Plankton motility patterns and encounter rates

Flow disturbances generated by feeding and swimming zooplankton

Interactions between planktonic organisms, such as detection of prey, predators, and mates, are often mediated by fluid signals. Consequently, many plankton predators perceive their prey from the fluid disturbances that it generates when it feeds and swims. Zooplankton should therefore seek to minimize the fluid disturbance that they produce. By means of particle image velocimetry, we describe the fluid disturbances produced by feeding and swimming in zooplankton with diverse propulsion mechanisms and ranging from 10-µm flagellates to greater than millimeter-sized copepods. We show that zooplankton, in which feeding and swimming are separate processes, produce flow disturbances during swimming with a much faster spatial attenuation (velocity u varies with distance r as u ∝ r(-3) to r(-4)) than that produced by zooplankton for which feeding and propulsion are the same process (u ∝ r(-1) to r(-2)). As a result, the spatial extension of the fluid disturbance produced by swimmers is an order of magnitude smaller than that produced by feeders at similar Reynolds numbers. The "quiet" propulsion of swimmers is achieved either through swimming erratically by short-lasting power strokes, generating viscous vortex rings, or by "breast-stroke swimming." Both produce rapidly attenuating flows. The more "noisy" swimming of those that are constrained by a need to simultaneously feed is due to constantly beating flagella or appendages that are positioned either anteriorly or posteriorly on the (cell) body. These patterns transcend differences in size and taxonomy and have thus evolved multiple times, suggesting a strong selective pressure to minimize predation risk.

Analysis of self-overlap reveals trade-offs in plankton swimming trajectories

Movement is a fundamental behaviour of organisms that not only brings about beneficial encounters with resources and mates, but also at the same time exposes the organism to dangerous encounters with predators. The movement patterns adopted by organisms should reflect a balance between these contrasting processes. This trade-off can be hypothesized as being evident in the behaviour of plankton, which inhabit a dilute three-dimensional environment with few refuges or orienting landmarks. We present an analysis of the swimming path geometries based on a volumetric Monte Carlo sampling approach, which is particularly adept at revealing such trade-offs by measuring the self-overlap of the trajectories. Application of this method to experimentally measured trajectories reveals that swimming patterns in copepods are shaped to efficiently explore volumes at small scales, while achieving a large overlap at larger scales. Regularities in the observed trajectories make the transition between these two regimes always sharper than in randomized trajectories or as predicted by random walk theory. Thus, real trajectories present a stronger separation between exploration for food and exposure to predators. The specific scale and features of this transition depend on species, gender and local environmental conditions, pointing at adaptation to state and stage-dependent evolutionary trade-offs.

Co-transport of polycyclic aromatic hydrocarbons by motile microorganisms leads to enhanced mass transfer under diffusive conditions

The environmental chemodynamics of hydrophobic organic chemicals (HOCs) are often rate-limited by diffusion in stagnant boundary layers. This study investigated whether motile microorganisms can act as microbial carriers that enhance mass transfer of HOCs through diffusive boundary layers. A new experimental system was developed that allows (1) generation of concentration gradients of HOCs under the microscope, (2) exposure and direct observation of microorganisms in such gradients, and (3) quantification of HOC mass transfer. Silicone O-rings were integrated into a Dunn chemotaxis chamber to serve as sink and source for polycyclic aromatic hydrocarbons (PAHs). This resulted in stable concentration gradients in water (>24 h). Adding the model organism Tetrahymena pyriformis to the experimental system enhanced PAH mass transfer up to hundred-fold (benzo[a]pyrene). Increasing mass transfer enhancement with hydrophobicity indicated PAH co-transport with the motile organisms. Fluorescence microscopy confirmed such transport. The effective diffusivity of T. pyriformis, determined by video imaging microscopy, was found to exceed molecular diffusivities of the PAHs up to four-fold. Cell-bound PAH fractions were determined to range from 28% (naphthalene) to 92% (pyrene). Motile microorganisms can therefore function as effective carriers for HOCs under diffusive conditions and might significantly enhance mobility and availability of HOCs.

