Behaviour-dependent predation risk in swimming zooplankters
© Uttieri et al.; licensee Springer. 2013
Received: 28 August 2013
Accepted: 25 September 2013
Published: 4 October 2013
The survival of zooplanktonic organisms is determined by their capability of moving in a fluid environment, trading off between the necessities of finding prey and avoiding predators. In previous numerical experiments, we concentrated on the relationship between natatorial modality and encounter success of a virtual copepod swimming in the presence of prey distributed either in patches or uniformly in the environment.
In this contribution, we extend this simulation framework to the encounter with chaetognaths, the primary copepod predators, considering different motion rules as a proxy of different swimming strategies and looking at the influence of the concentration of predators and the size of their detection radius in posing a risk on copepod survival. The outcomes of our simulations indicate that more convoluted trajectories are more vulnerable to predator encounter while straighter motions reduce predation risk.
Our results are then complemented with those obtained in our previous studies to perform a general cost-benefit analysis of zooplankton motion.
Movement behaviour is a central theme in assessing the ecological role of an animal and its adaptations to the environment, with effects propagating up to the population and metapopulation levels (Tilman and Kareiva 1997; Begon et al. 2006). Individual motion responds to the necessity of finding food resources (Bell 1991), but at the same time it has evolutionarily adapted to reduce predation risks (Lima and Dill 1990). In aquatic systems, pelagic zooplankters provide the link between the primary producers and the higher trophic levels (e.g. fish and whales). Copepods are the primary grazers of phytoplankton (Marshall 1973; Turner 2004), but meanwhile they are also the main food resource for chaetognaths (Feigenbaum and Maris 1984). These latter predators rank second in terms of abundance in the zooplankton (Feigenbaum and Maris 1984), are considered as the chief carnivores of marine ecosystems (Reeve 1970) and are in turn prey for a variety of larger animals (as reviewed in Feigenbaum and Maris 1984). The interactions between copepods and chaetognaths are thus remarkably critical for a deep insight into the functioning of pelagic food webs and for the evaluation of the mechanisms underlying the flux of energy and matter among the components of the ecosystem.
Chaetognaths are ambush predators using the fans of hair present over their body surface as distance mechanoreceptors (Horridge and Boulton 1967). Mobile prey that, through a distinctive signal, elicit the sensory hairs are attacked: the chaetognath makes a bout with a rapid flick of the tail, the prey is blocked with the grasping spines and swallowed (Feigenbaum 1991). Adult copepods and copepodites are the main food resource of adult chaetognaths (Øresland 1987), while younger developmental stages may prefer copepod nauplii (Feigenbaum 1991); on occasion, cladocerans can become the mainstay of chaetognaths (Kehayias and Ntakou 2008; Kehayias and Kourouvakalis 2010).
Feigenbaum (1977) hypothesised that the movement pattern and the swimming speed of the prey might affect its predation risk. Such speculation is backed up by the evidence that male copepods are often preyed more extensively than the conspecific females (Acartia clausi, Alvarez-Cadena 1993; Paracalanus parvus; Saito and Kiørboe , Saito and Kiørboe Saito and Kiørboe 2001). In fact, male copepods usually swim faster and over more tortuous paths (e.g. Doall et al. 1998; Weissburg et al. 1998; Nihongi et al. 2004; Uttieri et al. 2007b; Seuront 2011), peculiarities that may increase the possibility of entering the perceptive field of a predator, making them more vulnerable to predation.
Individual-based models (IBMs) are useful tools for addressing ecological questions, and they have proven to be successful in representing zooplankton motion (see Cianelli et al. 2012, for a recent review on this topic). The central actor of an IBM is the individual as unique component of the system, creating a population of individuals each with specific properties (Uchmański and Grimm 1996). Using a ‘bottom-up’ approach (Souissi et al. 2005), IBMs depict population-level emergent properties as the result of interactions between the individuals and their environment (Railsback 2001). In previous works (Uttieri et al. 2007a; Cianelli et al. 2009), model simulations have been carried out to evaluate how different search strategies, numerically represented by diverse typologies of random walks, may result in motion-specific encounters with uniformly (Uttieri et al. 2007a) and patchily (Cianelli et al. 2009) distributed prey. Here we expand upon these previous studies by addressing the problem of the encounter with a chaetognath predator. The present model aims at describing the encounter rate between a swimming copepod and an ambush chaetognath by relating the motion behaviour of a virtual copepod with the risk of entering the perceptive field of a chaetognath. The IBM explicitly accounted for realistic behavioural dynamics in the swimming motion of copepods and predators' perceptive field. In addition, to better evaluate the encounter probabilities, the model also considered the role of predator abundance in shaping predation risk. These aspects are of peculiar importance to understand the natatorial adaptations evolved by copepods to trade-off between encounter of food items and risk of being preyed upon. For comparative purposes, our simulations were complimentary compared with the encounter rates predicted by the theoretical model of Gerritsen and Strickler (1977), which explicitly accounts for the speed of both predators and prey, the encounter radius of the predator and the abundance of prey, but without considering the convolution of prey swimming tracks and the abundance of predators, which are instead realistically included in our simulations. This study constructs a comprehensive picture of the pros and cons linked to the different behavioural tactics employed by swimming copepods, and it is germane to a more exhaustive description of the predator–prey interactions in aquatic ecosystems.
