Diffusion about the mean drift location in a biased random walk

Random walks are used to model movement in a wide variety of contexts: from the movement of cells undergoing chemotaxis to the migration of animals. In a two-dimensional biased random walk, the diffusion about the mean drift position is entirely dependent on the moments of the angular distribution used to determine the movement direction at each step. Here we consider biased random walks using several different angular distributions and derive expressions for the diffusion coefficients in each direction based on either a fixed or variable movement speed, and we use these to generate a probability density function for the long-time spatial distribution. We demonstrate how diffusion is typically anisotropic around the mean drift position and illustrate these theoretical results using computer simulations. We relate these results to earlier studies of swimming microorganisms and explain how the results can be generalized to other types of animal movement.     

Detecting Lévy walks without turn designation

Many organisms have been reported to have movement patterns that are well approximated as Lévy walks. This is typically because distributions of straight line distances between consecutive significant turns in movement paths have heavy power law tails. This diagnostic tool has been called into question because there is currently no standard, unambiguous way to identify significant turns. Even if such a way could be found, statistical analyses based on significant turns cannot account for actual movements made between turns and as a consequence cannot distinguish between true Lévy walks and other fractal random walks such as Lévy modulated correlated random walks where organisms randomly meander rather than move in straight lines between consecutive reorientation events. Here, I show that structure functions (i.e. moments of net displacements made across fixed time intervals) can distinguish between different kinds of Lévy walks and between Lévy walks and random walks with a few scales such as composite correlated random walks and correlated random walks. Distinguishing between these processes will lead to a better understanding of how and why animals perform Lévy walks and help bridge the apparent divide between correlated random walks and Lévy walks. Structure functions do not require turn identification and instead take account of entire movement paths. Using this diagnostic tool, I bolster previous claims that honeybees use a movement strategy that can be approximated by Lévy walks when searching for their hive. I also show how structure functions can be used to establish the extent of self-similar behaviour in meandering Lévy walks.     

Distinguishing between Lévy walks and strong alternative models

Lévy walks are a widely used but contentious model of animal movement patterns. They are contentious because they have been wrongly ascribed to some animal species through use of incorrect statistical methods and because they have not been adequately compared against strong alternative models, such as composite correlated random walks. This lack of comparison has been partly because the strong alternative models do not have simple likelihood functions. Here I show that power-spectra and the distribution of the first significant digits (the leading non-zero digits) of the step lengths can distinguish between Lévy walks and composite correlated random walks. Using these diagnostic tools, I bolster previous claims that honey bees use a movement strategy that can be approximated by Lévy walks when searching for their hive or for a food source.     

How many animals really do the Lévy walk?

Lévy walks (LW) are superdiffusive and scale-free random walks that have recently emerged as a new conceptual tool for modeling animal search paths. They have been claimed to be more efficient than the "classical" random walks, and they also seem able to account for the actual search patterns of various species. This suggests that many animals may move using a LW process. LW patterns look like the actual search patterns displayed by animals foraging in a patchy environment, where extensive and intensive searching modes alternate, and which can be generated by a mixture of classical random walks. In this context, even elementary composite Brownian walks are more efficient than LW but may be confounded with them because they present apparent move-length-heavy tail distributions and superdiffusivity. The move-length "survival" distribution (i.e., the cumulative number of moves greater than any given threshold) appears to be a better means to highlight a LW pattern. Even once such a pattern has been clearly identified, it remains to determine how it was actually generated, because a LW pattern is not necessarily produced by a LW process but may emerge from the way the animal interacted with the environment structure through more classical movement processes. In any case, emergent movement patterns should not be confused with the processes that gave rise to them.     

Fitting animal survival models with temporal random effects

Estimating temporal variance in animal demographic parameters is of particular importance in population biology. We implement the Schall’s algorithm for incorporating temporal random effects in survival models using recovery data. Our frequentist approach is based on a formulation of band-recovery models with random effects as generalized linear mixed models and a linearization of the link function conditional on the random effects. A simulation study shows that our procedure provides unbiased and precise estimates. The method is then implemented on two case studies using recovery data on fish and birds.     

