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

Looking forwards and backwards: dynamics and genealogies of locally regulated populations