The IBM implemented in the present study presented two virtual entities: a swimming copepod and a still chaetognath. The former was allowed to move in a three-dimensional domain, while the latter was still according to its ambush strategy. The copepod was consequently characterised by two state variables: spatial coordinates (owing to its own movement, changing at each time step) and the number of predators encountered P E (if, during the motion, it entered the perceptive field of the chaetognath). For the chaetognath, two state variables were defined as well: prey encounter (not calculated as homologous to P E) and the detection radius R. All these state variables could be considered additive components, as defined in Thygesen et al. (2007). As the movement of the copepod and its interactions with a predator were considered, the model could also be assumed as a spatially explicit IBM sensu Berec (2002).
In Uttieri et al. (2007a) and Cianelli et al. (2009), the model was run considering virtual spatial and temporal units (vsu and vtu, respectively). For the sake of clarity and with the purpose of making direct comparisons with earlier results, in this study we considered 1 vsu ≡ 1 mm and 1 vtu ≡ 1 s; thus, the units will be interchangeably adopted throughout the text. The simulation frame was, as in Uttieri et al. (2007a) and Cianelli et al. (2009), a cubic field of 500 vsu of side length, corresponding to a domain volume V d = 0.125 m3. A semi-absorbing condition was set at the boundaries (Uttieri et al. 2007a; Cianelli et al. 2009).
Copepod swimming behaviour
according to which the displacement at time t along the direction i (i = 1, 2 or 3, respectively, for x, y and z) was due to a random impulse dη (i,t) extracted from a normal distribution with zero mean and second-order momentum 〈dη (i,t) · dη (i,t)〉 = 2dt multiplied by a diffusion coefficient m i = 1.
A second typology of motion was modelled as a self-avoiding random walk (SARW), a movement strategy according to which a position in the domain can be occupied only once (Orr 1947). Numerically, SARWs were generated using Equation 1 with the proviso of avoiding the re-occupancy of the same point. At each time step, the instantaneous position was compared with the previous ones; if the point had already been visited, the position was calculated again. To reduce the limitations imposed by a discretised environment, at each reiteration the walker could move into one of adjacent neighbouring cubic subdomains but not into that occupied previously. For more details on the implementations of SARW, the interested reader can refer to Uttieri et al. (2007a).
The velocity at each time step t was due to a random impulse (similarly to PRW), plus the inclusion of a term T (integral time scale; Kundu 1990) representing the memory of the process. Increasing T values brought to more correlated, less tortuous walks. In our work, we used three T values (CRW-T5, 5 time steps; CRW-T50, 50 time steps; CRW-T100, 100 time steps) to account for different memories in the motion.
Fifteen tracks for each class of motion were generated, each consisting of 50,000 points. At the beginning of each simulation (t 0), the copepod was allowed to start its track from any random position inside the domain. In each run, the motion of only one copepod at time was modelled, so as to avoid any copepod-copepod interference. Fractal and multifractal properties of zooplankton motion can be extremely useful in depicting specific features of the swimming behaviour of these organisms (e.g., Seuront et al. 2004a, b). As a proxy of track convolution, the three-dimensional fractal dimension D 3D (Uttieri et al. 2005) was evaluated for each trajectory through a box-counting approach (for further details on the methodologies, please refer to Uttieri et al. 2005 and Uttieri and Zambianchi 2012).
where X t+1 was the three-dimensional position of the virtual copepod at time t+1, and Δt was the integration time step (Δt = 1 vtu). To reconstruct more realistically the motion behaviour of copepods, our conceptual model contemplated an exponential increase of swimming speed as a function of D 3D, i.e. more tortuous tracks were described at the cost of higher swimming speed. Such an assumption was based on the evidence that zooplanktonic organisms often describe paths with higher fractal dimension by swimming faster, as demonstrated for both copepods (e.g. Uttieri et al. 2007b; Seuront 2006 2011) and cladocerans (e.g. Ziarek et al. 2011).
