The influence of turning angles on the success of non-oriented animal searches

Animal searches cover a full range of possibilities from highly deterministic to apparently completely random behaviors. However, even those stochastic components of animal movement can be adaptive, since not all random distributions lead to similar success in finding targets. Here we address the general problem of optimizing encounter rates in non-deterministic, non-oriented searches, both in homogeneous and patchy target landscapes. Specifically, we investigate how two different features related to turning angle distributions influence encounter success: (i) the shape (relative kurtosis) of the angular distribution and (ii) the correlations between successive relative orientations (directional memory). Such influence is analyzed in correlated random walk models using a proper choice of representative turning angle distributions of the recently proposed Jones and Pewsey class. We consider the cases of distributions with nearly the same shape but considerably distinct correlation lengths, and distributions with same correlation but with contrasting relative kurtosis. In homogeneous landscapes, we find that the correlation length has a large influence in the search efficiency. Moreover, similar search efficiencies can be reached by means of distinctly shaped turning angle distributions, provided that the resulting correlation length is the same. In contrast, in patchy landscapes the particular shape of the distribution also becomes relevant for the search efficiency, specially at high target densities. Excessively sharp distributions generate very inefficient searches in landscapes where local target density fluctuations are large. These results are of evolutionary interest. On the one hand, it is shown that equally successful directional memory can arise from contrasting turning behaviors, therefore increasing the likelihood of robust adaptive stochastic behavior. On the other hand, when target landscape is patchy, adequate tumbling may help to explore better local scale heterogeneities, being some details of the shape of the distribution also potentially adaptive.     

Modelling nematode movement using time-fractional dynamics

We use a correlated random walk model in two dimensions to simulate the movement of the slug parasitic nematode Phasmarhabditis hermaphrodita in homogeneous environments. The model incorporates the observed statistical distributions of turning angle and speed derived from time-lapse studies of individual nematode trails. We identify strong temporal correlations between the turning angles and speed that preclude the case of a simple random walk in which successive steps are independent. These correlated random walks are appropriately modelled using an anomalous diffusion model, more precisely using a fractional sub-diffusion model for which the associated stochastic process is characterised by strong memory effects in the probability density function.     

Turn costs change the value of animal search paths

The tortuosity of the track taken by an animal searching for food profoundly affects search efficiency, which should be optimised to maximise net energy gain. Models examining this generally describe movement as a series of straight steps interspaced by turns, and implicitly assume no turn costs. We used both empirical‐ and modelling‐based approaches to show that the energetic costs for turns in both terrestrial and aerial locomotion are substantial, which calls into question the value of conventional movement models such as correlated random walk or Lévy walk for assessing optimum path types. We show how, because straight‐line travel is energetically most efficient, search strategies should favour constrained turn angles, with uninformed foragers continuing in straight lines unless the potential benefits of turning offset the cost.     

Sampling rate effects on measurements of correlated and biased random walks

When observing the two-dimensional movement of animals or microorganisms, it is usually necessary to impose a fixed sampling rate, so that observations are made at certain fixed intervals of time and the trajectory is split into a set of discrete steps. A sampling rate that is too small will result in information about the original path and correlation being lost. If random walk models are to be used to predict movement patterns or to estimate parameters to be used in continuum models, then it is essential to be able to quantify and understand the effect of the sampling rate imposed by the observer on real trajectories. We use a velocity jump process with a realistic reorientation model to simulate correlated and biased random walks and investigate the effect of sampling rate on the observed angular deviation, apparent speed and mean turning angle. We discuss a method of estimating the values of the reorientation parameters used in the original random walk from the rediscretized data that assumes a linear relation between sampling time step and the parameter values.     

