Early investigations of pattern formation in fluid mechanical systems inspired some of the development of equivariant bifurcation theory. Recently, these pattern formation perspectives have been applied to modeling the vegetation in dryland ecosystems, where satellite images have revealed strikingly regular spatial patterns on large scales. Ecologists have proposed that characteristics of vegetation pattern formation in these water-limited ecosystems may serve as an early warning sign of impending desertification. We use the framework of equivariant bifurcation theory to investigate the mathematical robustness of this approach to probing an ecosystems robustness.