Chaetognath abundance and perceptive field
Abundances of chaetognaths used in the simulations and equivalent in situ values derived from literature
1 ind. V d −1
Approximately 10 ind. m−3 (typical year-round abundance)
5 ind. V d −1
Approximately 40 ind. m−3 (typical of peak periods)
17 ind. V d −1
Approximately 140 ind. m−3 (limit case of excess abundance)
The distribution of predators was determined stochastically by assigning a random position to the chaetognath. To account for the extension of the detection radius R, all individuals were always placed at a minimum distance of 10 mm from the boundaries.
where R is predator's encounter radius and N H is the density of prey, whereas and v are respectively the mean prey and predator speeds. To further appreciate the importance of track convolution in determining the effective risk of predator encounter, we also computed Z P considering an ambush predator (v = 0) with one prey at time (N H = 1) for the different R values (R 1, R 2 and R 3) used in the simulations and compared it to the P E resulting from our IBM. It is worth stressing that, since the model by Gerritsen and Strickler (1977) does not explicitly include predator abundance, we compared the trends in Z P and P E in dependence of motion speed for C 1 case only.
P E, D 3D and V values were statistically analysed with one-way analysis of variance (ANOVA) (Zar 1984). In particular, the statistical test was performed to evaluate any significant difference between the same class of motion under different conditions (i.e. different predator abundances and perceptive distances) and between the five motion strategies in the same simulation scenario. ANOVA was corroborated by a post hoc multiple comparison test using Tukey's honestly significant difference method (Hochberg and Tamhane 1987), a single-step procedure using the studentised range statistic to compare all pairs of means.
Results and discussion
Mean (± standard deviation) D 3D and swimming speed V (vsu/vtu) values of the simulated behaviour
1.923 ± 0.087
1.126 ± 0.002
1.942 ± 0.084
1.348 ± 0.002
1.765 ± 0.089
0.501 ± 0.002
1.559 ± 0.075
0.157 ± 0.003
1.520 ± 0.080
0.111 ± 0.003
Expectedly, for each class of motion P E increased as the perceptive radius R augmented for any given predator abundance C; similarly, P E augmented also as a function of the number of chaetognaths independently of the R considered. Generally, the same class of motion did not show significant differences at low C and with short R, whereas in the presence of a higher number of chaetognaths and/or of longer detection radii the dissimilarities were more pronounced. Only CRW-T50 and CRW-T100 showed a reduced difference even with high C and R values.
P E vs D 3D and P E vs V
P E vs D 3D
P E vs V
The theoretical encounter rate Z P between our virtual copepod and chaetognath sensu Gerritsen and Strickler (1977) was evaluated following Equation 4. Considering only the C 1 case owing to the limitations imposed by such model detailed above, an exponential increase in Z P with speed was noticed. This result contrasts with our simulations, which instead recorded a linear relationship between P E and V, pointing out that theoretical models based on speed as the only behavioural component do not adequately reconstruct the encounter process.
Predation is a major selective force in ecological systems (Barbosa and Castellanos 2005). Predation risk increases with dispersal, and dispersing animals reduce such a risk by adopting specific behavioural options (Zollner and Lima 2005, and references therein). The interactions between copepods and chaetognaths occupy a central position in the structure of pelagic food webs, governing the interaction between the lower and upper trophic levels (Reeve 1970). In this study, we reconstructed the motion of a virtual swimming copepod using five classes of random walks, simulating different modalities of exploration of the environment (Uttieri et al. 2007a; Cianelli et al. 2009). In particular, our simulations explicitly accounted for the empirically verified phenomenon of concomitant increase of swimming speed and fractal dimension in copepod swimming motion with the intention of mimicking a more realistic behaviour. Practical numerical limitations, however, hampered us from running the entire gamut of possible motion patterns, such as Lévy walks (Bartumeus et al. 2005) and biased random walks (Codling and Hill 2005), which might be included in future model implementations. We therefore deliberately focused on a subset of the possible movement rules (PRW, SARW and CRWs) which, on previous evidences, were adequate to represent zooplankton motion.
