nonnegative kernel. We pass to three different scaling
limits: an interacting superprocess, a nonlocal partial differential equation
(PDE), and a classical PDE. The classical PDE is obtained both by first scaling
time and population size to pass to the nonlocal PDE, and then scaling the
kernel that determines local population density; and also (when the limit is a
reaction-diffusion equation) by simultaneously scaling the kernel width,
timescale and population size in our individual based model. A novelty of our
model is that we explicitly model a juvenile phase: offspring are thrown off in
a Gaussian distribution around the location of the parent, and reach (instant)
maturity with a probability that can depend on the population density at the
location at which they land. Although we only record mature individuals, a
trace of this two-step description remains in our population models, resulting
in novel limits governed by a nonlinear diffusion. Using a lookdown
representation, we retain information about genealogies and, in the case of
deterministic limiting models, use this to deduce the backwards in time motion
of the ancestral lineage of a sampled individual. We observe that knowing the
history of the population density is not enough to determine the motion of
ancestral lineages in our model. We also investigate the behaviour of lineages
for three different deterministic models of a population expanding its range as
a travelling wave: the Fisher-KPP equation, the Allen-Cahn equation, and a
porous medium equation with logistic growth.
