Recent research on the intertemporal dynamics of poverty using microeconomic data often hints at the existence of poverty traps, where some find themselves trapped at a low-level stable equilibrium while others enjoy a higher stable equilibrium. Without a sizable positive shock to well-being, those trapped at the low equilibrium will not automatically outgrow destitution, but merely fluctuate around that low-level equilibrium. Given the dramatic policy consequences implied by such a theory, knowledge about the location of the different equilibria would be extremely helpful. In this paper, we explore the possibilities of threshold-type models to identify those crucial parameters. We illustrate the method by searching for traps in the dynamics of livestock asset holdings in rural Ethiopia. We find evidence of distribution-dependent dynamics and multiple equilibria for tropical livestock units.