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We study the frog model on Cayley graphs of groups with polynomial growth rate D ≥ 3. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one of these particles is active when the process begins. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices. We prove that the activation time of particles grows at least linearly and we show that in the abelian case with any finite generator set the set of activated sites has a limiting shape. [ABSTRACT FROM AUTHOR]

The Sierra Nevada Yellow-Legged Frog (Rana sierrae) has generally been viewed as a lake species, but it has increasingly been found in streams, including in the northern part of its range where it is particularly at risk. Developing effective conservation strategies has been hindered by a lack of knowledge of its basic ecological requirements in stream habitats. To address this information gap, we investigated the demography, habitat use, and movements of stream populations of this federally endangered species. We conducted capture–mark–recapture of adults, quantitatively described stream channel and riparian vegetation characteristics, and collected habitat use data at four northern Sierra Nevada mountain streams, counted egg masses at three central Sierra Nevada streams, and radio-tracked individuals at three central and southern Sierra Nevada streams. Stream populations in the northern range were very small with maximum abundances of <15 individuals, and apparent survival probability ranged from 0.57–0.81. In contrast, one southern Sierra Nevada stream had a large count of 547 adults. Egg mass counts ranged from 22–104 per stream. We found frogs in diverse headwater streams ranging from perennial to intermittent flow regimes, pool versus riffle dominated, and low to high channel gradient, and they used diverse microhabitats within these streams. In these stream habitats, frogs moved little over four-day survey periods but were capable of moving longer distances of up to 1248 m over the summer. Conservation and management of the at-risk R. sierrae are most likely to be effective when built on comprehensive quantitative information on basic ecological requirements in all habitats used by the species. [ABSTRACT FROM AUTHOR]