Gyrinid searching tactics: empirical observations and a tactical model

Summary Data on Gyrinus picipes were collected to examine the influence of prey distribution and capture on search path configuration. All paths were analyzed before and after prey capture. Parameters measured were turning angles, step lengths, and step direction. Three indices of search efficiency were calculated: linear displacement, thoroughness, and straightness. Prey distribution had little impact on search configuration. However, prey capture caused significant reductions in step length, linear displacement, and straightness. Thoroughnnes and turning angle increased significantly after prey capture.A simulation model was developed to analyze influences of search parameters on search efficiency. Search paths, simulated using empirically derived parameters, did not differ from actual search paths indicating that the model accurately describes gyrinid search paths and may be used in these analyses. Actual search paths were compared to a simulated Brownian (random) search. Search paths before prey capture were significantly different from random, but Brownian search can describe search paths after prey capture. Comparisons of simulations using various input parameters indicate that step length has the greatest effect on search paths. Turning angles and direction are important before prey capture but not after. Offprint requests to: D.L. Winkelman at his current address     

Estimator bias and efficiency for adaptive cluster sampling with order statistics and a stopping rule

Practical problems facing adaptive cluster sampling with order statistics (acsord) are explored using Monte Carlo simulation for three simulated fish populations and two known waterfowl populations. First, properties of an unbiased Hansen-Hurwitz (HH) estimator and a biased alternative Horvitz-Thompson (HT) estimator are evaluated. An increase in the level of population aggregation or the initial sample size increases the efficiencies of the two acsord estimators. For less aggregated fish populations, the efficiencies decrease as the order statistic parameter r (the number of units about which adaptive sampling is carried out) increases; for the highly aggregated fish and waterfowl populations, they increase with r. Acsord is almost always more efficient than simple random sampling for the highly aggregated populations. Positive bias is observed for the HT estimator, with the maximum bias usually occurring at small values of r. Secondly, a stopping rule at the Sth iteration of adaptive sampling beyond the initial sampling unit was applied to the acsord design to limit the otherwise open-ended sampling effort. The stopping rule induces relatively high positive bias to the HH estimator if the level of the population aggregation is high, the stopping level S is small, and r is large. The bias of HT is not very sensitive to the stopping rule and its bias is often reduced by the stopping rule at smaller values of r. For more aggregated populations, the stopping rule often reduces the efficiencies of the estimators compared to the non-stopping-rule scheme, but acsord still remains more efficient than simple random sampling. Despite its bias and lack of theoretical grounding, the HT estimator is usually more efficient than the HH estimator. In the stopping rule case, the HT estimator is preferable, because its bias is less sensitive to the stopping level.     

Site-selection bias and apparent population declines in long-term studies

Detecting population declines is a critical task for conservation biology. Logistical difficulties and the spatiotemporal variability of populations make estimation of population declines difficult. For statistical reasons, estimates of population decline may be biased when study sites are chosen based on abundance of the focal species. In this situation, apparent population declines are likely to be detected even if there is no decline. This site-selection bias is mentioned in the literature but is not well known. We used simulations and real population data to examine the effects of site-selection biases on inferences about population trends. We used a left-censoring method to detect population-size patterns consistent with site-selection bias. The site-selection bias is an important consideration for conservation biologists, and we offer suggestions for minimizing or mitigating it in study design and analysis. Article impact statement: Estimates of population declines are biased if studies begin in large populations, and time-series data show a signature of such an effect.     

Predicting [corrected] extinction risk in spite of predator-prey oscillations.

Most population viability analyses (PVA) assume that the effects of species interactions are subsumed by population-level parameters. We examine how robust five commonly used PVA models are to violations of this assumption. We develop a stochastic, stage-structured predator-prey model and simulate prey population vital rates and abundance. We then use simulated data to parameterize and estimate risk for three demographic models (static projection matrix, stochastic projection matrix, stochastic vital rate matrix) and two time series models (diffusion approximation [DA], corrupted diffusion approximation [CDA]). Model bias is measured as the absolute deviation between estimated and observed quasi-extinction risk. Our results highlight three generalities about the application of single-species models to multi-species conservation problems. First, our collective model results suggest that most single-species PVA models overestimate extinction risk when species interactions cause periodic variation in abundance. Second, the DA model produces the most (conservatively) biased risk forecasts. Finally, the CDA model is the most robust PVA to population cycles caused by species interactions. CDA models produce virtually unbiased and relatively precise risk estimates even when populations cycle strongly. High performance of simple time series models like the CDA owes to their ability to effectively partition stochastic and deterministic sources of variation in population abundance.     