The virtual copepod was allowed to swim in the presence of a chaetognath, recreating several possible realistic conditions of predator abundance and perceptive field. Overall, the simulations showed that the predation risk P E increased with the tortuosity (evaluated in terms of D3D) and the speed of copepod motion. This effect was more noticeable at high predator abundances and when considering large detection radii, as postulated in the theoretical model by Gerritsen and Strickler (1977). In addition, our results confirm the conjectures by Feigenbaum (1977), Duró and Saiz (2000) and Saito and Kiørboe (2001) who hypothesised that prey swimming behaviour might influence the predation by chaetognaths. The evidence that more conspicuous motion (i.e. motion with higher D 3D) is susceptible to higher predation risk provides a rationale for the observation that male copepods, which usually swim faster and describe tracks with higher D3D than conspecific females, are preyed more intensely by chaetognaths (Alvarez-Cadena 1993; Saito and Kiørboe 2001).
The mechanistic encounter model here implemented incorporates both prey behaviour (copepod swimming modes) and predator perception (chaetognath perceptive field). Our simulations elucidate a linear dependence of P E with V in place of the exponential increase predicted by the theoretical model of Gerritsen and Strickler (1977). Such a discrepancy buttresses that speed only is not an exhaustive behavioural descriptor of the encounter process, but the complexity of the motion must be integrated to gain a more comprehensive view. It should be emphasised that our model intrinsically included also the effect of D 3D, which cannot be incorporated explicitly in Equation 4, but which nonetheless empirically affects the encounter rate between planktonic organisms as demonstrated by our model results.
The perceptive radii R used in our simulations have been based upon indirect measurements available in the literature (as summarised in Table 2). It is expected that specifically conceived laboratory experiments may help obtaining direct estimates of chaetognath reactive distances, which could be integrated in next model versions. Such information, together with the simulation of realistic copepod species-specific movement behaviour (as, for example, done in Uttieri et al. 2010, for Clausocalanus furcatus and Oithona plumifera), would provide a window of insight into the phenomenon of encounter risk in zooplankton organisms.
The outcomes of the present set of simulations can be integrated with previous models exploring the encounter efficiency of the chosen classes of motion at finding prey items (E R ) in uniform (Uttieri et al. 2007a) and patchy (Cianelli et al. 2009) distributions. These numerical runs highlighted that ER ∝ αD3D, α = f(cm , NH) being a multiplicative factor dependent on motion complexity (c m) and prey abundance (N H ) (Cianelli et al. 2009). Such relationship is fundamentally the same relating P E and D 3D (Equation 5) and P E and V (Equation 6), indicating that both the encounter of predators (P E) and of prey (E R) are dependent on the tortuosity and on the speed of the motion. The complementation of these results can be used to define a cost-benefit analysis of zooplankton motion. Uncorrelated motions (PRW and SARW) turn out as not beneficial in an evolutionary perspective: while these movement rules guarantee high number of encounters of prey, they are also associated with a high predation risk. On the opposite extreme, motions with high correlation may reduce P E but at the cost of low E R . CRW-T5 seems to balance between these two opposite conditions, providing a trade-off between the different needs of copepods. This conclusion is supported by the Lagrangian characterisation of the swimming tracks of the marine copepod C. furcatus (Uttieri et al. 2008) and of the freshwater cladoceran Daphnia pulex (Uttieri et al. 2004), which both show the features of a moderately correlated random walk.
Several works have demonstrated that small-scale turbulence may influence the encounter rate between two planktonic organisms (Rothschild and Osborn 1988; Evans 1989; Lewis and Pedley 2000) by affecting the relative speeds of both prey and predator but without any direct effect on the subsequent processes (recognition, capture and ingestion). In addition, of peculiar relevance is also the duration of the contact between a plankter and its predator: for an encounter to determine a capture, the prey must reside inside the perceptive field of the marauder for a time long enough to elicit the attack (e.g. Lewis and Pedley 2000; Mariani et al. 2005 2008). In our simulations, turbulence was deliberately excluded to avoid the addition of a further element of variability in the encounter process.
In the description of the model, we made some basic assumptions, but the conclusions thus derived have some degree of adaptability to more general conditions. In our simulations, we considered a generic pelagic chaetognath. However, the main deductions on the role of predator abundance and perceptive radius can be applied also to the benthic genus Spadella, which shows an attack behaviour similar to that of pelagic chaetognaths (Horridge and Boulton 1967). We also assumed a continuous swimming copepod as a prey, but other zooplankters can be important elements in the diet of chaetognaths as well (Feigenbaum and Maris 1984; Kehayias and Ntakou 2008). Our results can therefore be extended by considering the classes of motion analysed as different exploration strategies of a generic zooplanktonic organisms, such as cladocerans.
