Planktonic encounter rates in homogeneous isotropic turbulence: the case of predators with limited fields of sensory perception

It is a well-established fact that encounter rates between different species of planktonic microorganism, either swimming, or passively advected by the flow, are enhanced in the presence of turbulence. However, due to the complexity of the various calculations involved, current encounter rate theories are based on a number of simplifying approximations, which do not reflect reality. In particular, a typical planktonic predator is usually assumed to have perfect 'all round vision', i.e. it can perceive a prey particle at any relative orientation, provided it lies within some given contact radius R. Unfortunately, there is a wide body of experimental evidence that this is not the case. In this study the encounter problem for a predator with a limited field of sensory perception, swimming in a turbulent flow, is examined from first principles and a number of new modelling ideas proposed. A wide range of kinematic simulations are also undertaken to test these predictions. Particular attention is paid to the swimming strategy such a predator might undertake to enhance its encounter rate. It turns out that the predicted optimum swimming strategies differ radically from the results of previous work. Empirical evidence is also presented which appears to support these new findings.     

Plankton predation rates in turbulence: a study of the limitations imposed on a predator with a non-spherical field of sensory perception

This paper presents an extension to previously published work which studied encounter rates of planktonic predators with restricted perception fields, to examine the related problems of prey capture and predation rates. Small-scale turbulence influences planktonic predation in two ways: the extra energy of the flow enhances the number of encounter events between individual predator and prey meso/micro-zooplankton, but it lowers the capture probability (because the time spent by the predator and prey in close proximity is reduced). Typically, an 'encounter' has usually been defined as an event when a potential prey swims (or is advected) to within a distance R of the predator in any direction. However, there is a considerable body of experimental evidence showing that predators perception fields are far from spherical; often they are wedge shaped (e.g. fish larvae), or strongly aligned with the directions of sensory antennae (e.g. copepods); and this is certain to influence optimal predation strategies. This paper presents a theoretical model which for the first time examines the combined problems of both encounter and capture for a predator with a restricted perception field swimming in a turbulent flow. If such a predator adopts a cruising strategy (continuous swimming, possibly with direction changes) the model predictions suggest that predation rates actually vary little with swimming speed, in contrast to predictions made for spherical perception fields. Consequently, cruising predators are predicted to swim at relatively low speeds whilst foraging. However, application of the model to examine the net energy gain of a typical pause-travel predator (the Atlantic cod larva), does predict the existence of an optimal ratio of the length of pauses to time spent swimming (specifically one pause phase to every two travel phases), in line with experimental observations. Kinematic simulations are presented which support these findings.     

The influence of turbulence on plankton predation strategies

The importance of predation in regulating the size of competing plankton and larval fish populations has long been appreciated. However, it has only recently been recognized that turbulence must have a significant influence on predator-prey interactions because most rival species of microorganisms co-exist in oceanic or fast moving fresh water flows. Turbulence is likely to influence predation strategies in two ways. The extra energy imparted to a micro-organism from the flow field will enhance the number of encounters or "contacts" between predators and prey. At the same time, because the velocity of a predator relative to its potential prey will be increased, the time-scale over which a capture must be completed is reduced. Balancing the benefits of extra encounters with the drawbacks of more difficult captures, will dictate an optimal predation strategy, either foraging behaviour or ambush feeding, on the predator. This will depend on its own and the prey's swimming capabilities, as well as the characteristics of the turbulent environment. In this paper some previous work, examining the increased encounter rate in turbulence, will be extended to look at the capture problem. The main proposal is that the capture event should be encapsulated in a capture probability function, from which the optimal predation strategy can be derived. As an illustration, plausible capture probability functions will be postulated and the resulting predictions tested against numerical simulations carried out in a turbulent-like flow field. Good agreement between the predictions and the simulations is demonstrated.     

Numerical streamline patterns at swimmer's surface using RANS equations

The objective of this article is to perform a numerical modeling on the flow dynamics around a competitive female swimmer during the underwater swimming phase for a velocity of 2.2 m/s corresponding to national swimming levels. Flow around the swimmer is assumed turbulent and simulated with a computational fluid dynamics method based on a volume control approach. The 3D numerical simulations have been carried out with the code ANSYS FLUENT and are presented using the standard k-ω turbulence model for a Reynolds number of 6.4 × 10(6). To validate the streamline patterns produced by the simulation, experiments were performed in the swimming pools of the National Institute of Sports and Physical Education in Paris (INSEP) by using the tufts method.     

