We consider spatial population models with local (self-)interactions. The interaction can be induced by a common random (spatial) environment in which the individuals live (branching random walk in random environment). It can also be given by a local feed-back mechanism which regulates an individual's offspring law depending on the configuration in a certain neighbourhood around it. These models are more realistic than classical branching random walks, where particles move and branch completely independently. In the first case, they take into account inhomogeneities in the habitat. In the second case, they describe a self-regulation of the population coming for instance from a competition for resources. They are also harder to analyse: in the latter case, they are typically not "monotone" systems, since adding additional individuals at some time point can diminish the population at a later time. We focus on (1) extinction versus survival regimes, (2) the linear speed of the front, and the corresponding fluctuations, (3) genealogies and interfaces. A common connecting tool is the study of the spatial embedding of ancestral lines. They are random walks in a dynamic random environment, and the study of such random walks has an independent interest.

We consider spatial population models with local (self-)interactions. The interaction can be either induced by a common random (spatial) environment in which the individuals live or it can be modelled explicitly by a local feed-back mechanism which regulates an individual’s offspring law depending on the configuration in a certain neighbourhood around it. These models are more realistic than classical branching random walks, where particles move and branch completely independently; they are also harder to analyse because of this lack of independence and also because they typically are not “monotone” systems, since adding additional individuals can be actually harmful for the present population.

We focus on (1) extinction versus survival regimes, (2) in an equilibrium, when including neutraltypes: growth of clusters versus local diversity, (3) front speed and fluctuations in expanding populations in a random environment. A common connecting tool is the study of the spatial embedding of ancestral lines, which are random walks in a dynamic random environment.