  This paper is concerned with spreading properties of space-time heterogeneous
Fisher--KPP equations in one space dimension. We focus on the case of
everywhere favorable environment with three different zones, a left half-line
with slow or intermediate growth, a central patch with fast growth and a right
half-line with slow or intermediate growth. The central patch moves at various
speeds. The behavior of the front changes drastically depending on the speed of
the central patch. Among other things, intriguing phenomena such as nonlocal
pulling and locking may occur, which would make the behavior of the front
further complicated. The problem we discuss here is closely related to
questions in biomathematical modelling. By considering several special cases,
we illustrate the remarkable diversity of possible behaviors. In particular,
the case of a central patch with constant size and constant speed is entirely
settled.