The forward rate of binding of surface-tethered reactants: effect of relative motion between two surfaces

The reaction of molecules confined to two dimensions is of interest in cell adhesion, specifically for the reaction between cell surface receptors and substrate-bound ligand. We have developed a model to describe the overall rate of reaction of species that are bound to surfaces under relative motion, such that the Peclet number is order one or greater. The encounter rate between reactive species is calculated from solution of the two-dimensional convection-diffusion equation. The probability that each encounter will lead to binding depends on the intrinsic rate of reaction and the encounter duration. The encounter duration is obtained from the theory of first passage times. We find that the binding rate increases with relative velocity between the two surfaces, then reaches a plateau. This plateau indicates that the increase in the encounter rate is counterbalanced by the decrease in the encounter duration as the relative velocity increases. The binding rate is fully described by two dimensionless parameters, the Peclet number and the Damk?hler number. We use this model to explain data from the cell adhesion literature by incorporating these rate laws into "adhesive dynamics" simulations to model the binding of a cell to a surface under flow. Leukocytes are known to display a "shear threshold effect" when binding selectin-coated surfaces under shear flow, defined as an increase in bind rate with shear; this effect, as calculated here, is due to an increase in collisions between receptor and ligand with increasing shear. The model can be used to explain other published data on the effect of wall shear rate on the binding of cells to surfaces, specifically the mild decrease in binding within a fixed area with increasing shear rate.     

A Model for Bacterial Colonization of Sinking Aggregates

Sinking aggregates provide important nutrient-rich environments for marine bacteria. Quantifying the rate at which motile bacteria colonize such aggregations is important in understanding the microbial loop in the pelagic food web. In this paper, a simple analytical model is presented to predict the rate at which bacteria undergoing a random walk encounter a sinking aggregate. The model incorporates the flow field generated by the sinking aggregate, the swimming behavior of the bacteria, and the interaction of the flow with the swimming behavior. An expression for the encounter rate is computed in the limit of large Péclet number when the random walk can be approximated by a diffusion process. Comparison with an individual-based numerical simulation is also given.     

Rapid behavioral responses of an invertebrate larva to dissolved settlement cue

Larvae of the nudibranch Phestilla sibogae were used to study whether a natural dissolved settlement cue (from their prey, Porites compressa, an abundant coral on Hawaiian reefs) induces behavioral responses that can affect larval transport to suitable settlement sites. As cue and larvae are mixed in the turbulent flow over a reef, cue is distributed in fine-scale filaments that the larva experiences as rapid (seconds) on/off encounters. To examine larval responses in this setting, individual larvae were tethered in a small flume with flow simulating water velocity relative to a freely swimming larva, and their responses to realistic temporal patterns of cue encounter were videotaped. Competent larvae quickly ceased swimming in cue filaments and resumed swimming after exiting filaments. The threshold cue concentration eliciting a response was 3%-17% of concentrations within heads of P. compressa in nature. When moving freely in filtered seawater, competent larvae swam along straight paths in all directions at approximately 0.2 cm s(-1), whereas in water conditioned by P. compressa, most ceased swimming and sank at approximately 0.1 cm s(-1). The ability of larvae to rapidly respond (by sinking) to brief encounters with dissolved settlement cues can enhance their rapid transport to the substratum, even in wave-driven turbulent flow.     

Mechanisms and rates of bacterial colonization of sinking aggregates

Quantifying the rate at which bacteria colonize aggregates is a key to understanding microbial turnover of aggregates. We used encounter models based on random walk and advection-diffusion considerations to predict colonization rates from the bacteria's motility patterns (swimming speed, tumbling frequency, and turn angles) and the hydrodynamic environment (stationary versus sinking aggregates). We then experimentally tested the models with 10 strains of bacteria isolated from marine particles: two strains were nonmotile; the rest were swimming at 20 to 60 microm s(-1) with different tumbling frequency (0 to 2 s(-1)). The rates at which these bacteria colonized artificial aggregates (stationary and sinking) largely agreed with model predictions. We report several findings. (i) Motile bacteria rapidly colonize aggregates, whereas nonmotile bacteria do not. (ii) Flow enhances colonization rates. (iii) Tumbling strains colonize aggregates enriched with organic substrates faster than unenriched aggregates, while a nontumbling strain did not. (iv) Once on the aggregates, the bacteria may detach and typical residence time is about 3 h. Thus, there is a rapid exchange between attached and free bacteria. (v) With the motility patterns observed, freely swimming bacteria will encounter an aggregate in <1 day at typical upper-ocean aggregate concentrations. This is faster than even starving bacteria burn up their reserves, and bacteria may therefore rely solely on aggregates for food. (vi) The net result of colonization and detachment leads to a predicted equilibrium abundance of attached bacteria as a function of aggregate size, which is markedly different from field observations. This discrepancy suggests that inter- and intraspecific interactions among bacteria and between bacteria and their predators may be more important than colonization in governing the population dynamics of bacteria on natural aggregates.     