The physics of broadcast spawning in benthic invertebrates

Most benthic invertebrates broadcast their gametes into the sea, whereupon successful fertilization relies on the complex interaction between the physics of the surrounding fluid flow and the biological properties and behavior of eggs and sperm. We present a holistic overview of the impact of instantaneous flow processes on fertilization across a range of scales. At large scales, transport and stirring by the flow control the distribution of gametes. Although mean dilution of gametes by turbulence is deleterious to fertilization, a variety of instantaneous flow phenomena can aggregate gametes before dilution occurs. We argue that these instantaneous flow processes are key to fertilization efficiency. At small scales, sperm motility and taxis enhance contact rates between sperm and chemoattractant-releasing eggs. We argue that sperm motility is a biological adaptation that replaces molecular diffusion in conventional mixing processes and enables gametes to bridge the gap that remains after aggregation by the flow.

Effect of grazing-mediated dimethyl sulfide (DMS) production on the swimming behavior of the copepod Calanus helgolandicus

Chemical interactions play a fundamental role in the ecology of marine foodwebs. Dimethyl sulfide (DMS) is a ubiquitous marine trace gas that acts as a bioactive compound by eliciting foraging behavior in a range of marine taxa including the copepod Temora longicornis. Production of DMS can rapidly increase following microzooplankton grazing on phytoplankton. Here, we investigated whether grazing-induced DMS elicits an increase in foraging behavior in the copepod Calanus helgolandicus. We developed a semi-automated method to quantify the effect of grazing-mediated DMS on the proportion of the time budget tethered females allocate towards slow swimming, typically associated with feeding. The pooled data showed no differences in the proportion of the 25 min time budget allocated towards slow swimming between high (23.6 ± 9.74%) and low (29.1 ± 18.33%) DMS treatments. However, there was a high degree of variability between behavioral responses of individual copepods. We discuss the need for more detailed species-specific studies of individual level responses of copepods to chemical signals at different spatial scales to improve our understanding of chemical interactions between copepods and their prey.

Unexpected Regularity in Swimming Behavior of Clausocalanus furcatus Revealed by a Telecentric 3D Computer Vision System

Planktonic copepods display a large repertoire of motion behaviors in a three-dimensional environment. Two-dimensional video observations demonstrated that the small copepod Clausocalanus furcatus, one the most widely distributed calanoids at low to medium latitudes, presented a unique swimming behavior that was continuous and fast and followed notably convoluted trajectories. Furthermore, previous observations indicated that the motion of C. furcatus resembled a random process. We characterized the swimming behavior of this species in three-dimensional space using a video system equipped with telecentric lenses, which allow tracking of zooplankton without the distortion errors inherent in common lenses. Our observations revealed unexpected regularities in the behavior of C. furcatus that appear primarily in the horizontal plane and could not have been identified in previous observations based on lateral views. Our results indicate that the swimming behavior of C. furcatus is based on a limited repertoire of basic kinematic modules but exhibits greater plasticity than previously thought.

Females roam while males patrol: divergence in breeding season movements of pack-ice polar bears (Ursus maritimus)

Intraspecific differences in movement behaviour reflect different tactics used by individuals or sexes to favour strategies that maximize fitness. We report movement data collected from n = 23 adult male polar bears with novel ear-attached transmitters in two separate pack ice subpopulations over five breeding seasons. We compared movements with n = 26 concurrently tagged adult females, and analysed velocities, movement tortuosity, range sizes and habitat selection with respect to sex, reproductive status and body mass. There were no differences in 4-day displacements or sea ice habitat selection for sex or population. By contrast, adult females in all years and both populations had significantly more linear movements and significantly larger breeding range sizes than males. We hypothesized that differences were related to encounter rates, and used observed movement metrics to parametrize a simulation model of male-male and male-female encounter. The simulation showed that the more tortuous movement of males leads to significantly longer times to male-male encounter, while having little impact on male-female encounter. By contrast, linear movements of females are consistent with a prioritized search for sparsely distributed prey. These results suggest a possible mechanism for explaining the smaller breeding range sizes of some solitary male carnivores compared to females.