Copepods are well known for their ability to discriminate between hydrodynamic signals created by their own movement from that due to the motion of prey or predators (Hwang and Strickler 2001). This ability allows them to perform efficient escapes from an approaching predator, and they may use a ‘fluid-dynamical camouflage’ (Hwang and Strickler 2001) to reduce predation risk.
Copepods are staple food for both chaetognaths and fish larvae, which thus compete for the same resource (Coston-Clements et al. 2009). Petrik et al. (2009) implemented an IBM to investigate the foraging of haddock and cod larvae on copepods displaying species-specific escape reactions. Their results highlight that escape abilities are more crucial than encounter rate in determining the predation risk and thus represent an important selective determinant of prey selection by fish larvae. Future implementations of our model will include the realistic description of copepod movement and their escape behaviour with the aim of reconstructing more faithfully the interactions between copepods and chaetognaths.
The interactions between zooplanktonic organisms are mediated by their swimming behaviour. Pelagic copepods display a great variety of natatorial modes, used to exploit at best the environmental conditions encountered. At the same time, however, by moving in the environment, these organisms are subject to the encounter with predators. Swimming behaviour can thus be considered an ecological descriptor of the adaptations developed by individuals to proliferate.
Previous modelling simulations have concentrated on the swimming-dependent encounters with prey (Uttieri et al. 2007a; Cianelli et al. 2009). In this work, we extend this numerical framework to the process of predator encounter. The results demonstrate that more tortuous tracks are subject to higher predation risk and that the encounter with marauders is dependent on their abundance and perceptive field. In view of a cost-benefit analysis, uncorrelated and highly correlated motions result as not beneficial, as either determining an excessive predation risk (uncorrelated motions) or a too low number of encounters with prey (highly correlated walks). Motions with a low degree of correlation emerge as efficiently balancing the costs and benefits of predator and prey encounter and thus can be considered as an efficient search strategy. The results collected in the framework of the present investigation improve our understanding of the behavioural plasticity of zooplanktonic organisms.
The research of Marco Uttieri was supported by the MOKA project (Modelling and Observation of zooplanKtonic orgAnisms; ID: RBFR10VF6M) financed by the Italian Ministry of Education, University and Research. Marco Uttieri is grateful to T. Kiørboe and G. Kehayias for sharing information on chaetognath biology and ecology and to M. Pottek for the design of the MOKA cartoon.
- Alvarez-Cadena JN: Feeding of the chaetognath Sagitta elegans Verrill. Estuar Coast Shelf Sci 1993,36(2):195–206. 10.1006/ecss.1993.1013View ArticleGoogle Scholar
- Barbosa P, Castellanos I: Ecology of predator–prey interactions. New York: Oxford University Press; 2005.Google Scholar
- Bartumeus F, Da Luz MGE, Viswanathan GM, Catalan J: Animal search strategies: a quantitative random-walk analysis. Ecology 2005,86(11):3078–3087. 10.1890/04-1806View ArticleGoogle Scholar
- Begon M, Townsend CR, Harper JL: Ecology: from individuals to ecosystems. Malden: Blackwell Publishing Ltd.; 2006.Google Scholar
- Bell WJ: Searching behaviour. The behavioural ecology of finding resources. London: Chapman & Hall; 1991.Google Scholar
- Berec L: Techniques of spatially explicit individual-based models: construction, simulation, and mean-field analysis. Ecol Model 2002,150(1–2):55–81.View ArticleGoogle Scholar
- Chandrasekhar S: Stochastic problems in physics and astronomy. Rev Mod Phys 1943,15(1):1–89. 10.1103/RevModPhys.15.1View ArticleGoogle Scholar