A model-independent test for the presence of regulatory equilibrium and non-random structure in island species trajectories

Aims To test a key prevision of the dynamic equilibrium theory of island biogeography, namely that changes in species numbers on islands over time (hereafter, species trajectories) are equilibrial, and to characterize aspects of the dynamical properties of species change over time using a model‐independent test. Methods We tested for regulatory equilibrium and non‐random structure in species numbers through time by comparing observed correlation coefficients at lag‐k for species trajectories from four true islands and two habitat islands. First, we estimated the shape of the autocorrelation function for each observed species trajectory by calculating correlation coefficients of the observed data between pairs of values N_t?k and N_t separated by lag‐k (k = 1, 2, …, N ? 1). Second, we tested the observed correlation coefficients at each lag against a distribution of correlation coefficients generated by randomly ordering observed numbers in the species trajectories. Results The patterns of autocorrelation functions for all but one of the observed species trajectories did not exhibit evidence of regulatory equilibrium, and, in fact, closely matched what would be expected from a non‐stationary or ‘random walk’ process. The majority of the correlation coefficients generated from the observed species trajectories did not deviate significantly from correlation coefficients produced by the randomized trajectories. However, there was strong evidence of unusual positive autocorrelation at small time lags for birds on islands measured annually (2‐ to 4‐year lags) and for arthropods on islands measured weekly (7‐ to 8‐week lags), suggesting some degree of structure in change in species richness over time. Main conclusions The autocorrelation function patterns for all but one of the observed species trajectories showed various forms of non‐stationarity. These types of patterns suggest that the numbers of species through time gradually wandered away from their initial sizes. Our model‐independent test of individual correlation coefficients revealed significant structure in the observed species trajectories. These trajectories appear to be non‐random at relatively short lag intervals, indicating a process with short memory.     

From random walks to informed movement

The analysis of animal movement is a large and continuously growing field of research. Detailed knowledge about movement strategies is of crucial importance for understanding eco‐evolutionary dynamics at all scales – from individuals to (meta‐)populations. This and the availability of detailed movement and dispersal data motivated Nathan and colleagues to published their much appreciated call to base movement ecology on a more thorough mechanistic basis. So far, most movement models are based on random walks. However, even if a random walk might describe real movement patterns acceptably well, there is no reason to assume that animals move randomly. Therefore, mechanistic models of foraging strategies should be based on information use and memory in order to increase our understanding of the processes that lead to animal movement decisions. We present a mechanistic movement model of an animal with a limited perceptual range and basic information storage capacities. This ‘spatially informed forager’ constructs an internal map of its environment by using perception, memory and learned or evolutionarily acquired assumptions about landscape attributes. We analyse resulting movement patterns and search efficiencies and compare them to area restricted search strategies (ARS) and biased correlated random walks (BCRW) of omniscient individuals. We show that, in spite of their limited perceptual range, spatially informed individuals boost their foraging success and may perform much better than the best ARS. The construction of an internal map and the use of spatial information results in the emergence of a highly correlated walk between patches and a rather systematic search within resource clusters. Furthermore, the resulting movement patterns may include foray search behaviour. Our work highlights the strength of mechanistic modelling approaches and sets the stage for the development of more sophisticated models of memory use for movement decisions and dispersal.     