Prey detection and prey capture in copepod nauplii

Copepod nauplii are either ambush feeders that feed on motile prey or they produce a feeding current that entrains prey cells. It is unclear how ambush and feeding-current feeding nauplii perceive and capture prey. Attack jumps in ambush feeding nauplii should not be feasible at low Reynolds numbers due to the thick viscous boundary layer surrounding the attacking nauplius. We use high-speed video to describe the detection and capture of phytoplankton prey by the nauplii of two ambush feeding species (Acartia tonsa and Oithona davisae) and by the nauplii of one feeding-current feeding species (Temora longicornis). We demonstrate that the ambush feeders both detect motile prey remotely. Prey detection elicits an attack jump, but the jump is not directly towards the prey, such as has been described for adult copepods. Rather, the nauplius jumps past the prey and sets up an intermittent feeding current that pulls in the prey from behind towards the mouth. The feeding-current feeding nauplius detects prey arriving in the feeding current but only when the prey is intercepted by the setae on the feeding appendages. This elicits an altered motion pattern of the feeding appendages that draws in the prey.

Predator-induced fleeing behaviors in phytoplankton: a new mechanism for harmful algal bloom formation?

In the plankton, heterotrophic microbes encounter and ingest phytoplankton prey, which effectively removes >50% of daily phytoplankton production in the ocean and influences global primary production and biochemical cycling rates. Factors such as size, shape, nutritional value, and presence of chemical deterrents are known to affect predation pressure. Effects of movement behaviors of either predator or prey on predation pressure, and particularly fleeing behaviors in phytoplankton are thus far unknown. Here, we quantified individual 3D movements, population distributions, and survival rates of the toxic phytoplankton species, Heterosigma akashiwo in response to a ciliate predator and predator-derived cues. We observed predator-induced defense behaviors previously unknown for phytoplankton. Modulation of individual phytoplankton movements during and after predator exposure resulted in an effective separation of predator and prey species. The strongest avoidance behaviors were observed when H. akashiwo co-occurred with an actively grazing predator. Predator-induced changes in phytoplankton movements resulted in a reduction in encounter rate and a 3-fold increase in net algal population growth rate. A spatially explicit population model predicted rapid phytoplankton bloom formation only when fleeing behaviors were incorporated. These model predictions reflected field observations of rapid H. akashiwo harmful algal bloom (HAB) formation in the coastal ocean. Our results document a novel behavior in phytoplankton that can significantly reduce predation pressure and suggests a new mechanism for HAB formation. Phytoplankton behaviors that minimize predatory losses, maximize resource acquisition, and alter community composition and distribution patterns could have major implications for our understanding and predictive capacity of marine primary production and biochemical cycling rates.

Seascapes: the world of aquatic organisms as determined by their particulate natures

The interactions between planktonic organisms and their aquatic environment are mediated by diffusive processes on the scale of millimeters and smaller. The uptake of nutrients and food and the release of metabolic products creates localized patchiness that diffusive processes homogenize. Organism size determines, to a large extent, the character of these interactions. This paper builds a framework for considering these interactions, starting with an analysis of the diffusive environment around individual organisms and concluding with an integration over a range of organism sizes to describe aggregated properties. Several examples show the importance of the size-based abundance of organisms and other particles in determining the duration, spatial extent and frequency of environmental cues. One implication of these results is that microbial chemotactic behavior does not help them to find and utilize the majority of nutrient releases.

