Table 3 Overview of long transient (LT) classification and mechanisms.
Type of LTRelationship to
invariant set
Relationship to
bifurcation
Dynamics mimicked
by LT
Possibility of
recurrent LTs?
Biological
example
Ghost (Fig. 2)No invariant setOccurs past a
bifurcation
where stable
equilibrium
is lost
Equilibrium, cycles,
chaos
NoForage fish (27)
(Fig. 3B)
Crawl-by
(Fig. 3, C and D)
Caused by
saddle-type
invariant set
None necessaryEquilibrium, cycles,
chaos
YesPhytoplankton-grazer
models (26)
Slow-fast systems
(Fig. 3, E and F)
None necessaryMultiple time scalesPeriodic or
aperiodic cycles
Yes, if invariant
set(s) present
Univoltine insects (2)
(Fig. 3C)
High dimension
(e.g., time delays,
space) (Fig. 4A)
None necessaryNone necessaryEquilibrium, cycles,
chaos
YesChemostat microbial
communities (57)
Stochasticity
(Fig. 4B)
If invariant set present:None necessaryAperiodic cycles, chaosYesCancer crabs (53)
If invariant set absent:Past a bifurcation
where cycles/chaos
are lost
Quasi-periodic cycles