Flow- and substratum-mediated movement by a stream insect

1. In streams subject to frequent hydrologic disturbance, the ability of benthic invertebrates to disperse within the channel is key to understanding the mechanisms of flow refugium use and population persistence. This study focuses on crawling invertebrates, the effects on movement of abiotic factors (specifically, flow near the stream bed and bed micro‐topography) and the consequences for dispersal. 2. In a large flume, we observed individual cased caddisflies, Potamophylax latipennis, moving in fully turbulent flows over a precise replica of a water‐worked surface. From maps of movement paths, we quantified crawling behaviour and entrainment, and the influence of bed micro‐topography. We manipulated discharge and tested its effect on movement, linear displacement and areal dispersal. The highest discharge treatment was a disturbance to the caddis; the lowest discharge was not. Crawling behaviours were used to parameterise random walk models and estimate population dispersal, and to test the effects of abiotic factors on movement. 3. Bed micro‐topography influenced crawling in several ways. Caddis spent most of their time at the junctions between proud particles and the adjacent plane bed. The frequency distribution of turn angles was bimodal, with modal values approximating the angle required to travel around median‐sized particles. Larvae generally crawled downstream, but crawling direction relative to the flow was skewed by bed micro‐topography and was not directly downstream, unlike drift. 4. Caddis crawled for most of the time and discharge affected almost every aspect of their movement. As discharge increased, caddis crawled less often, more slowly and over shorter distances; they also became entrained more frequently and over greater distances. With increased discharge, caddis spent proportionately less time at the junctions between proud particles and the adjacent plane bed, and more time on the tops and sides of proud clasts. This is curious as most entrainment occurred from the tops and sides of clasts and entrainment is generally considered to be disadvantageous during disturbances. 5. Linear displacement (drift and entrainment combined) was downstream, but the relation between total displacement and discharge was complex. Total displacement decreased at intermediate discharge as crawling decreased, but increased at high discharge as entrainment and drift played a greater role in movement. 6. Within‐stream dispersal via crawling contained elements of both a correlated random walk (we observed directional persistence in turn angles) and a biased random walk (we observed downstream bias in move direction angles) and was best described as a biased correlated random walk. Dispersal was inversely related to discharge, suggesting that the ability of P. latipennis to crawl into flow refugia on the streambed is reduced at high flow.     

Analyzing fish movement as a persistent turning walker

The trajectories of Kuhlia mugil fish swimming freely in a tank are analyzed in order to develop a model of spontaneous fish movement. The data show that K. mugil displacement is best described by turning speed and its auto-correlation. The continuous-time process governing this new kind of displacement is modelled by a stochastic differential equation of Ornstein–Uhlenbeck family: the persistent turning walker. The associated diffusive dynamics are compared to the standard persistent random walker model and we show that the resulting diffusion coefficient scales non-linearly with linear swimming speed. In order to illustrate how interactions with other fish or the environment can be added to this spontaneous movement model we quantify the effect of tank walls on the turning speed and adequately reproduce the characteristics of the observed fish trajectories.     

Comparative analysis of leaf angle and sclerophylly of Aspidosperma quebracho‐blanco on a water deficit gradient

Abstract Aspidosperma quebracho‐blanco is found throughout the Chaco (17°?33°S) in Argentina, and it is the dominant tree species in the arid Chaco. Under the hypothesis that morpho‐physiological features of A. quebracho‐blanco change as a function of its geographical position on a water deficit gradient, it was predicted that with increasing water stress, leaf angles (specifically horizontal) would be greater and mean values of the leaf mass per area would increase. These leaf characteristics were compared at three points on a water deficit gradient extending from the humid Chaco through semi‐arid Chaco to the arid Chaco of Argentina (south‐west to north‐east rainfall gradient, from 350 to 1200 mm annual mean precipitation). Twig and leaf positions were modified and water potentials were measured at the highest heating hour of the day at a site of the arid Chaco. Daily and seasonal water potential variations of untreated twigs were also observed. Leaf angle modification towards horizontal produced more negative twig water potentials with respect to those of leaves in non‐horizontal positions. The comparison of the three sites along the gradient showed contrasting patterns of leaf‐angle frequency distribution of adults. In Chancaní (mean annual temperature: 18–24°C, mean annual precipitation: 450 mm, arid) there was a higher frequency of angles near 90° for non‐pendulous and about 270° for pendulous trees. Leaf angles in Copo (semi‐arid) and Chaco National Park (mean annual temperature: 20–23°C, mean annual precipitation: 1300 mm humid) were widely distributed with higher frequency towards the angles near 0° and 180°. This sclerophyllous tree species showed plasticity in its leaf traits along the precipitation gradient. Plasticity in leaf mass per area and leaf position enables plants to develop efficiently in contrasting environmental conditions of humidity and aridity.     