Intermittency in processing explains the diversity and shape of functional grazing responses

Central to theoretical studies of trophic interactions is the formulation of the consumer response to varying food availability. Response functions, however, are only rarely derived in mechanistic ways. As a consequence, the uncertainty in the functional representation of feeding remains large, as, e.g., evident from the ongoing debate on the usage of Ivlev, or Holling type I, II, and III functions in aquatic ecosystem models. Here, I refer to the work of Sjöberg in Ecol Model 10:215-225 (1980) who proposed to apply elements of the queuing theory developed in operational research to plankton-plankton interactions. Within this frame, food item processing is subdivided into two major stages which may operate with variable synchronicity. Asynchronous phasing of the two stages enhances the probability of long total processing times. This phenomenon is here termed feeding intermittency. Intermittency is assumed to determine the functional form of grazing kinetics, for which a novel grazing function containing a "shape" parameter is derived. Using this function, I evaluate the hypotheses that intermittency is influenced by (1) patchiness in the prey field (e.g., related to turbulence), and (2) the ratio of actual prey size to optimal prey size. Evidence for the first hypothesis arises from explaining reported variations in clearance rates of Acartia tonsa under different turbulence regimes. Further model applications to ingestion data for rotifers, copepods, and ciliates support the view that an increasing food size enhances intermittency and, this way, affects functional grazing responses. In the application to ciliate grazing, a possible prey density effect appears, possibly due to an intermittent activation of a feeding sub-stage. Queueing theory offers mechanistic explanations for transitions between Holling I-, II-, and Ivlev-type grazing. In doing so for variable prey size ratios, it may also refine size-based ecosystem models which are increasingly emerging in plankton ecology.

Plankton and particle size and packaging: from determining optical properties to driving the biological pump

Understanding pelagic ecology and quantifying energy fluxes through the trophic web and from the surface to the deep ocean requires the ability to detect and identify all organisms and particles in situ and in a synoptic manner. An idealized sensor should observe both the very small living or dead particles such as picoplankton and detritus, respectively, and the large particles such as aggregates and meso- to macroplankton. Such an instrument would reveal an astonishing amount and diversity of living and nonliving particles present in a parcel of water. Unfortunately such sensors do not exist. However, complex interactions constrain the space, temporal, and size distributions of these objects in such ways that general rules can be inferred from the measurement of their optical properties. Recent technological developments allow for the in situ measurement of the optical properties and size distributions of particles and plankton in a way such that synoptic surveys are possible. This review deals with particle and plankton size distributions (PSDs) as well as how particles' geometry and nature affect their optical properties. Finally, we propose the integration of the PSD into size-structured mathematical models of biogeochemical fluxes.

Hydrocarbon contamination decreases mating success in a marine planktonic copepod

The mating behavior and the mating success of copepods rely on chemoreception to locate and track a sexual partner. However, the potential impact of the water-soluble fraction of hydrocarbons on these aspects of copepod reproduction has never been tested despite the widely acknowledged acute chemosensory abilities of copepods. I examined whether three concentrations of the water-soluble fraction of diesel oil (0.01%, 0.1% and 1%) impacts (i) the swimming behavior of both adult males and females of the widespread calanoid copepod Temora longcornis, and (ii) the ability of males to locate, track and mate with females. The three concentrations of the water-soluble fraction of diesel oil (WSF) significantly and non-significantly affect female and male swimming velocities, respectively. In contrast, both the complexity of male and female swimming paths significantly decreased with increasing WSF concentrations, hence suggesting a sex-specific sensitivity to WSF contaminated seawater. In addition, the three WSF concentrations impacted both T. longicornis mating behavior and mating success. Specifically, the ability of males to detect female pheromone trails, to accurately follow trails and to successfully track a female significantly decreased with increasing WSF concentrations. This led to a significant decrease in contact and capture rates from control to WSF contaminated seawater. These results indicate that hydrocarbon contamination of seawater decreases the ability of male copepods to detect and track a female, hence suggest an overall impact on population fitness and dynamics.

Characteristic spatial and temporal scales unify models of animal movement

Animal movements have been modeled with diffusion at large scales and with more detailed movement models at smaller scales. We argue that the biologically relevant behavior of a wide class of movement models can be efficiently summarized with two parameters: the characteristic temporal and spatial scales of movement. We define these scales so that they describe movement behavior both at short scales (through the velocity autocorrelation function) and at long scales (through the diffusion coefficient). We derive these scales for two types of commonly used movement models: the discrete-step correlated random walk, with either constant or random step intervals, and the continuous-time correlated velocity model. For a given set of characteristic scales, the models produce very similar trajectories and encounter rates between moving searchers and stationary targets. Thus, we argue that characteristic scales provide a unifying currency that can be used to parameterize a wide range of ecological phenomena related to movement.