- Cianelli D, Uttieri M, Strickler JR, Zambianchi E: Zooplankton encounters in patchy particle distributions. Ecol Model 2009,220(5):596–604. 10.1016/j.ecolmodel.2008.10.015View ArticleGoogle Scholar
- Cianelli D, Uttieri M, Zambianchi E: Individual based modelling of planktonic organisms. In Ecological modeling. Edited by: Zhang W-J. New York: Nova Science Publishers, Inc.; 2012:83–96.Google Scholar
- Codling EA, Hill NA: Sampling rate effects on measurements of correlated and biased random walks. J Theor Biol 2005,233(4):573–588. 10.1016/j.jtbi.2004.11.008View ArticlePubMedGoogle Scholar
- Coston-Clements L, Wagget RJ, Tester PA: Chaetognaths of the United States South Atlantic Bight: distribution, abundance and potential interactions with newly spawned larval fish. J Exp Mar Biol Ecol 2009,373(2):111–123. 10.1016/j.jembe.2009.03.008View ArticleGoogle Scholar
- Daponte MC, Capitanio FL, Nahabedian DE, Viñas MD, Negri RM: Sagitta friderici Ritter-Záhony (Chaetognatha) from South Atlantic waters: abundance, population structure, and life cycle. ICES J Mar Sci 2004,61(4):680–686. 10.1016/j.icesjms.2004.03.006View ArticleGoogle Scholar
- Daponte MC, Calcagno JA, Acevedo-Luque MJJ, Martos P, Machinandiarena L, Esnal GB: Composition, density, and biomass of Salpidae and Chaetognatha in the southwestern Atlantic Ocean (34.5°S-39°S). Bull Mar Sci 2011,87(3):437–461. 10.5343/bms.2010.1014View ArticleGoogle Scholar
- Doall MH, Colin SP, Strickler JR, Yen J: Locating a mate in 3D: the case of Temora longicornis . Phil Trans R Soc London B 1998, 353: 681–689. 10.1098/rstb.1998.0234View ArticleGoogle Scholar
- Duró A, Saiz E: Distribution and trophic ecology of chaetognaths in the western Mediterranean in relation to an inshore–offshore gradient. J Plankton Res 2000,22(2):339–361. 10.1093/plankt/22.2.339View ArticleGoogle Scholar
- Evans GT: The encounter speed of moving predator and prey. J Plankton Res 1989,11(2):415–417. 10.1093/plankt/11.2.415View ArticleGoogle Scholar
- Feigenbaum DL: Nutritional ecology of the Chaetognatha with particular reference to external hair patterns, prey detection, and feeding. PhD thesis: University of Miami; 1977.Google Scholar
- Feigenbaum D: Food and feeding behaviour. In The biology of chaetognaths. Edited by: Bone Q, Kapp H, Pierrot-Bults AC. Oxford: Oxford University Press; 1991:45–54.Google Scholar
- Feigenbaum D, Reeve MR: Prey detection in the Chaetognatha: response to a vibrating probe and experimental determination of attack distance in large aquaria. Limnol Oceanogr 1977,22(6):1052–1058. 10.4319/lo.1922.214.171.1242View ArticleGoogle Scholar
- Feigenbaum DL, Maris RC: Feeding in the Chaetognatha. In Oceanography and marine biology: an annual review, vol 22. Edited by: Barnes M. Aberdeen: Aberdeen University Press; 1984:343–392.Google Scholar
- Gerritsen J, Strickler JR: Encounter probabilities and community structure in zooplankton: a mathematical model. J Fish Res Board Can 1977, 34: 73–82. 10.1139/f77-008View ArticleGoogle Scholar
- Hochberg Y, Tamhane AC: Multiple comparison procedures. New York: John Wiley & Sons, Inc.; 1987.View ArticleGoogle Scholar
- Holling CS: The functional response of invertebrate predators to prey density. Mem Entomol Soc Can 1966, 48: 1–86.View ArticleGoogle Scholar
- Horridge GA, Boulton PS: Prey detection by Chaetognatha via a vibration sense. Proc R Soc Lond B 1967,168(1013):413–419. 10.1098/rspb.1967.0072View ArticleGoogle Scholar
- Hwang J-S, Strickler JR: Can copepods differentiate prey from predator hydromechanically? Zool Stud 2001,40(1):1–6.Google Scholar
- Kareiva PM, Shigesada N: Analyzing insect movement as a correlated random walk. Oecologia 1983, 56: 234–238. 10.1007/BF00379695View ArticleGoogle Scholar
- Kehayias G, Kourouvakalis D: Diel vertical migration and feeding of chaetognaths in coastal waters of the eastern Mediterranean. Biologia 2010,65(2):301–308. 10.2478/s11756-010-0024-8View ArticleGoogle Scholar
