  Ecosystems are complex systems, currently experiencing several threats
associated with global warming, intensive exploitation, and human-driven
habitat degradation. Such threats are pushing ecosystems to the brink of
collapse. Because of a general presence of multiple stable states, including
states involving population extinction, and due to intrinsic nonlinearities
associated with feedback loops, collapse can occur in a catastrophic manner.
Such catastrophic shifts have been suggested to pervade many of the future
transitions affecting ecosystems at many different scales. Many studies have
tried to delineate potential warning signals predicting such ongoing shifts but
little is known about how such transitions might be effectively prevented. It
has been recently suggested that a potential path to prevent or modify the
outcome of these transitions would involve designing synthetic organisms and
synthetic ecological interactions that could push these endangered systems out
of the critical boundaries. Four classes of such ecological engineering designs
or {\em Terraformation motifs} have been defined in a qualitative way. Here we
develop the simplest mathematical models associated with these motifs, defining
the expected stability conditions and domains where the motifs shall properly
work.