How zooplankton feed: mechanisms, traits and trade-offs

Zooplankton is a morphologically and taxonomically diverse group and includes organisms that vary in size by many orders of magnitude, but they are all faced with the common problem of collecting food from a very dilute suspension. In order to maintain a viable population in the face of mortality, zooplankton in the ocean have to clear daily a volume of ambient water for prey particles that is equivalent to about 10(6) times their own body volume. While most size-specific vital rates and mortality rates decline with size, the clearance requirement is largely size-independent because food availability also declines with size. There is a limited number of solutions to the problem of concentrating dilute prey from a sticky medium: passive and active ambush feeding; feeding-current feeding, where the prey is either intercepted directly, retained on a filter, or individually perceived and extracted from the feeding current; cruise feeding; and colonization of large particles and marine snow aggregates. The basic mechanics of these food-collection mechanisms are described, and it is shown that their efficiencies are inherently different and that each of these mechanisms becomes less efficient with increasing size. Mechanisms that compensate for this decline in efficiency are described, including inflation of feeding structures and development of vision. Each feeding mode has implications beyond feeding in terms of risk of encountering predators and chance of meeting mates, and they partly target different types of prey. The main dichotomy is between (inefficient) ambush feeding on motile prey and the more efficient active feeding modes; a secondary dichotomy is between (efficient) hovering and (less efficient) cruising feeding modes. The efficiencies of the various feeding modes are traded off against feeding-mode-dependent metabolic expenses, predation risks, and mating chances. The optimality of feeding strategies, evaluated as the ratio of gain over risk, varies with the environment, and may explain both size-dependent and spatio-temporal differences in distributions of various feeding types as well as other aspects of the biology of zooplankton (mating behaviour, predator defence strategies).

Fluctuation-driven Turing patterns

Models of diffusion-driven pattern formation that rely on the Turing mechanism are utilized in many areas of science. However, many such models suffer from the defect of requiring fine tuning of parameters or an unrealistic separation of scales in the diffusivities of the constituents of the system in order to predict the formation of spatial patterns. In the context of a very generic model of ecological pattern formation, we show that the inclusion of intrinsic noise in Turing models leads to the formation of "quasipatterns" that form in generic regions of parameter space and are experimentally distinguishable from standard Turing patterns. The existence of quasipatterns removes the need for unphysical fine tuning or separation of scales in the application of Turing models to real systems.

Mechanisms of temperature-dependent swimming: the importance of physics, physiology and body size in determining protist swimming speed

Body temperatures and thus physiological rates of poikilothermic organisms are determined by environmental temperature. The power an organism has available for swimming is largely dependent on physiological rates and thus body temperature. However, retarding forces such as drag are contingent on the temperature-dependent physical properties of water and on an organism's size. Consequently, the swimming ability of poikilotherms is highly temperature dependent. The importance of the temperature-dependent physical properties of water (e.g. viscosity) in determining swimming speed is poorly understood. Here we propose a semi-mechanistic model to describe how biological rates, size and the physics of the environment contribute to the temperature dependency of microbial swimming speed. Data on the swimming speed and size of a predatory protist and its protist prey were collected and used to test our model. Data were collected by manipulating both the temperature and the viscosity (independently of temperature) of the organism's environment. Protists were either cultured in their test environment (for several generations) or rapidly exposed to their test environment to assess their ability to adapt or acclimate to treatments. Both biological rates and the physics of the environment were predicted to and observed to contribute to the swimming speed of protists. Body size was not temperature dependent, and protists expressed some ability to acclimate to changes in either temperature or viscosity. Overall, using our parameter estimates and novel model, we are able to suggest that 30 to 40% (depending on species) of the response in swimming speed associated with a reduction in temperature from 20 to 5°C is due to viscosity. Because encounter rates between protist predators and their prey are determined by swimming speed, temperature- and viscosity-dependent swimming speeds are likely to result in temperature- and viscosity-dependent trophic interactions.

