  Critical transitions are of great interest to scientists in many fields. Most
knowledge about these transitions comes from systems exhibiting the
multistability of spatially uniform states. In spatially extended and,
particularly, in pattern-forming systems, there are many possible scenarios for
transitions between alternative states. Quenched disorder may affect the
dynamics, bifurcation diagrams and critical transitions in nonlinear systems.
However, only a few studies have explored the effects of quenched disorder on
pattern-forming systems, either experimentally or by using theoretical models.
Here, we use a fundamental model describing pattern formation, the
Swift-Hohenberg model and a well-explored mathematical model describing the
dynamics of vegetation in drylands to study the effects of quenched disorder on
critical transitions in pattern-forming systems. We find that the disorder
affects the patterns formed by introducing an interplay between the imposed
pattern and the self-organized one. We show that, in both systems considered
here, the disorder significantly increases the durability of the patterned
state and makes the transition between the patterned state and the uniform
state more gradual. In addition, the disorder induces hysteresis in the
response of the system to changes in the bifurcation parameter well before the
critical transition occurs. We also show that the cross-correlation between the
disordered parameter and the dynamical variable can serve as an early indicator
for an imminent critical transition.
