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We consider the first-passage problem for N identical independent particles that are initially released uniformly in a finite domain Ω and then diffuse toward a reactive area Γ, which can be part of the outer boundary of Ω or a reaction centre in the interior of Ω. For both cases of perfect and partial reactions, we obtain the explicit formulas for the first two moments of the fastest first-passage time (fFPT), i.e., the time when the first out of the N particles reacts with Γ. Moreover, we investigate the full probability density of the fFPT. We discuss a significant role of the initial condition in the scaling of the average fFPT with the particle number N, namely, a much stronger dependence (1/N and 1/N2 for partially and perfectly reactive targets, respectively), in contrast to the well known inverse-logarithmic behaviour found when all particles are released from the same fixed point. We combine analytic solutions with scaling arguments and stochastic simulations to rationalise our results, which open new perspectives for studying the relevance of multiple searchers in various situations of molecular reactions, in particular, in living cells.

We study the probability density function (PDF) of the first-reaction times between a diffusive ligand and a membrane-bound, immobile imperfect target region in a restricted ‘onion-shell’ geometry bounded by two nested membranes of arbitrary shapes. For such a setting, encountered in diverse molecular signal transduction pathways or in the narrow escape problem with additional steric constraints, we derive an exact spectral form of the PDF, as well as present its approximate form calculated by help of the so-called self-consistent approximation. For a particular case when the nested domains are concentric spheres, we get a fully explicit form of the approximated PDF, assess the accuracy of this approximation, and discuss various facets of the obtained distributions. Our results can be straightforwardly applied to describe the PDF of the terminal reaction event in multi-stage signal transduction processes.

We consider a lognormal diffusion process having a multisigmoidal logistic mean, useful to model the evolution of a population which reaches the maximum level of the growth after many stages. Referring to the problem of statistical inference, two procedures to find the maximum likelihood estimates of the unknown parameters are described. One is based on the resolution of the system of the critical points of the likelihood function, and the other is on the maximization of the likelihood function with the simulated annealing algorithm. A simulation study to validate the described strategies for finding the estimates is also presented, with a real application to epidemiological data. Special attention is also devoted to the first-passage-time problem of the considered diffusion process through a fixed boundary.

Prey species must consistently balance the need for resource acquisition with the threat of predation. This balance is particularly true for gallinaceous birds, such as the northern bobwhite (Colinus virginianus), which are ground foragers and are often exposed to increased predation risk compared to arboreal foragers. We studied how bobwhites might mitigate the threat of human hunting pressure by altering their foraging strategy. We directly monitored hunting encounters with bobwhites, during October 2014–March 2015, and collected daily movement paths (n = 514) using radio-telemetry. We performed first-passage time analysis to infer foraging behavior by calculating bout frequency, area, duration, and timing. We found bobwhites mitigated exposure to human hunters by increasing foraging frequency coupled with a decrease in duration (15.4%) and area (7.1%) in response to hunting pressure. We observed a temporal shift in foraging away from peak hunting hours by 30 minutes when birds were recently exposed to a discharged firearm. Our results imply that hunting game species can disrupt timing of foraging and therefore influence allocation to competing activities such as anti-predator vigilance. We propose implementing a dynamic harvest management regime that distributes hunting activity temporally and spatially to mitigate high hunting pressure and reduce behavioral effects on bobwhites. Careful planning of hunting activity should reduce its additive effects on natural mortality while improving hunter satisfaction.

We consider the first-passage time problem for the Feller-type diffusion process, having infinitesimal drift B1(x,t)=α(t)x+β(t) and infinitesimal variance B2(x,t)=2r(t)x, defined in the space state [0,+∞), with α(t)∈R, β(t)>0, r(t)>0 continuous functions. For the time-homogeneous case, some relations between the first-passage time densities of the Feller process and of the Wiener and the Ornstein–Uhlenbeck processes are discussed. The asymptotic behavior of the first-passage time density through a time-dependent boundary is analyzed for an asymptotically constant boundary and for an asymptotically periodic boundary. Furthermore, when β(t)=ξr(t), with ξ>0, we discuss the asymptotic behavior of the first-passage density and we obtain some closed-form results for special time-varying boundaries.

