Dispersal, settling and layer formation

Motivated by examples in developmental biology and ecology, we develop a model for convection-dominated invasion of a spatial region by initially motile agents which are able to settle permanently. The motion of the motile agents and their rate of settling are affected by the local concentration of settled agents. The model can be formulated as a nonlinear partial differential equation for the time-integrated local concentration of the motile agents, from which the instantaneous density of settled agents and its long-time limit can be extracted. In the limit of zero diffusivity, the partial differential equation is of first order; for application-relevant initial and boundary-value problems, shocks arise in the time-integrated motile agent density, leading to delta-function components in the motile agent density. Furthermore, there are simple solutions for a model of successive layer formation. In addition some analytic results for a one-dimensional system with non-zero diffusivity can also be obtained. A case study, both with and without diffusion, is examined numerically. Some important predictions of the model are insensitive to the specific settling law used and the model offers insight into biological processes involving layered growth or overlapping generations of colonization.