Locusts and some noctuid moths exhibit polyphenism whereby they can change their " phase " from a solitary ("solitarious") condition to a gregarious one. Gregarious phase insects are often migratory travelling from recession areas into larger invasion zones and, among locusts, occur in swarms. Difference equation models of the population dynamics of insects that take account of such changes between solitarious or gregarious phases in relation to predation, both with and without time delays, are described. Solutions of the models are non-linear. Chaotic solutions are obtainable under some circumstances even with very low values for the intrinsic rate of increase in the prey population, in contrast to previous conclusions from models without predation. Comparisons with the results obtained for single species with those obtained in this paper show that predation can reduce (i) the average density of the prey, (ii) durations of periods when the populations stay in the gregarious phase , and (iii) the frequency of their shifts from the solitarious state to the gregarious form. Similar results are obtained if a time delay is introduced to mimic a transient phase . With a wide range of parameter values, models including predation with or without random perturbation reveal several stable attractors for phase diagrams of the populations, which are biologically meaningful compared with empirical datasets and which were unobtainable without predation, suggesting that inclusion of predation and time delays improved the realism of the models. However, comparisons between autocorrelation analyses of locust time-series, but of swarms only, with those of model outputs suggest that inclusion of the time delay leads to less, not more, realism. The prediction of non-linearity in the dynamics of migrant insects with phase changes and its significance for forecasting to aid control is briefly discussed in relation to published data on the desert locust Schistocerca gregaria .