Predators directly impact prey populations through lethal encounters, but understanding nonlethal, indirect effects is also critical because foraging animals often face trade‐offs between predator avoidance and energy intake. Quantifying these indirect effects can be difficult even when it is possible to monitor individuals that regularly interact. Our goal was to understand how movement and resource selection of a predator (wolves; Canis lupus) influence the movement behavior of a prey species (moose; Alces alces). We tested whether moose avoided areas with high predicted wolf resource use in two study areas with differing prey compositions, whether avoidance patterns varied seasonally, and whether daily activity budgets of moose and wolves aligned temporally. We deployed GPS collars on both species at two sites in northern Minnesota. We created seasonal resource selection functions (RSF) for wolves and modeled the relationship between moose first‐passage time (FPT), a method that discerns alterations in movement rates, and wolf RSF values. Larger FPT values suggest rest/foraging, whereas shorter FPT values indicate travel/fleeing. We found that the movements of moose and wolves peaked at similar times of day in both study areas. Moose FPTs were 45% lower in areas most selected for by wolves relative to those avoided. The relationship between wolf RSF and moose FPT was nonlinear and varied seasonally. Differences in FPT between low and high RSF values were greatest in winter (−82.1%) and spring (−57.6%) in northeastern Minnesota and similar for all seasons in the Voyageurs National Park ecosystem. In northeastern Minnesota, where moose comprise a larger percentage of wolf diet, the relationship between moose FPT and wolf RSF was more pronounced (ave. across seasons: −60.1%) than the Voyageurs National Park ecosystem (−30.4%). These findings highlight the role wolves can play in determining moose behavior, whereby moose spend less time in areas with higher predicted likelihood of wolf resource selection. Quantifying indirect effects of predation risk is often difficult but important for understanding trade‐offs between predator avoidance and energy intake. Our goal was to understand how the likelihood of encountering a predator (wolves) influences the movement behavior of a prey species (moose). Moose spent ~45% less time in areas with the highest predicted areas of wolf presence relative to those with the least.

With the increasing capacity of doubly fed induction generators (DFIGs), the diameter of the wind turbine is increasing, the blades are getting longer and longer, and in the process of power generation, the tower shadow effect as well as the role of wind shear are more obvious, and the random blade stresses caused by this is also getting bigger. Random blade stresses cause random and cyclic fluctuations in the power generated by the wind turbine, and power fluctuations often cause voltage flicker, which affects the control system and power quality. To address the impact of random blade stresses on the grid‐connected stability of DFIG, the results of the traditional stability analysis methods may be too conservative or lead to too high a dimensionality to be analyzed. To solve the above problems, this paper proposes a grid‐connected stochastic stability analysis method for DFIG sets considering random blade stresses based on the stochastic averaging method under the Hamiltonian system. a stochastic dynamics model of the doubly fed wind farm was established by considering random blade stress. Subsequently, using the proposed generalized Hamiltonian principle, the model and energy functions in the proposed Hamiltonian form H, based on the stochastic averaging method (SAM), were established to obtain the system energy diffusion equation. Probability density function and regional stability probability were obtained from explicit expressions of the mean and regression square root processes. The drift and diffusion coefficients were obtained using the SAM, and the backward Kolmogorov equation was derived from the Ito equation to obtain the conditional reliability function and the probability density of the first crossing time. Finally, the effects of torque fluctuations with different stochastic intensities on the grid‐connected stability of doubly fed wind farms were investigated, and the effectiveness of the proposed generalized Hamiltonian SAM applied to the stochastic stability analysis of DFIG was verified by numerical analysis and Monte Carlo simulation. This provides a theoretical foundation for analyzing the grid‐connected stability of DFIG affected by random blade stresses. In this paper, based on the stochastic averaging method under the Hamiltonian system, a grid‐connected stability analysis method for doubly fed generator sets considering random blade stress is proposed. The effectiveness of the proposed generalized Hamiltonian stochastic averaging method applied to the stochastic stability analysis of doubly fed wind turbines is also verified by numerical analysis and Monte Carlo simulation.

