An rth-order extremal process Δ(r) = (Δ(r)t)t≥0 is a continuous-time analogue of the rth partial maximum sequence of a sequence of independent and identically distributed random variables. Studying maxima in continuous time gives rise to the notion of limiting properties of Δt(r) as t ↓ 0. Here we describe aspects of the small-time behaviour of Δ(r) by characterising its upper and lower classes relative to a nonstochastic nondecreasing function bt > 0 with limt↓bt = 0. We are then able to give an integral criterion for the almost sure relative stability of Δt(r) as t ↓ 0, r = 1, 2, . . ., or, equivalently, as it turns out, for the almost sure relative stability of Δt(1) as t ↓ 0.

In this paper we characterize the mean and the distribution of the first exit time of a Lévy flight from a bounded region in N-dimensional spaces. We characterize tight upper and lower bounds on the tail distribution of the first exit time, and provide the exact asymptotics of the mean first exit time for a given range of step-length distribution parameters.

Animal movements in heterogeneous environments shape most ecological processes from individuals to ecosystems. The identification of the processes underlying animal movements thus has critical implications in a wide range of fields. Changes in the motion behaviour of free-ranging species have mainly been reported across different spatial scales but have been less investigated over time. Here the processes potentially triggering temporal changes in movement, microhabitat occupation and distribution patterns of the intertidal herbivorous gastropod Nerita atramentosa were examined on a south Australian rocky shore during 8 successive daytime low tides considered in four different months. The observed temporal shift in microhabitat occupation and aggregation behaviour was likely an adaptation to both abiotic and biotic stressors such as temperature and food distribution. In contrast, the temporal changes observed in N. atramentosa motion behaviour are consistent with optimal foraging strategies driven by the presumed variability in both microalgal density and distribution. Individuals seemed to switch from an extensive foraging strategy in February and May to an intensive strategy in August and October. Specifically, N. atramentosa seemed to adopt two different foraging strategies, a Lévy-like foraging strategy in May and a ballistic foraging strategy in February when the resources are expected to be respectively non-depleted and depleted. The Brownian foraging strategy theoretically expected under high food density conditions was however not observed. The potential role of the behavioural flexibility observed in N. atramentosa motion behaviour is discussed in relation to the persistence of this species under disrupted environmental conditions.