homeostatic swimming of zooplankton upon crowding: the
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royalsocietypublishing.org/journal/rsif
Research
Cite this article: Uttieri M, Hinow P, Pastore
R, Bianco G, Ribera d’Alcalá M, Mazzocchi MG.
2021 Homeostatic swimming of zooplankton
upon crowding: the case of the copepod
Centropages typicus. J. R. Soc. Interface 18:20210270.
https://doi.org/10.1098/rsif.2021.0270
Received: 31 March 2021
Accepted: 1 June 2021
Subject Category:Life Sciences–Physics interface
Subject Areas:ecosystems, biophysics
Keywords:Centropages typicus, crowding, random walk,
fractal dimension, mean square displacement,
ecological temperature
Authors for correspondence:Marco Uttieri
e-mail: [email protected]
Peter Hinow
e-mail: [email protected]
Raffaele Pastore
e-mail: [email protected]
© 2021 The Author(s) Published by the Royal Society. All rights reserved.
Electronic supplementary material is available
online at https://doi.org/10.6084/m9.figshare.
c.5478066.
Homeostatic swimming of zooplanktonupon crowding: the case of the copepodCentropages typicus
Marco Uttieri1,2, Peter Hinow3, Raffaele Pastore4, Giuseppe Bianco5,Maurizio Ribera d’Alcalá1 and Maria Grazia Mazzocchi1
1Department of Integrative Marine Ecology, Stazione Zoologica Anton Dohrn, Villa Comunale,Naples 80121, Italy2CoNISMa, ULR Partehnope, Piazzale Flaminio 9, Rome 00196, Italy3Department of Mathematical Sciences, University of Wisconsin–Milwaukee, Milwaukee, WI 53201, USA4Department of Chemical, Materials and Production Engineering, University of Naples Federico II,P.le Tecchio 80, Napoli 80125, Italy5Department of Biology, Lund University, Sölvegatan 37, Lund 22362, Sweden
MU, 0000-0001-7026-0156; PH, 0000-0001-5875-1470; RP, 0000-0002-8335-9961;GB, 0000-0002-6144-2959; MRdA, 0000-0002-5492-9961
Crowding has a major impact on the dynamics of many material andbiological systems, inducing effects as diverse as glassy dynamics andswarming. While this issue has been deeply investigated for a variety ofliving organisms, more research remains to be done on the effect of crowd-ing on the behaviour of copepods, the most abundant metazoans on Earth.To this aim, we experimentally investigate the swimming behaviour, used asa dynamic proxy of animal adaptations, of males and females of the calanoidcopepod Centropages typicus at different densities of individuals (10, 50 and100 ind. l−1) by performing three-dimensional single-organism tracking. Wefind that the C. typicus motion is surprisingly unaffected by crowding overthe investigated density range. Indeed, the mean square displacements asa function of time always show a crossover from ballistic to Fickianregime, with poor variations of the diffusion constant on increasing thedensity. Close to the crossover, the displacement distributions display expo-nential tails with a nearly density-independent decay length. The trajectoryfractal dimension, D3D≅ 1.5, and the recently proposed ‘ecological tempera-ture’ also remain stable on increasing the individual density. This suggeststhat, at least over the range of animal densities used, crowding does notimpact on the characteristics of C. typicus swimming motion, and that ahomeostatic mechanism preserves the stability of its swimming performance.
1. IntroductionCrowding dramatically affects the dynamics of a wide variety of material andliving systems, from colloidal suspensions to animal assemblages. Crowdingleads, indeed, to complex effects, even in systems as simple as Brownianhard-sphere suspensions [1], where the dynamics is uniquely determined bythe excluded volume and the entropy. On increasing the volume fraction in anarrow range, in fact, particle motion becomes sluggish and develops long-range correlations [1], up to a structural arrest. Such a ‘glassy’ slow-down oncrowding also characterizes complex fluids and a variety of living and activesystems, including microswimmers, cells and ants [2–4]. However, antagonisticdynamics may also be at play upon crowding. For example, swarming is a spec-tacular collective effect preventing the slow-down in crowded conditions foranimals capable of coordinated response [5,6].