Numerical simulations of undulatory swimming at moderate Reynolds number

We perform numerical simulations of the swimming of a three-linkage articulated system in a moderately viscous regime. The computational methodology focuses on the creation, diffusion and transport of vorticity from the surface of the bodies into the fluid. The simulations are dynamically coupled, in that the motion of the three-linkage swimmer is computed simultaneously with the dynamics of the fluid. The novel coupling scheme presented in this work is the first to exploit the relationship between vorticity creation and body dynamics. The locomotion of the system, when subject to undulatory inputs of the hinges, is computed at Reynolds numbers of 200 and 1000. It is found that the forward swimming speed increases with the Reynolds number, and that in both cases the swimming is slower than in an inviscid medium. The vortex shedding is examined, and found to exhibit behavior consistent with experimental flow visualizations of fish.     

The feeding behaviour of juvenile Atlantic salmon in relation to turbulent flow

The feeding behaviour of juvenile Atlantic salmon Salmo salar in the Sainte‐Marguerite River, Quebec, Canada, varied with the characteristics of turbulent flow. Simulations indicated that juveniles would decrease their swimming costs during attacks by 19·8% in low and by 31·1% in high turbulent conditions by initiating movements in low‐speed flow events. The real swimming costs did not differ from the swimming costs estimated for a situation where fish initiate their movements at randomly selected flow velocities. The juvenile Atlantic salmon did not seem to prefer low‐speed flow events when initiating their movements. The proportion of time used for movements by fish decreased with an increase in the mean and the s . d . of the flow velocity.     

Mechanisms and Rates of Bacterial Colonization of Sinking Aggregates

Quantifying the rate at which bacteria colonize aggregates is a key to understanding microbial turnover of aggregates. We used encounter models based on random walk and advection-diffusion considerations to predict colonization rates from the bacteria's motility patterns (swimming speed, tumbling frequency, and turn angles) and the hydrodynamic environment (stationary versus sinking aggregates). We then experimentally tested the models with 10 strains of bacteria isolated from marine particles: two strains were nonmotile; the rest were swimming at 20 to 60 μm s⁻¹ with different tumbling frequency (0 to 2 s⁻¹). The rates at which these bacteria colonized artificial aggregates (stationary and sinking) largely agreed with model predictions. We report several findings. (i) Motile bacteria rapidly colonize aggregates, whereas nonmotile bacteria do not. (ii) Flow enhances colonization rates. (iii) Tumbling strains colonize aggregates enriched with organic substrates faster than unenriched aggregates, while a nontumbling strain did not. (iv) Once on the aggregates, the bacteria may detach and typical residence time is about 3 h. Thus, there is a rapid exchange between attached and free bacteria. (v) With the motility patterns observed, freely swimming bacteria will encounter an aggregate in <1 day at typical upper-ocean aggregate concentrations. This is faster than even starving bacteria burn up their reserves, and bacteria may therefore rely solely on aggregates for food. (vi) The net result of colonization and detachment leads to a predicted equilibrium abundance of attached bacteria as a function of aggregate size, which is markedly different from field observations. This discrepancy suggests that inter- and intraspecific interactions among bacteria and between bacteria and their predators may be more important than colonization in governing the population dynamics of bacteria on natural aggregates.     

Turbulence: does vorticity affect the structure and shape of body and fin propulsors?

Over the past century, many ideas have been developed on the relationships between water flow and the structure and shape of the body and fins of fishes, largely during swimming in relatively steady flows. However, both swimming by fishes and the habitats they occupy are associated with vorticity, typically concentrated as eddies characteristic of turbulent flow. Deployment of methods to examine flow in detail suggests that vorticity impacts the lives of fishes. First, vorticity near the body and fins can increase thrust and smooth variations in thrust that are a consequence of using oscillating and undulating propulsors to swim. Second, substantial mechanical energy is dissipated in eddies in the wake and adaptations that minimize these losses would be anticipated. We suggest that such mechanisms may be found in varying the length of the propulsive wave, stiffening propulsive surfaces, and shifting to using median and paired fins when swimming at low speeds. Eddies in the flow encountered by fishes may be beneficial, but when eddy radii are of the order of 0.25 of the fish's total length, negative impacts occur due to greater difficulties in controlling stability. The archetypal streamlined "fish" shape reduces destabilizing forces for fishes swimming into eddies.     