- Kehayias G, Ntakou E: Abundance, vertical distribution and feeding of chaetognaths in the upper 50 m layer of the eastern Aegean Sea. J Nat Hist 2008,42(5–8):633–648.View ArticleGoogle Scholar
- Kundu PJ: Fluid mechanics. San Diego: Academic Press; 1990.Google Scholar
- Lewis DM, Pedley TJ: Planktonic contact rates in homogeneous isotropic turbulence: theoretical predictions and kinematic simulations. J Theor Biol 2000, 205: 377–408. 10.1006/jtbi.2000.2073View ArticlePubMedGoogle Scholar
- Lima SL, Dill M: Behavioral decisions made under the risk of predation: a review and prospectus. Can J Zool 1990,68(4):619–640. 10.1139/z90-092View ArticleGoogle Scholar
- Marazzo A, Nogueira CSR: Composition, spatial and temporal variations of Chaetognatha in Guanabara Bay, Brazil. J Plankton Res 1996,18(12):2367–2376. 10.1093/plankt/18.12.2367View ArticleGoogle Scholar
- Mariani P, Botte V, Ribera d'Alcalà M: An object-oriented model for the prediction of turbulence effects on plankton. Deep Sea Res II 2005,52(9–10):1287–1307.View ArticleGoogle Scholar
- Mariani P, Botte V, Ribera d'Alcalà M M: A numerical investigation of the impact of turbulence on the feeding rates of Oithona davisae . J Mar Syst 2008,70(3–4):273–286.View ArticleGoogle Scholar
- Marshall SM: Respiration and feeding in copepods. Adv Mar Biol 1973, 11: 57–120.View ArticleGoogle Scholar
- Nihongi A, Lovern SB, Strickler JR: Mate-searching behaviors in the freshwater calanoid copepod Leptodiaptomus ashlandi . J Mar Syst 2004,49(1–4):65–74.View ArticleGoogle Scholar
- Nishii S: Hydrodynamical perception of chaetognaths. MSc thesis: Mie University; 1998.Google Scholar
- Noblezada MMP, Campos WL: Spatial distribution of chaetognaths off the northern Bicol Shelf. Philippines (Pacific coast) 2008,65(3):484–494.Google Scholar
- Noblezada MMP, Campos WL: Chaetognath assemblages along the Pacific Coast and adjacent inland waters of the Philippines: relative importance of oceanographic and biological factors. ICES J Mar Sci 2012,69(3):410–420. 10.1093/icesjms/fsr209View ArticleGoogle Scholar
- Okubo A: Diffusion and ecological problems: mathematical models. Berlin: Springer; 1980.Google Scholar
- Øresland V: Feeding of the chaetognaths Sagitta elegans and S. setosa at different seasons in Gullmarsfjorden, Sweden. Mar Ecol Prog Ser 1987, 39: 69–79.View ArticleGoogle Scholar
- Orr WJC: Statistical treatment of polymer solutions at infinite dilution. Trans Faraday Soc 1947, 43: 12–27.View ArticleGoogle Scholar
- Parry DA: Structure and functioning of the gut in Spadella cephaloptera and Sagitta setosa . J Mar Biol Assoc UK 1944,26(01):16–36. 10.1017/S0025315400014430View ArticleGoogle Scholar
- Petrik CM, Kristiansen T, Lough RG, Davis CD: Prey selection by larval haddock and cod on copepods with species-specific behavior: an individual-based model analysis. Mar Ecol Prog Ser 2009, 396: 123–143.View ArticleGoogle Scholar
- Railsback SF: Concepts from complex adaptive systems as a framework for individual-based modelling. Ecol Model 2001,139(1):47–62. 10.1016/S0304-3800(01)00228-9View ArticleGoogle Scholar
- Reeve MR: The biology of chaetognaths. I. Quantitative aspects of growth and egg reproduction in Sagitta hispida . In Marine food chains. Edited by: Steele JH. Edinburgh: Oliver & Boyd; 1970:168–189.Google Scholar
- Rothschild BJ, Osborn TR: Small-scale turbulence and plankton contact rates. J Plankton Res 1988,10(3):465–474. 10.1093/plankt/10.3.465View ArticleGoogle Scholar
- Saito H, Kiørboe T: Feeding rates in the chaetognath Sagitta elegans : effects of prey size, prey swimming behaviour and small-scale turbulence. J Plankton Res 2001,23(12):1385–1398. 10.1093/plankt/23.12.1385View ArticleGoogle Scholar
- Seuront L: Effect of salinity on the swimming behaviour of the estuarine calanoid copepod Eurytemora affinis . J Plankton Res 2006,28(9):805–813. 10.1093/plankt/fbl012View ArticleGoogle Scholar