Brief Communication: Lumbar lordosis in extinct hominins: Implications of the pelvic incidence

Recently, interest has peaked regarding the posture of extinct hominins. Here, we present a new method of reconstructing lordosis angles of extinct hominin specimens based on pelvic morphology, more specifically the orientation of the sacrum in relation to the acetabulum (pelvic incidence). Two regression models based on the correlation between pelvic incidence and lordosis angle in living hominoids have been developed. The mean values of the calculated lordosis angles based on these models are 36°?45° for australopithecines, 45°?47° for Homo erectus, 27°?34° for the Neandertals and the Sima de los Huesos hominins, and 49°?51° for fossil H. sapiens. The newly calculated lordosis values are consistent with previously published values of extinct hominins (Been et al.: Am J Phys Anthropol 147 (2012) 64–77). If the mean values of the present nonhuman hominoids are representative of the pelvic and lumbar morphology of the last common ancestor between humans and nonhuman hominoids, then both pelvic incidence and lordosis angle dramatically increased during hominin evolution from 27° ± 5 to 22° ± 3 (respectively) in nonhuman hominoids to 54° ± 10 and 51° ± 11 in modern humans. This change to a more human‐like configuration appeared early in the hominin evolution as the pelvis and spines of both australopithecines and H. erectus show a higher pelvic incidence and lordosis angle than nonhuman hominoids. The Sima de los Huesos hominins and Neandertals show a derived configuration with a low pelvic incidence and lordosis angle. Am J Phys Anthropol 154:307–314, 2014. © 2014 Wiley Periodicals, Inc.     

Stochastic epidemics: the probability of extinction of an infectious disease at the end of a major outbreak

 The aim of this study is to derive an asymptotic expression for the probability that an infectious disease will disappear from a population at the end of a major outbreak (‘fade-out’). The study deals with a stochastic SIR-model. Local asymptotic expansions are constructed for the deterministic trajectories of the corresponding deterministic system, in particular for the deterministic trajectory starting in the saddle point. The analytical expression for the probability of extinction is derived by asymptotically solving a boundary value problem based on the Fokker-Planck equation for the stochastic system. The asymptotic results are compared with results obtained by random walk simulations. Received 20 July 1995; received in revised form 6 May 1996     

The role of spatial scale and the perception of large-scale species-richness patterns

Despite two centuries of exploration, our understanding of factors determining the distribution of life on Earth is in many ways still in its infancy. Much of the disagreement about governing processes of variation in species richness may be the result of differences in our perception of species‐richness patterns. Until recently, most studies of large‐scale species‐richness patterns assumed implicitly that patterns and mechanisms were scale invariant. Illustrated with examples and a quantitative analysis of published data on altitudinal gradients of species richness (n = 204), this review discusses how scale effects (extent and grain size) can influence our perception of patterns and processes. For example, a hump‐shaped altitudinal species‐richness pattern is the most typical (c. 50%), with a monotonic decreasing pattern (c. 25%) also frequently reported, but the relative distribution of patterns changes readily with spatial grain and extent. If we are to attribute relative impact to various factors influencing species richness and distribution and to decide at which point along a spatial and temporal continuum they act, we should not ask only how results vary as a function of scale but also search for consistent patterns in these scale effects. The review concludes with suggestions of potential routes for future analytical exploration of species‐richness patterns.     

Accounting for spatial autocorrelation in null models of tree species association