Grazer cues induce stealth behavior in marine dinoflagellates

Chain formation is common among phytoplankton organisms but the underlying reasons and consequences are poorly understood. Here we show that chain formation is strongly impaired by waterborne cues from copepod grazers in the dinoflagellate Alexandrium tamarense. Chains of Alexandrium cells exposed to copepod cues responded by splitting into single cells or shorter chains. Motion analysis revealed significantly lower swimming velocities for single cells compared with chains, with two- to fivefold higher simulated predator encounter rates for two- and four-cell chains, respectively. In addition, the few remaining two-cell chains in grazed treatments were swimming at approximately half the speed of two-cell chains in treatments without grazers, which reduced encounter rates with grazers to values similar to that of single cells. Chain length plasticity and swimming behavior constitute unique mechanisms to reduce encounters with grazers. We argue that dinoflagellates can regulate the balance between motility and predator avoidance by adjusting chain length. The high predator encounter rate for motile chains may have contributed to the low prevalence of chain formation in motile phytoplankton compared with in nonmotile phytoplankton where chain formation is more common.

Inherent high correlation of individual motility enhances population dispersal in a heterotrophic, planktonic protist

Quantitative linkages between individual organism movements and the resulting population distributions are fundamental to understanding a wide range of ecological processes, including rates of reproduction, consumption, and mortality, as well as the spread of diseases and invasions. Typically, quantitative data are collected on either movement behaviors or population distributions, rarely both. This study combines empirical observations and model simulations to gain a mechanistic understanding and predictive ability of the linkages between both individual movement behaviors and population distributions of a single-celled planktonic herbivore. In the laboratory, microscopic 3D movements and macroscopic population distributions were simultaneously quantified in a 1L tank, using automated video- and image-analysis routines. The vertical velocity component of cell movements was extracted from the empirical data and used to motivate a series of correlated random walk models that predicted population distributions. Validation of the model predictions with empirical data was essential to distinguish amongst a number of theoretically plausible model formulations. All model predictions captured the essence of the population redistribution (mean upward drift) but only models assuming long correlation times (minute), captured the variance in population distribution. Models assuming correlation times of 8 minutes predicted the least deviation from the empirical observations. Autocorrelation analysis of the empirical data failed to identify a de-correlation time in the up to 30-second-long swimming trajectories. These minute-scale estimates are considerably greater than previous estimates of second-scale correlation times. Considerable cell-to-cell variation and behavioral heterogeneity were critical to these results. Strongly correlated random walkers were predicted to have significantly greater dispersal distances and more rapid encounters with remote targets (e.g. resource patches, predators) than weakly correlated random walkers. The tendency to disperse rapidly in the absence of aggregative stimuli has important ramifications for the ecology and biogeography of planktonic organisms that perform this kind of random walk.

Organism movement is fundamental to how organisms interact with each other and the environment. Such movements are also important on the population level and determine the spread of disease and invasion, reproduction, consumption, and mortality. Theoretical ecologists have sought to predict population dispersal rates, which are often hard to measure, from individual movement behaviors, which are often easier to measure. This problem has been non-trivial. This manuscript contributes seldom available, simultaneously measured movement behaviors and population distributions of a single celled planktonic organism. The empirical data are used to distinguish amongst a set of plausible theoretical modeling approaches to suggest that organism movements are highly correlated, meaning movement direction and speed is consistent over several minutes. Previous estimates suggested persistence only lasted several seconds. Minute-scale correlations result in much more rapid organism dispersal and greater dispersal distance, indicating that organisms encounter and impact a greater portion of their surrounding habitat than previously suspected.

