The phenomenon of perceptual multistability in which alternate interpretations of a fixed stimulus are perceived intermittently is an active area of research and the underlying mechanisms that gate perception are little understood. Numerical tools from bifurcation analysis are applied to the study of a competition model posed as a feature-only neural field equation. In the absence of input the model with a spike-frequency adaptation mechanism has been shown to produce an array of complex spatio-temporal dynamics local to a Bogdanov-Takens point. Here, we demonstrate how, with the introduction of an input, the organisation of solutions in parameter space changes via symmetry breaking. The model is then used to investigate a more complex stimulus in the context of motion integration that is multistable in terms of its perceived direction of motion, the so-called multistable barber pole, which has been the subject of concurrent psychophysics experiments. We bring the model to an operating regime where known physiological response properties are reproduced whilst also working close to bifurcation. We find that in this regime the model is able to account for characteristic behaviour from experiment in terms of the type of switching observed and changes in the rate of switching with respect to contrast. In this way, the modelling study sheds light on the underlying mechanisms that drive perceptual switching in different contrast regimes. The general approach presented is applicable to a broad range of perceptual competition problems in which spatial interactions play a role.