Magnetic compass orientation in the yellow-faced honeyeater,Lichenostomus chrysops,a day migrating bird from Australia

Summary In Australia, the southern populations of the yellow-faced honeyeater, Lichenostomus chrysops (Meliphagidae), perform annual migrations, with routes following the eastern coastline. In order to assess the role of magnetic cues in the migratory orientation of this diurnal migrant, its directional behaviour was recorded in recording cages under natural and experimentally manipulated magnetic-field conditions. During autumn the birds tested indoors in the local geomagnetic field showed a directional change from north initially to northwest later in the season (Fig. 1 a, b), which corresponds well with the general pattern of movement of this species in the field. Deflecting magnetic north to ESE resulted in a clockwise shift of the mean direction by 77° and 71°, respectively (Fig. 1 c, d), while no significant directional tendencies were observed in a magnetic field with a compensated horizontal component (Fig. 1 e, f; see Table 1). In outdoor tests in spring, the birds preferred southerly directions when tested in the local geo-magnetic field. In a magnetic field with a reversed vertical component (i.e. with an inclination pointing down instead of upwards) the birds reversed their directional tendencies and oriented northward (Fig. 2, Table 2). These results clearly show: (1) that yellow-faced honeyeaters can use the magnetic field for direction finding, and (2) that their magnetic compass functions as an inclination compass, as has been shown for several holarctic migrants.Correspondence to: W. Wiltschko     

Estimation of population parameters from aerial counts of North American mallards: a cautionary tale.

The accuracy of population estimates strongly interferes with our ability to obtain unbiased estimates of population parameters based on analyses of time series of population fluctuations. Here we use long-term data on fluctuations in the size of Mallard populations collected as part of the May Breeding Waterfowl Survey covering a large section of North America. We assume a log-linear model of density dependence and use a hierarchical Bayesian state-space approach in which all parameters are assumed to be realizations from a common underlying distribution. Thus, parameters for different populations are not allowed to vary independently of each other. We then simulated independent time series of aerial counts, using the estimated parameters and adding various levels of observation error. These simulations showed that the estimates of stochastic population growth rate and strength of density dependence were biased even when moderate sampling errors were present. In contrast, the estimates of the environmental stochasticity and the carrying capacity were unbiased even for short time series and large observation error. Our results underline the importance of reducing the magnitude of sampling error in the design of large-scale monitoring programs of population fluctuations.     

The Reliability of Using Population Viability Analysis for Risk Classification of Species

I examine whether or not it is appropriate to use extinction probabilities generated by population viability analyses, based on best estimates for model parameters, as criteria for listing species in Red Data Book categories as recently proposed by the World Conservation Union. Such extinction probabilities are influenced by how accurately model parameters are estimated and by how accurately the models depict actual population dynamics. I evaluate the effect of uncertainty in parameter estimation through simulations. Simulations based on Steller sea lions were used to evaluate bias and precision in estimates of probability of extinction and to consider the performance of two proposed classification schemes. Extinction time estimates were biased (because of violation of the assumption of stable age distribution) and underestimated the variability of probability of extinction for a given time (primarily because of uncertainty in parameter estimation). Bias and precision in extinction probabilities are important when these probabilities are used to compare the risk of extinction between species. Suggestions are given for population viability analysis techniques that incorporate parameter uncertainty. I conclude that testing classification schemes with simulations using quantitative performance objectives should precede adoption of quantitative listing criteria.     

