Two-phase flow models have been used previously to model cell motility. In order to reduce the complexity inherent with describing the many physical processes, we formulate a minimal model. Here we demonstrate that even the simplest 1D, two-phase, poroviscous, reactive flow model displays various types of behaviour relevant to cell crawling. We present stability analyses that show that an asymmetric perturbation is required to cause a spatially uniform, stationary strip of cytoplasm to move, which is relevant to cell polarization. Our numerical simulations identify qualitatively distinct families of travellingwave solutions that coexist at certain parameter values. Within each family, the crawling speed of the strip has a bell-shaped dependence on the adhesion strength. The model captures the experimentally observed behaviour that cells crawl quickest at intermediate adhesion strengths, when the substrate is neither too sticky nor too slippy. © The Author 2012. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.