1. In order to study and predict population distribution, it is crucial to identify and understand factors affecting individual movement decisions at different scales. Movements of foraging animals should be adjusted to the hierarchical spatial distribution of resources in the environment and this scale-dependent response to environmental heterogeneity should differ according to the forager's characteristics and exploited habitats. 2. Using First-Passage Time analysis, we studied scales of search effort and habitat used by individuals of seven sympatric Indian Ocean Procellariiform species fitted with satellite transmitters. We characterized their search effort distribution and examined whether species differ in scale-dependent adjustments of their movements according to the marine environment exploited. 3. All species and almost all individuals (91% of 122 individuals) exhibited an Area-Restricted Search (ARS) during foraging. At a regional scale (1000s km), foraging ranges showed a large spatial overlap between species. At a smaller scale (100s km, at which an increase in search effort occurred), a segregation in environmental characteristics of ARS zones (where search effort is high) was found between species. 4. Spatial scales at which individuals increased their search effort differed between species and also between exploited habitats, indicating a similar movement adjustment for predators foraging in the same habitat. ARS zones of the two populations of wandering albatross Diomedea exulans (Crozet and Kerguelen) were similar in their adjustments (i.e. same ARS scale) as well as in their environmental characteristics. These two populations showed a weak spatial overlap in their foraging distribution, with males foraging in more southerly waters than females in both populations. 5. This study demonstrates that predators of several species adjust their foraging behaviour to the heterogeneous environment and these scale-dependent movement adjustments depend on both forager and environment characteristics.

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

1. It is predicted that the movements of foraging animals are adjusted to the hierarchical spatial distribution of resources in the environment, and that decisions to modify movement in response to heterogeneous resource distribution are scale-dependent. Thus, controlling for spatial scales of interaction with environment is critical for a better understanding of habitat selection, which is likely to follow scale-dependent processes. 2. Here we study the scales of interactions and habitat selection in a long-ranging marine predator foraging from a central place, the yellow-nosed albatross. We use first-passage time analysis to identify the scales of interaction with environmental variables and compositional analysis to study habitat selection. 3. Of 26 birds, 22 adopted an area restricted search (ARS) at a scale of 130 ± 85 km, and 11 of these 22 birds adopted a second, nested ARS scale at 34 ± 20 km. Habitat use differed according to the spatial scale considered. At the oceanic basin macro-scale, birds foraged in pelagic, subtropical waters. Birds commuted to the ARS zones after a c. 1500-km trip to reach predictable turbulence zones from Agulhas return current, where primary productivity was enhanced at large scale. At a smaller, meso-scale, birds increased their search effort according to sea surface height anomalies (SSHa) and chlorophyll-a concentrations (Chl-a), indicating association with productive cyclonic eddies. 4. Among birds, differences in search pattern were noted: 11 birds concentrated their search effort directly at a small scale of 77 ± 22 km, avoiding anticyclonic eddies. The 11 other birds showed two scales of ARS pattern: (i) first at 180 ± 90 km with a preference for high Chl-a concentrations but unrelated to SSHa; and (ii) secondly at a nested scale at 34 ± 20 km related exclusively to SSHa where prey patches were expected to be distributed at this scale. This second group of birds appeared to be less efficient, spending more time at sea for the same mass gain than the first group. 5. Our study is the first to demonstrate scale-dependent adjustments, with interindividual variability, in relation to environmental features for predators with a central-place constraint.