Offspring sex ratios reflect lack of repayment by auxiliary males in a cooperatively breeding passerine

The repayment hypothesis posits that primary sex ratios in cooperative species should be biased towards the helping sex because these offspring “repay” a portion of their cost through helping behavior and therefore are less expensive to produce. However, many cooperatively breeding birds and mammals do not show the predicted bias in the primary sex ratio. Recent theoretical work has suggested that the repayment hypothesis should only hold when females gain a large fitness advantage from the presence of auxiliary adults in the group. When auxiliaries provide little or no fitness advantage, competition between relatives should lead to sex ratios biased towards the dispersing (non-helping) sex. We examined the benefits auxiliaries provide to females and corresponding offspring sex ratios in the red-backed fairy-wren (Malurus melanocephalus), a cooperatively breeding Australian bird with male auxiliary helpers. We found that auxiliaries provide little or no benefit to female reproductive success or survival. As predicted, the population primary sex ratio was biased towards daughters, the dispersing sex, and females with auxiliaries produced female-biased broods whereas females without auxiliaries produced unbiased broods. Moreover, offspring sex ratios were more strongly biased toward females in years when auxiliaries were more common in the population. These results suggest that offspring sex ratios are associated with competition among the non-dispersing sex in this species, and also that females may use cues to assess local breeding opportunities for their offspring.     

Interactive effects of disturbance and dispersal directionality on species richness and composition in metacommunities

Dispersal among ecological communities is usually assumed to be random in direction, or to vary in distance or frequency among species. However, a variety of natural systems and types of organisms may experience dispersal that is biased by directional currents or by gravity on hillslopes. We developed a general model for competing species in metacommunities to evaluate the role of directionally biased dispersal on species diversity, abundance, and traits. In parallel, we tested the role of directionally biased dispersal on communities in a microcosm experiment with protists and rotifers. Both the model and experiment independently demonstrated that diversity in local communities was reduced by directionally biased dispersal, especially dispersal that was biased away from disturbed patches. Abundance of species (and composition) in local communities was a product of disturbance intensity but not dispersal directionality. High disturbance selected for species with high intrinsic growth rates and low competitive abilities. Overall, our conclusions about the key role of dispersal directionality in (meta)communities seem robust and general, since they were supported both by the model, which was set in a general framework and not parameterized to fit to a specific system, and by a specific experimental test with microcosms.     

Analysis of ecosystem structure and function: extended path and flow analysis of a steady-state oyster reef model

An analysis of the extended path and flow structure of a six compartment steady-state oyster reef model was conducted. The extended path and flow structure were analyzed in the context of a refined canonical path classification system based on the systems theory methods of environ and network unfolding analyses. A computer implementation of an operational path classification system facilitated investigation of a finite portion (path length ≤17 arcs) of the direct and indirect path structure of the oyster reef model. Important results of the path structure analysis include: (1) few simple paths and large numbers of compound paths enumerated; (2) dominance of path numbers by subsequent passage terminal cycle paths; (3) structural evidence in support of feedback control in ecosystems; (4) results provide evidence by analogy to support the hypothesis of network homogenization first described using the systems analysis methods of environ analysis and network unfolding; (5) constancy of the pattern of origin–destination path counts with increasing path length; (6) importance of nonliving compartments in the extended path structure of ecosystems. Simultaneous path and flow analysis of the oyster reef model assessed the flow contributions of the fundamental path categories for this model using a modification of a path-based network unfolding method. First passage paths contribute most of the flow; however, multiple passage cyclic paths also provide a large (22%) flow contribution. Because of cycling in the system, the numerous long paths in the extended path structure of this ecosystem model are significant in its function as represented by the flows. These results provide microscopic evidence for the macroscopic results of environ analysis that implicate cycling as a key ecosystem attribute in the mechanisms of holistic system determination. The principles enunciated here for a model with a low cycling index (11%) carry over to, and would be even more significant for, models with high cycling indexes. These results also serve to form a link between the extended structure of food webs and their functioning as represented by energy-matter flows. The present analysis demonstrates that extended path structure, and the component articulation from which it is generated, have significant consequences for ecosystem function.     

Old models explain new observations of butterfly movement at patch edges

Understanding movement in heterogeneous environments is central to predicting how landscape changes affect animal populations. Several recent studies point out an intriguing and distinctive looping behavior by butterflies at habitat patch edges and hypothesize that this behavior requires a new framework for analyzing animal movement. We show that this looping behavior could be caused by a longstanding movement model, biased correlated random walk, with bias toward habitat patches. The ability of this longstanding model to explain recent observations reinforces the point that butterflies respond to habitat heterogeneity and do not move randomly through heterogeneous environments. We discuss the implications of different movement models for predicting butterfly responses to landscape change, and our rationale for retaining longstanding movement models, rather than developing new modeling frameworks for looping behavior at patch edges.     

