Exploring extensions to multi-state models with multiple unobservable states

Many biological systems include a portion of the target population that is unobservable during certain life history stages. Transition to and from an unobservable state may be of primary interest in many ecological studies and such movements are easily incorporated into multi-state models. Several authors have investigated properties of open-population multi-state mark-recapture models with unobservable states, and determined the scope and constraints under which parameters are identifiable (or, conversely, are redundant), but only in the context of a single observable and a single unobservable state (Schmidt et al. 2002; Kendall and Nichols 2002; Schaub et al. 2004; Kendall 2004). Some of these constraints can be relaxed if data are collected under a version of the robust design (Kendall and Bjorkland 2001; Kendall and Nichols 2002; Kendall 2004; Bailey et al. 2004), which entails >1 capture period per primary period of interest (e.g., 2 sampling periods within a breeding season). The critical assumption shared by all versions of the robust design is that the state of the individual (e.g. observable or unobservable) remains static for the duration of the primary period (Kendall 2004). In this paper, we extend previous work by relaxing this assumption to allow movement among observable states within primary periods while maintaining static observable or unobservable states. Stated otherwise, both demographic and geographic closure assumptions are relaxed, but all individuals are either observable or unobservable within primary periods. Within these primary periods transitions are possible among multiple observable states, but transitions are not allowed among the corresponding unobservable states. Our motivation for this work is exploring potential differences in population parameters for pond-breeding amphibians, where the quality of habitat surrounding the pond is not spatially uniform. The scenario is an example of a more general case where individuals move between habitats both during the breeding season (within primary periods; transitions among observable states only) and during the non-breeding season (between primary periods; transitions between observable and unobservable states). Presumably, habitat quality affects demographic parameters (e.g. survival and breeding probabilities). Using this model we are able to test this prediction for amphibians and determine if individuals move to more favorable habitats to increase survival and breeding probabilities.