  Complex systems such as ecosystems, electronic circuits, lasers or chemical
reactions can be modelled by dynamical systems which typically experience
bifurcations. Transients typically suffer extremely long delays at the vicinity
of bifurcations and it is also known that these transients follow scaling laws
as the bifurcation parameter gets closer the bifurcation value in deterministic
systems. The mechanisms involved in local bifurcations are well-known. However,
for saddle-node bifurcations, the relevant dynamics after the bifurcation occur
in the complex phase space. Hence, the mechanism responsible for the delays and
the associated inverse-square root scaling law for this bifurcation can be
better understood by looking at the dynamics in the complex space. We follow
this approach and complexify a simple ecological system undergoing a
saddle-node bifurcation. The discrete model describes a biological system with
facilitation (cooperation) under habitat destruction for species with
non-overlapping generations. We study the complex (as opposed to real) dynamics
once the bifurcation has occurred. We identify the fundamental mechanism
causing these long delays (called ghosts), given by two repellers in the
complex space. Such repellers appear to be extremely close to the real line,
thus forming a narrow channel close to the two new fixed points and responsible
for the slow passage of the orbits, which remains tangible in the real numbers
phase space. We analytically provide the relation between the inverse
square-root scaling law and the multipliers of these repellers. We finally
prove that the same phenomenon occurs for more general i.e., non-necessarily
polynomial, models.