- Seuront L: Hydrocarbon contamination decreases mating success in a marine planktonic copepod. PLoS ONE 2011,6(10):e26283. 10.1371/journal.pone.0026283PubMed CentralView ArticlePubMedGoogle Scholar
- Seuront L, Schmitt FG, Brewer MC, Strickler JR, Souissi S: From random walk to multifractal random walk in zooplankton swimming behavior. Zool Stud 2004,43(2):498–510.Google Scholar
- Seuront L, Yamazaki H, Souissi S: Hydrodynamic disturbance and zooplankton swimming behaviour. Zool Stud 2004,43(2):376–387.Google Scholar
- Souissi S, Seuront L, Schmitt FG, Ginot V: Describing space-time patterns in aquatic ecology using IBMs and scaling and multi-scaling approaches. Nonlinear Anal-Real 2005,6(4):705–730. 10.1016/j.nonrwa.2004.12.013View ArticleGoogle Scholar
- Thygesen UH, Nilsson LAF, Andersen KH: Eulerian techniques for individual-based models with additive components. J Mar Syst 2007,67(1–2):179–188.View ArticleGoogle Scholar
- Tilman D, Kareiva PM: Spatial ecology: the role of space in population dynamics and interspecific interactions. San Diego: Academic Press; 1997.Google Scholar
- Turner JT: The importance of small planktonic copepods and their roles in pelagic marine food webs. Zool Stud 2004,43(2):255–266.Google Scholar
- Uchmański J, Grimm V: Individual-based modelling in ecology: what makes the difference? TREE 1996,11(10):437–441.PubMedGoogle Scholar
- Uttieri M, Zambianchi E: On the fractal characterisation of zooplankton motion: applications and perspectives. In Classification and application of fractals: new research. Edited by: Mitchell EW, Murray SR. New York: Nova Science Publishers, Inc.; 2012:113–129.Google Scholar
- Uttieri M, Mazzocchi MG, Nihongi A, Ribera d'Alcalà M, Strickler JR, Zambianchi E: Lagrangian description of zooplankton swimming trajectories. J Plankton Res 2004,26(1):99–105. 10.1093/plankt/fbg116View ArticleGoogle Scholar
- Uttieri M, Zambianchi E, Strickler JR, Mazzocchi MG: Fractal characterization of three-dimensional zooplankton swimming trajectories. Ecol Model 2005,185(1):51–63. 10.1016/j.ecolmodel.2004.11.015View ArticleGoogle Scholar
- Uttieri M, Cianelli D, Strickler JR, Zambianchi E: On the relationship between fractal dimension and encounters in three-dimensional trajectories. J Theor Biol 2007,247(3):480–491. 10.1016/j.jtbi.2007.03.026View ArticlePubMedGoogle Scholar
- Uttieri M, Nihongi A, Mazzocchi MG, Strickler JR, Zambianchi E: Pre-copulatory swimming behaviour of Leptodiaptomus ashlandi (Copepoda: Calanoida): a fractal approach. J Plankton Res 2007,29(suppl 1):i17-i26.Google Scholar
- Uttieri M, Paffenhöfer G-A, Mazzocchi MG: Prey capture in Clausocalanus furcatus (Copepoda: Calanoida). The role of swimming behaviour. Mar Biol 2008,153(5):925–935.Google Scholar
- Uttieri M, Sabia L, Cianelli D, Strickler JR, Zambianchi E: Lagrangian modelling of swimming behaviour and encounter success in co-occurring copepods: Clausocalanus furcatus vs. Oithona plumifera. J Mar Syst 2010,81(1–2):112–121.View ArticleGoogle Scholar
- Weissburg MJ, Doall MH, Yen J: Following the invisible trail: kinematic analysis of mate-tracking in the copepod Temora longicornis . Phil Trans R Soc London B 1998, 353: 701–712. 10.1098/rstb.1998.0236View ArticleGoogle Scholar
- Zambianchi E, Griffa A: Effects of finite scales of turbulence on dispersion estimates. J Mar Res 1994, 52: 129–148. 10.1357/0022240943076731View ArticleGoogle Scholar
- Zar JH: Biostatistical analysis. Englewood Cliffs, NJ: Prentice Hall; 1984.Google Scholar
- Ziarek JJ, Nihongi A, Nagai T, Uttieri M, Strickler JR: Seasonal adaptations of Daphnia pulicaria swimming behaviour: the effect of water temperature. Hydrobiologia 2011,661(1):317–327. 10.1007/s10750-010-0540-0View ArticleGoogle Scholar
- Zollner PA, Lima SL: Behavioral tradeoffs when dispersing across a patchy landscape. Oikos 2005,108(2):219–230. 10.1111/j.0030-1299.2005.13711.xView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.