Little is known, however, about the effect of crowding on copepods, themost abundant metazoans on Earth [7]. These crustaceans play a key role inthe functioning of aquatic ecosystems, sustaining the vertical fluxes of matterand energy and linking lower and higher trophic levels [8]. In natural conditions,
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copepod density is in the range of few individuals per litre.However, copepods can also occur in dense aggregationswith densities greater than 103–104 ind. l−1 [9–11]. Similarvalues can be reached also in artificial conditions when alarge number of individuals is needed for experimental trials[12], for aquaculture purposes [13–15] or when recording themotion of copepods [16]. As a drawback, high densities mayimpair copepod fitness, reducing food availability and eggproduction while increasing mortality [17–19], boosting theconcentration of metabolic products and reducing water qual-ity [20], and decreasing the respiration rate [15]. In addition,increasing copepod abundance can favour the transfer of oildroplets to lower trophic level by direct manipulation andegestion [21]. These arguments emphasize the importance ofimproving current knowledge about the effects of crowdingon the fitness of these tiny crustaceans.
The calanoid copepod Centropages typicus Krøyer, 1849 isa temperate Atlantic species widespread in coastal and neriticwaters over a wide latitudinal range, including the adjacentMediterranean Sea, where it is abundant especially in theWestern Basin [22]. It is one of the best known copepodspecies (approx. 1.5mm total body length in both adultsexes), thanks to extensive studies investigating its ecology,biology and behaviour [23,24]. In the present contribution,we investigate the potential effects of different populationdensities on the swimming performance of this species. Incopepods, changes in motion patterns as a result of endogen-ous and/or exogenous stresses may affect the search for foodand mates, as well as the escape from predators [25,26].Alterations in motion features can also be employed toassess, on short temporal scales, sublethal behavioural end-points [27]. The goal of this study is to evaluate if and towhat extent does crowding alter the characteristics of C. typi-cus swimming motion, and consequently the overall fitness ofthis species. To address this topic, we carried out an inte-grated analysis by combining multiple descriptors to gathera detailed characterization of the three-dimensional (3D) C.typicus swimming behaviour under different population den-sities. First, length and time scales characterizing C. typicusmotion are quantified through the mean square displacement(MSD) and the displacement distributions [28]. Next, the frac-tal dimension of the trajectories is estimated [29,30] in order toprobe their degree of space-occupancy. Finally, the ecologicaltemperature [31] is exploited to evaluate the activity level ofC. typicus under different crowding levels. The integration ofthese indicators leads to join the macroscopic (ensembleaveraged) properties of the system to the microscopic(single-organism) behaviour of its components.
2. Methods2.1. Experimental designIndividuals of Centropages typicus were collected at thecoastal station LTER-MC (40.8080° N; 14.2500° E) in the Gulf ofNaples (Tyrrhenian Sea, western Mediterranean Sea), wherethis species is abundant and shows a clear seasonal cycle withregular peaks in late spring–early summer [32,33]. It is one ofthe most abundant copepod species at LTER-MC, where itaccounts for an annual mean of 6:8+ 2:5% of the totaland very diversified copepod assemblage, and reaches peakconcentrations of 2.4 × 103 ind. l−1 [32].
Zooplankton samples were collected in spring 2008 throughgentle vertical tows in the upper 50m of the water column usinga Nansen net (200 μm mesh size) mounting a 5 l non-filteringcod end. In the laboratory, the samples were diluted and C. typi-cus individuals were manually sorted with a large-bore glasspipette and checked for stage, sex and overall integrity underan Olympus SZH10 dissecting microscope. Only undamagedand healthy adult males and females were kept for thesubsequent video recordings.
Freely swimming individuals were recorded using a 3Dsystem, equipped with two orthogonal digital CCD cameras(Sony XCD-X700, operating at 15 fps) mounting custom tele-centric lenses; the filming was carried out in the absence ofvisible light, using only two background LED infrared panels(780 nm, 18mW) (see [34]). The system guaranteed a spatial resol-ution of 78 μm, sufficient to discriminate between singleindividuals. The cameras overlooked an experimental volume of80 × 80 × 60 mm (384ml), representing a fraction (approx. 38%)of the entire aquarium (1 l) where the animals could swim freelywithout any wall effect. Apart from the small volume, theexperimental conditions were similar to the environment ofC. typicus. The aquarium was filled with seawater sampledin situ, enriched with 7:0� 103 microzooplankton cells l�1 and1:5� 107 phytoplankton cells l�1. All experiments were carriedout at 17°C, reproducing environmental temperature.
