Random flow on random graphs: Amphibian Movements in the Playas



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This thesis examines, through a computer simulation, the movement of Ambystoma tigrinum or Tiger Salamanders through the playa lakes of the High Plains of Texas. Due to the local nature of precipitation in the High Plains of Texas, playas often fill or dry out. During the wet seasons, tiger salamanders often move out of one playa into surrounding playa lakes. It may be possible to track genetic adaptations or track movements of local populations throughout the playa lake system, however only limited research has been done to date in this area. Furthermore, genetic adaptation among a population can be extremely localized, for example, present in only a single playa lake. A population can then spread throughout the playa lake system.

The goal, in this thesis, is to acquire a time estimate for the time required for a trait or population propagation starting from one starting playa to the most distant playa lake in the system. This is accomplished by constructing a simple stochastic model simulating the movement of a population carrying the genetic trait. The model will construct a grid to take into account that populations have the most success when propagating to nearest neighbors. The model also accommodates for the drying out and filling of playas on a random basis depending on ecological variables. Based on the average number of time steps for propagation, a time estimate is then applied to the average number of time steps to determine an approximate time required for a trait or population to populate from on playa to the next. From this data, a time estimate can be constructed for any system size or given specific environmental constraints incorporated into the model.

The problem reduces to a stochastic flow on a graph that is changing randomly. For small sets of nodes (playas), the problem has the potential for an analytic solution. However, for systems of the same order of magnitude as the west Texas playa system (approximately 20,000 nodes) an analytic solution is infeasible, but computational results give good approximations. Archaeological records suggest the playa system was populated from a few playas on the border of the region during the last 3,000 years. Computational results and archaeological records allow for the construction of a time estimate.