Assessing the roles of glutamatergic and cholinergic synaptic drive in the control of fictive swimming frequency in young Xenopus tadpoles

This paper investigates the proposal that the frequency of the swimming central pattern generator in young Xenopus tadpoles is partly determined by the population of glutamatergic premotor interneurons active on each cycle. During fictive swimming spinal neurons also receive cholinergic and electrotonic excitation from motoneurons. As frequency changes during swimming we make two predictions: first, since most motoneurons fire very reliably at all frequencies, the electrotonic and nicotinic drive from motoneurons should remain constant, and second, when swimming frequency decreases, the glutamatergic drive should decrease as the number of active premotor excitatory interneurons decreases. We have tested these predictions by measuring the excitatory synaptic drive to motoneurons as frequency changes during fictive swimming. The components of synaptic drive were revealed by the local microperfusion of strychnine together with different excitatory antagonists. After blocking the nicotinic acetylcholine receptor, the mainly glutmatergic excitatory synaptic drive still changed with frequency. However, when glutamate receptors or all chemical transmission was blocked, excitation did not change with frequency. Our predictions are confirmed, suggesting that premotor excitatory interneurons are a major factor in frequency control in the tadpole central pattern generator and that motoneurons provide a stable background excitation. Accepted: 14 August 1998     

Skin-friction drag analysis from the forced convection modeling in simplified underwater swimming

This study deals with skin-friction drag analysis in underwater swimming. Although lower than profile drag, skin-friction drag remains significant and is the second and only other contribution to total drag in the case of underwater swimming. The question arises whether varying the thermal gradient between the underwater swimmer and the pool water may modify the surface shear stress distribution and the resulting skin-friction drag acting on a swimmer's body. As far as the authors are aware, such a question has not previously been addressed. Therefore, the purpose of this study was to quantify the effect of this thermal gradient by using the integral formalism applied to the forced convection theory. From a simplified model in a range of pool temperatures (20-30 degrees C) it was demonstrated that, whatever the swimming speeds, a 5.3% reduction in the skin-friction drag would occur with increasing average boundary-layer temperature provided that the flow remained laminar. However, as the majority of the flow is actually turbulent, a turbulent flow analysis leads to the major conclusion that friction drag is a function of underwater speed, leading to a possible 1.5% reduction for fast swimming speeds above 1m/s. Furthermore, simple correlations between the surface shear stress and resulting skin-friction drag are derived in terms of the boundary-layer temperature, which may be readily used in underwater swimming situations.     

The Effect of Density and Temperature on the Swimming and Aggregating Behaviour of the Backswimmer,Anisops deanei (Heteroptera: Notonectidae) and Subsequent Encounter Rate with a Sit-and-Wait Predator

Four different swimming behaviours of the backswimmer, Anisops deanei, each characterized by a different angular position of the metathoracic legs and speed, are described along with the effect of water temperature and density on (1) the number of each behaviour exhibited and (2) total time spent swimming, per unit time. Generally, as density increased the swimming activity in each behaviour decreased while as temperature increased swimming activity increased. The direction of the effects were the same when total time spent swimming per unit time was examined. Nearest neighbour analysis showed that A. deanei form obvious aggregations in laboratory tanks as density increases, and this effect remains significant at least to the third neighbour. Density had a significant effect on neighbour distances whereas temperature did not. It was hypothesized that the primary function of these aggregations, which are also commonly observed in the field, is an anti-predator behaviour. Laboratory experiments using models of a common sit-and-wait predator of A. deanei, showed that although the encounter rate increased with both water temperature and density the observed encounter rate at the higher densities was significantly less than expected with a random distribution of prey. It is suggested that this defence effect would also exist with more mobile predators like fish.     

Optimization of diameters and bifurcation angles in lung and vascular tree structures

An optimization model is described for lung and vascular tree structures. The model extends Murray's model, which is derived from minimal power dissipation due to the frictional resistance of laminar flow and the volume of the duct system. Instead of just laminar flow, it takes into account all types of steady flow, e.g. turbulent and laminar flow, and predicts which structural changes will occur among different parts of trees having different types of flow. The sensitivity of the optimal values is indicated and the model, predictions are compared with literature data.     