A commonly used null model for species association among forest trees is a well‐mixed community (WMC). A WMC represents a non‐spatial, or spatially implicit, model, in which species form nearest‐neighbor pairs at a rate equal to the product of their community proportions. WMC models assume that the outcome of random dispersal and demographic processes is complete spatial randomness (CSR) in the species’ spatial distributions. Yet, stochastic dispersal processes often lead to spatial autocorrelation (SAC) in tree species densities, giving rise to clustering, segregation, and other nonrandom patterns. Although methods exist to account for SAC in spatially‐explicit models, its impact on non‐spatial models often remains unaccounted for. To investigate the potential for SAC to bias tests based upon non‐spatial models, we developed a spatially‐heterogeneous (SH) modelling approach that incorporates measured levels of SAC. Using the mapped locations of individuals in a tropical tree community, we tested the hypothesis that the identity of nearest‐neighbors represents a random draw from neighborhood species pools. Correlograms of Moran's I confirmed that, for 50 of 51 dominant species, stem density was significantly autocorrelated over distances ranging from 50 to 200 m. The observed patterns of SAC were consistent with dispersal limitation, with most species occurring in distinct patches. For nearly all of the 106 species in the community, the frequency of pairwise association was statistically indistinguishable from that projected by the null models. However, model comparisons revealed that non‐spatial models more strongly underestimated observed species‐pair frequencies, particularly for conspecific pairs. Overall, the CSR models projected more significant facilitative interactions than did SH models, yielding a more liberal test of niche differences. Our results underscore the importance of accounting for stochastic spatial processes in tests of association, regardless of whether spatial or non‐spatial models are employed.     

Stochastic modelling of animal movement

Modern animal movement modelling derives from two traditions. Lagrangian models, based on random walk behaviour, are useful for multi-step trajectories of single animals. Continuous Eulerian models describe expected behaviour, averaged over stochastic realizations, and are usefully applied to ensembles of individuals. We illustrate three modern research arenas. (i) Models of home-range formation describe the process of an animal ‘settling down’, accomplished by including one or more focal points that attract the animal''s movements. (ii) Memory-based models are used to predict how accumulated experience translates into biased movement choices, employing reinforced random walk behaviour, with previous visitation increasing or decreasing the probability of repetition. (iii) Lévy movement involves a step-length distribution that is over-dispersed, relative to standard probability distributions, and adaptive in exploring new environments or searching for rare targets. Each of these modelling arenas implies more detail in the movement pattern than general models of movement can accommodate, but realistic empiric evaluation of their predictions requires dense locational data, both in time and space, only available with modern GPS telemetry.     

Cell surface hydrophobicity of colistin-susceptible vs resistant Acinetobacter baumannii determined by contact angles: methodological considerations and implications

Contact angle analysis of cell surface hydrophobicity (CSH) describes the tendency of a water droplet to spread across a lawn of filtered bacterial cells. Colistin‐induced disruption of the Gram‐negative outer membrane necessitates hydrophobic contacts with lipopolysaccharide (LPS). We aimed to characterize the CSH of Acinetobacter baumannii using contact angles, to provide insight into the mechanism of colistin resistance. Contact angles were analysed for five paired colistin‐susceptible and resistant Ac. baumannii strains. Drainage of the water droplet through bacterial layers was demonstrated to influence results. Consequently, measurements were performed 0·66 s after droplet deposition. Colistin‐resistant cells exhibited lower contact angles (38·8±2·8–46·8±1·3°) compared with their paired colistin‐susceptible strains (40·7±3·0–48·0±1·4°; anova ; P < 0·05). Contact angles increased at stationary phase (50·3±2·9–61·5±2·5° and 47·4±2·0–50·8±3·2°, susceptible and resistant, respectively, anova ; P < 0·05) and in response to colistin 32 mg l^?1 exposure (44·5±1·5–50·6±2·8° and 43·5±2·2–48·0±2·2°, susceptible and resistant, respectively; anova ; P < 0·05). Analysis of complemented strains constructed with an intact lpxA gene, or empty vector, highlighted the contribution of LPS to CSH. Compositional outer‐membrane variations likely account for CSH differences between Ac. baumannii phenotypes, which influence the hydrophobic colistin–bacterium interaction. Important insight into the mechanism of colistin resistance has been provided. Greater consideration of contact angle methodology is necessary to ensure accurate analyses are performed.     

