  We introduce two models of biological aggregation, based on randomly moving
particles with individual stochasticity depending on the perceived average
population density in their neighbourhood. In the first-order model the
location of each individual is subject to a density-dependent random walk,
while in the second-order model the density-dependent random walk acts on the
velocity variable, together with a density-dependent damping term. The main
novelty of our models is that we do not assume any explicit aggregative force
acting on the individuals; instead, aggregation is obtained exclusively by
reducing the individual stochasticity in response to higher perceived density.
We formally derive the corresponding mean-field limits, leading to nonlocal
degenerate diffusions. Then, we carry out the mathematical analysis of the
first-order model, in particular, we prove the existence of weak solutions and
show that it allows for measure-valued steady states. We also perform linear
stability analysis and identify conditions for pattern formation. Moreover, we
discuss the role of the nonlocality for well-posedness of the first-order
model. Finally, we present results of numerical simulations for both the first-
and second-order model on the individual-based and continuum levels of
description.