Filtration of submicrometer particles by pelagic tunicates

Salps are common in oceanic waters and have higher per-individual filtration rates than any other zooplankton filter feeder. Although salps are centimeters in length, feeding via particle capture occurs on a fine, mucous mesh (fiber diameter d approximately 0.1 microm) at low velocity (U = 1.6 +/- 0.6 cmxs(-1), mean +/- SD) and is thus a low Reynolds-number (Re approximately 10(-3)) process. In contrast to the current view that particle encounter is dictated by simple sieving of particles larger than the mesh spacing, a low-Re mathematical model of encounter rates by the salp feeding apparatus for realistic oceanic particle-size distributions shows that submicron particles, due to their higher abundances, are encountered at higher rates (particles per time) than larger particles. Data from feeding experiments with 0.5-, 1-, and 3-microm diameter polystyrene spheres corroborate these findings. Although particles larger than 1 microm (e.g., flagellates, small diatoms) represent a larger carbon pool, smaller particles in the 0.1- to 1-microm range (e.g., bacteria, Prochlorococcus) may be more quickly digestible because they present more surface area, and we find that particles smaller than the mesh size (1.4 microm) can fully satisfy salp energetic needs. Furthermore, by packaging submicrometer particles into rapidly sinking fecal pellets, pelagic tunicates can substantially change particle-size spectra and increase downward fluxes in the ocean.

Sampling rate and misidentification of Lévy and non-Lévy movement paths

A large number of empirical studies have attributed Lévy search patterns to the foraging movements of animals. Typically, this is done by fitting a power-law distribution with an exponent of 1 < mu < or = 3 to the observed step lengths. Most studies record the animal's location at equally spaced time intervals, which are sometimes significantly longer than the natural time scale of the animal's movements. The collected data thus represent a subsample of the animal's movement. In this paper, the effect of subsampling on the observed properties of both Lévy and non-Lévy simulated movement paths is investigated. We find that the apparent properties of the observed movement path can be sensitive to the sampling rate even though Lévy search patterns are supposedly scale-independent. We demonstrate that, in certain contexts and dependent on the sampling rate used in observation, it is possible to misidentify a non-Lévy movement path as being a Lévy path. We also demonstrate that a Lévy movement path can be misidentified as a non-Lévy path, but this is dependent on the value of mu of the original simulated path, with the greatest uncertainty for mu = 2. We discuss the implications of these results in the context of studies of animal movements and foraging behavior.

Fractal reorientation clocks: Linking animal behavior to statistical patterns of search

The movement ecology framework depicts animal movement as the result of the combined effects of internal and external constraints on animal navigation and motion capacities. Nevertheless, there are still fundamental problems to understand how these modulations take place and how they might be translated into observed statistical properties of animal trajectories. Of particular interest, here, is the general idea of intermittence in animal movement. Intermittent locomotion assumes that animal movement is, in essence, discrete. The existence of abrupt interruptions in an otherwise continuous flow of movement allows for the possibility of reorientations, that is, to break down previous directional memories of the trajectory. In this study, we explore the potential links between intermittent locomotion, reorientation behavior, and search efficiency. By means of simulations we show that the incorporation of Lévy intermittence in an otherwise nonintermittent search strongly modifies encounter rates. The result is robust to different types of landscapes (i.e., target density and spatial distribution), and spatial dimensions (i.e., 2D, 3D). We propose that Lévy intermittence may come from reorientation mechanisms capable of organizing directional persistence on time (i.e., fractal reorientation clocks), and we rationalize that the explicit distinction between scanning and reorientation mechanisms is essential to make accurate statistical inferences from animal search behavior. Finally, we provide a statistical tool to judge the existence of episodic and strong reorientation behaviors capable of modifying relevant properties of stochastic searches, ultimately controlling the chances of finding unknown located items.

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.