Diagnosing and assessing uncertainties of terrestrial ecosystem models in a multimodel ensemble experiment: 1. Primary production

We conducted an ensemble modeling exercise using the Terrestrial Observation and Prediction System (TOPS) to evaluate sources of uncertainty in carbon flux estimates resulting from structural differences among ecosystem models. The experiment ran public‐domain versions of biome‐bgc, lpj, casa , and tops‐bgc over North America at 8 km resolution and for the period of 1982–2006. We developed the Hierarchical Framework for Diagnosing Ecosystem Models (HFDEM) to separate the simulated biogeochemistry into a cascade of three functional tiers and sequentially examine their characteristics in climate (temperature–precipitation) and other spaces. Analysis of the simulated annual gross primary production (GPP) in the climate domain indicates a general agreement among the models, all showing optimal GPP in regions where the relationship between annual average temperature (T, °C) and annual total precipitation (P, mm) is defined by P=50T+500. However, differences in simulated GPP are identified in magnitudes and distribution patterns. For forests, the GPP gradient along P=50T+500 ranges from ～50 g C yr^?1 m^?2 °C^?1 (casa ) to ～125 g C yr^?1 m^?2 °C^?1 (biome‐bgc ) in cold/temperate regions; for nonforests, the diversity among GPP distributions is even larger. Positive linear relationships are found between annual GPP and annual mean leaf area index (LAI) in all models. For biome‐bgc and lpj , such relationships lead to a positive feedback from LAI growth to GPP enhancement. Different approaches to constrain this feedback lead to different sensitivity of the models to disturbances such as fire, which contribute significantly to the diversity in GPP stated above. The ratios between independently simulated NPP and GPP are close to 50% on average; however, their distribution patterns vary significantly between models, reflecting the difficulties in estimating autotrophic respiration across various climate regimes. Although these results are drawn from our experiments with the tested model versions, the developed methodology has potential for other model exercises.     

Global distribution of fin whales Balaenoptera physalus in the post‐whaling era (1980–2012)

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A Model for a Correlated Random Walk Based on the Ordered Extension of Pseudopodia

Cell migration in the absence of external cues is well described by a correlated random walk. Most single cells move by extending protrusions called pseudopodia. To deduce how cells walk, we have analyzed the formation of pseudopodia by Dictyostelium cells. We have observed that the formation of pseudopodia is highly ordered with two types of pseudopodia: First, de novo formation of pseudopodia at random positions on the cell body, and therefore in random directions. Second, pseudopod splitting near the tip of the current pseudopod in alternating right/left directions, leading to a persistent zig-zag trajectory. Here we analyzed the probability frequency distributions of the angles between pseudopodia and used this information to design a stochastic model for cell movement. Monte Carlo simulations show that the critical elements are the ratio of persistent splitting pseudopodia relative to random de novo pseudopodia, the Left/Right alternation, the angle between pseudopodia and the variance of this angle. Experiments confirm predictions of the model, showing reduced persistence in mutants that are defective in pseudopod splitting and in mutants with an irregular cell surface.     

Patch exploitation in two dimensions: from Daphnia to simulated foragers

We explore the variability that animals display in their movement choices as they forage in a finite-sized food patch with a uniform food distribution, and present a framework for how these choices may be adjusted to optimize foraging efficiency. Inspired by experimental studies of the zooplankton Daphnia, we model foraging animals as “agents” moving in two dimensions in repeated and successive sequences of hops, pauses, and turns. For Daphnia and other species, critical movement parameters such as hop lengths, pause times, and turning angles are typically reported as probability density functions. Similarly, the agents in our simulations choose their movement parameters at random from such distributions. Each distribution is defined by a characteristic width, which we interpret as a “noise width,” available to be tuned for increased foraging efficiency. We investigate the sensitivity of the system by measuring the food gathered by the agents as the turning angle and hop length noise widths are varied. In all cases, we find a maximum in food gathered at some particular value of the noise width in question, suggesting that these results can be considered robust examples of natural stochastic resonance.