Spatial methods for plot-based sampling of wildlife populations

Classical sampling methods can be used to estimate the mean of a finite or infinite population. Block kriging also estimates the mean, but of an infinite population in a continuous spatial domain. In this paper, I consider a finite population version of block kriging (FPBK) for plot-based sampling. The data are assumed to come from a spatial stochastic process. Minimizing mean-squared-prediction errors yields best linear unbiased predictions that are a finite population version of block kriging. FPBK has versions comparable to simple random sampling and stratified sampling, and includes the general linear model. This method has been tested for several years for moose surveys in Alaska, and an example is given where results are compared to stratified random sampling. In general, assuming a spatial model gives three main advantages over classical sampling: (1) FPBK is usually more precise than simple or stratified random sampling, (2) FPBK allows small area estimation, and (3) FPBK allows nonrandom sampling designs.     

A Voronoi diagram based population model for social species of wildlife

A population model is presented that accounts for spatial structure within habitat patches. It is designed for social species of wildlife that form social group home ranges that are much smaller than patch size. The model represents social group home ranges by Voronoi regions that tessellate a patch to form a Voronoi diagram. Neighbouring social groups are linked with habitat-confined shortest paths and form a dispersal network. The model simulates population dynamics and makes use of Voronoi diagrams and dispersal networks as a spatial component. It then produces density maps as outputs. These are maps that show predicted animal densities across the patches of a landscape. A construction procedure for the particular Voronoi diagram type used by the model is described. As a test case, the model is run for the squirrel glider (Petaurus norfolcensis), a small arboreal marsupial native to Australia. A time series of density maps are produced that show squirrel glider density changing across a landscape through time.     

Corticosterone levels in relation to migratory readiness in red-eyed vireos (<Emphasis Type="Italic">Vireo olivaceus</Emphasis>)

We examined the relationship between plasma levels of corticosterone and the migratory activity and directional preference of red-eyed vireos during fall migration at the northern coast of the Gulf of Mexico. Corticosterone is thought to play a role in physiological and behavioural processes before, during, and after long-distance migratory flights. An increase in corticosterone at the onset of migratory flights can be expected in birds that are energetically prepared to migrate in a seasonally appropriate southerly direction. Red-eyed vireos ( Vireo olivaceus) were tested in orientation cages under clear twilight skies. Just prior to the orientation experiments, blood was sampled to assay baseline corticosterone levels. Average corticosterone level for all birds was 22.8 ng/ml. Red-eyed vireos with higher than average baseline levels of corticosterone were significantly more active in orientation cages compared to birds with lower levels of corticosterone. Moreover, birds with higher than average levels oriented in a southwesterly direction, which is consistent with a trans-Gulf flight, whereas individuals with levels below average showed a NNW mean direction. Although there was no significant difference in baseline levels of corticosterone between fat and lean birds, individual mass loss between capture and test was negatively correlated with corticosterone levels. Results from this study clearly demonstrate that corticosterone influences departure decisions and the choice of direction during migration.     

Group navigation and the "many-wrongs principle" in models of animal movement

Traditional studies of animal navigation over both long and short distances have usually considered the orientation ability of the individual only, without reference to the implications of group membership. However, recent work has suggested that being in a group can significantly improve the ability of an individual to align toward and reach a target direction or point, even when all group members have limited navigational ability and there are no leaders. This effect is known as the "many-wrongs principle" since the large number of individual navigational errors across the group are suppressed by interactions and group cohesion. In this paper, we simulate the many-wrongs principle using a simple individual-based model of movement based on a biased random walk that includes group interactions. We study the ability of the group as a whole to reach a target given different levels of individual navigation error, group size, interaction radius, and environmental turbulence. In scenarios with low levels of environmental turbulence, simulation results demonstrate a navigational benefit from group membership, particularly for small group sizes. In contrast, when movement takes place in a highly turbulent environment, simulation results suggest that the best strategy is to navigate as individuals rather than as a group.