Optimal swimming strategies in mate-searching pelagic copepods

Male copepods must swim to find females, but swimming increases the risk of meeting predators and is expensive in terms of energy expenditure. Here I address the trade-offs between gains and risks and the question of how much and how fast to swim using simple models that optimise the number of lifetime mate encounters. Radically different swimming strategies are predicted for different feeding behaviours, and these predictions are tested experimentally using representative species. In general, male swimming speeds and the difference in swimming speeds between the genders are predicted and observed to increase with increasing conflict between mate searching and feeding. It is high in ambush feeders, where searching (swimming) and feeding are mutually exclusive and low in species, where the matured males do not feed at all. Ambush feeding males alternate between stationary ambush feeding and rapid search swimming. Swimming speed and the fraction of time spent searching increase with food availability, as predicted. This response is different from the pattern in other feeding types. The swimming speeds of non-feeding males are predicted and observed to be independent of the magnitude of their energy storage and to scale with the square root of body length in contrast to the proportionality scaling in feeding copepods. Suspension feeding males may search and feed at the same time, but feeding is more efficient when hovering than when cruising. Therefore, females should mainly be hovering and males cruising, which is confirmed by observations. Differences in swimming behaviour between genders and feeding types imply different mortality rates and predict well the observed patterns in population sex ratios. Sex ratios may become so female-biased that male abundances limit population growth, demonstrating that behaviours that are optimal to the individual may be suboptimal to the population.

On the relationship between fractal dimension and encounters in three-dimensional trajectories

The encounter of individuals-prey, predators and mates-living in the surrounding environment is a fundamental process in the life of an organism. Along with the sensory abilities, this process will be regulated by the movement rules adopted by the individual. In this work we discuss the encounter-enhancement effect due to different natatorial modes by calculating the number of encounters realised by differently convoluted trajectories in two homogeneous distributions of particles. Using numerically generated trajectories representative of specific swimming behaviour, we demonstrate that high values of three-dimensional fractal dimension D(3D)(>1.9) are beneficial only at high concentration, whereas at low concentration less tortuous tracks (D(3D) approximately 1.5) are almost equally efficient. In the light of our results it is possible to better understand the behavioural adaptations evolved by individuals to thrive in their environment.

Optimal foraging by zooplankton within patches: the case of Daphnia

The motions of many physical particles as well as living creatures are mediated by random influences or 'noise'. One might expect that over evolutionary time scales internal random processes found in living systems display characteristics that maximize fitness. Here we focus on animal random search strategies [G.M. Viswanathan, S.V. Buldyrev, S. Havlin, M.G.E. Da Luz, E.P. Raposo, H.E. Stanley, Optimizing the success of random searches, Nature 401 (1999) 911-914; F. Bartumeus, J. Catalan, U.L. Fulco, M.L. Lyra, G.M. Viswanathan, Optimizing the encounter rate in biological interactions: Lévy versus Brownian stratagies, Phys. Rev. Lett. 88 (2002) 097901 and 89 (2002) 109902], and we describe experiments with the following Daphnia species: D. magna, D. galeata, D. lumholtzi, D. pulicaria, and D. pulex. We observe that the animals, while foraging for food, choose turning angles from distributions that can be described by exponential functions with a range of widths. This observation leads us to speculate and test the notion that this characteristic distribution of turning angles evolved in order to enhance survival. In the case of theoretical agents, some form of randomness is often introduced into search algorithms, especially when information regarding the sought object(s) is incomplete or even misleading. In the case of living animals, many studies have focused on search strategies that involve randomness [H.C. Berg, Random Walks in Biology, Princeton University, Princeton, New Jersey, 1993; A. Okubo, S.A. Levin (Eds.), Diffusion and Ecological Problems: Modern Perspectives, second ed., Springer, New York, 2001]. A simple theory based on stochastic differential equations of the motion backed up by a simulation shows that the collection of material (information, energy, food, supplies, etc.) by an agent executing Brownian-type hopping motions is optimized while foraging for a finite time in a supply patch of limited spatial size if the agent chooses turning angles taken from an exponential distribution with a specific stochastic intensity or 'noise width'. Search strategies that lead to optimization is a topic of high current interest across many disciplines [D. Wolpert, W. MacReady, No free lunch theorems for optimization, IEEE Transactions on Evolutionary Computation 1 (1997) 67].