A model for switching between particulate-feeding and filter-feeding in the common bream,Abramis brama

Synopsis A model has been developed to describe the process of switching between particulate- and filter-feeding in common bream, Abramis brama, in relation to fish size and zooplankton density. The model assumes that the encounter rate of fish and zooplankton is determined by the density of zooplankton and the swimming speed of fish. However, if zooplankton density is so high as to allow at least one prey to be engulfed per random snap, the encounter rate is determined by the volume of the buccal cavity and by zooplankton density, but is independent of swimming speed. The snapping frequency will be maximal at the time of switching, decreasing with increasing zooplankton density because of the extra time needed for intra-oral prey handling. The model predicted switching from particulate- to filter-feeding only for bream> 15 cm standard length at zooplankton densities < 500 l^-1. The snap frequency of six size classes of bream (7.5, 10.4, 12.5, 15, 24 and 29.5 cm) was measured at varying densities of Daphnia. The model predictions for snap frequencies of all size classes corresponded to the highest values observed. The average of the observed snap frequencies was only 50% of the predicted values, probably because the calculated average distance between prey animals assumed an ideal swimming route of the fish and error-free vision for particulate-feeding, and the handling time was ignored.     

Biomechanics of Larval Morphology Affect Swimming: Insights from the Sand Dollars Dendraster excentricus

Most planktonic larvae of marine invertebrates are denser than sea water, and rely on swimming to locate food, navigate advective currents, and avoid predators. Therefore, swimming behaviors play important roles in larval survival and dispersal. Larval bodies are often complex and highly variable across developmental stages and environmental conditions. These complex morphologies reflect compromises among multiple evolutionary pressures, including maintaining the ability to swim. Here, I highlight metrics of swimming performance, their relationships with morphology, and the roles of behavior in modulating larval swimming within biomechanical limits. Sand dollars have a representative larval morphology using long ciliated projections for swimming and feeding. Observed larval sand dollars fell within a narrow range of key morphological parameters that maximized their abilities to maintain directed upward movement over the most diverse flow fields, outperforming hypothetical alternatives in a numerical model. Ontogenetic changes in larval morphology also led to different vertical movements in simulated flow fields, implying stage-dependent vertical distributions and lateral transport. These model outcomes suggest a tight coupling between larval morphology and swimming. Environmental stressors, such as changes in temperature and pH, can therefore affect larval swimming through short-term behavioral adjustments and long-term changes in morphology. Larval sand dollars reared under elevated pCO(2) conditions had significantly different morphology, but not swimming speeds or trajectories. Geometric morphometric analysis showed a pH-dependent, size-mediated change in shape, suggesting a coordinated morphological adjustment to maintain swimming performance under acidified conditions. Quantification of the biomechanics and behavioral aspects of swimming improves predictions of larval survival and dispersal under present-day and future environmental conditions.     

Hydrodynamics of Sperm Cells near Surfaces

Sperm are propelled by an actively beating tail, and display a wide variety of swimming patterns. When confined between two parallel walls, sperm swim either in circles or on curvilinear trajectories close to the walls. We employ mesoscale hydrodynamics simulations in combination with a mechanical sperm model to study the swimming behavior near walls. The simulations show that sperm become captured at the wall due to the hydrodynamic flow fields which are generated by the flagellar beat. The circular trajectories are determined by the chiral asymmetry of the sperm shape. For strong (weak) chirality, sperm swim in tight (wide) circles, with the beating plane of the flagellum oriented perpendicular (parallel) to the wall. For comparison, we also perform simulations based on a local anisotropic friction of the flagellum. In this resistive force approximation, surface adhesion and circular swimming patterns are obtained as well. However, the adhesion mechanism is now due to steric repulsion, and the orientation of the beating plane is different. Our model provides a theoretical framework that explains several distinct swimming behaviors of sperm near and far from a wall. Moreover, the model suggests a mechanism by which sperm navigate in a chemical gradient via a change of their shape.     

From individuals to aggregations: the interplay between behavior and physics

This paper analyses the processes by which organisms form groups and how social forces interact with environmental variability and transport. For aquatic organisms, the latter is especially important-will sheared or turbulent flows disrupt organism groups? To analyse such problems, we use individual-based models to study the environmental and social forces leading to grouping. The models are then embedded in turbulent flow fields to gain an understanding of the interplay between the forces acting on the individuals and the transport induced by the fluid motion. Instead of disruption of groups, we find that flows often enhance grouping by increasing the encounter rate among groups and thereby promoting merger into larger groups; the effect breaks down for strong flows. We discuss the transformation of individual-based models into continuum models for the density of organisms. A number of subtle difficulties arise in this process; however, we find that a direct comparison between the individual model and the continuum model is quite favorable. Finally, we examine the dynamics of group statistics and give an example of building an equation for the spatial and temporal variations of the group-size distribution from the individual-based simulations. These studies lay the groundwork for incorporating the effects of grouping into models of the large scale distributions of organisms as well as for examining the evolutionary consequences of group formation.