Three different copepod densities were tested, namely 10, 50and 100 ind. l−1. These values were 4 to 42 times higher than thehighest C. typicus abundance in situ [32]. The first value tested(10 ind. l−1) is in the range of densities used in the literature incopepod behavioural studies [16]. Such densities increase theprobability of recording a sufficient number of tracks, while atthe same time leaving enough free space to the individuals tomove independently of the others [16]. The other two values,50 and 100 ind. l−1, were instead selected to test the effect ofincreasing densities on the individual behaviour. Each sex wasrecorded separately to avoid any mate-seeking behaviour.Copepods were acclimated for a minimum of 15 min beforerecording. The acclimatization was done each time a newgroup of individuals was added to increase the density.
Once recorded, the tracks described by single individualswere digitized using a custom software [34], providing the(x, y, z) coordinates at each time step t. The length of the tracksvaried, depending on the time each copepod was in focus inthe arena. For the evaluation of the fractal dimension and forthe calculation of the ecological temperature, only tracks lastingapproximately 60 s were considered. Such threshold was necess-ary to ensure a statistically meaningful time span to characterizethe tracks efficiently. For the calculation of the mean squared dis-placement and the displacement distributions, a wider datasetwas instead used, including all tracks lasting more than 5 s.
2.2. Characterization of C. typicus swimming behaviourThe 3D tracks digitized were numerically characterized by inte-grating different descriptors, each focusing on a specific aspectof C. typicus motion. In the following, we illustrate briefly theparameters used; a more detailed description is given in theelectronic supplementary material.
The MSD and the associated distribution of displacementswere calculated over several decades in time, length and prob-ability. These descriptors have been used in different fields ofinvestigation, including zooplankton behaviour [28,35]. TheMSD represents the average square distance ⟨Δr2(Δt)⟩ coveredby an individual over a time interval Δt, whereas the displace-ment distribution Gs(l, Δt) is the probability density that anindividual has moved a distance l over the time Δt. These metricswere measured both on the 3D tracks, as well as separately on thehorizontal plane (x, y) and vertical direction (z) to distinguish
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Figure 1. Representative 3D trajectories of freely swimming Centropages typicus females (a) and males (b) at the experimental concentrations of 10 (blue), 50(green) and 100 (red) ind. l−1. For each condition and sex, the trajectories of two randomly chosen individuals are plotted. The starting point of each track is markedwith an orange circle. The trajectories are plotted in an isometric reference frame resembling the observation window of the experimental set-up.
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any possible effect induced by gravity. Since there are no relevantdifferences, we find it more concise and instructive to focusonly on the 3D case. These descriptors allow discriminatingbetween standard (Fickian) and anomalous (ballistic) diffusionin C. typicus motion [36] (see also §1.1 of electronicsupplementary material).
Zooplankton tracks have been robustly described in terms oftheir fractal properties [30]. The fractal dimension D provides anestimate of the space-filling tendency of an object. In this work,the 3D fractal dimension D3D of C. typicus tracks was estimatedto discern any possible behavioural change induced by crowding,using the protocol described in [29].D3Dwas also used to comparethe motion of males and females exposed to the same density oforganisms. The fractal dimension was also estimated on the 2Dprojections of the tracks (Dxy,Dxz andDyz) to evaluate any possibleanisotropy in the motion. D3D, Dxy, Dxz and Dyz values were stat-istically analysed to ascertain any significant difference among thegroups. The groups were compared using a non-parametric Krus-kal–Wallis H test, if necessary complemented with a post hocmultiple pairwise comparison with Dunn–Šidàk’s correction.The accuracy in the estimations was assessed through the coeffi-cient of determination (R2) of the regression lines and the rootmean square error (RMSE).
The notion of ecological temperature is inspired by theMaxwell–Boltzmann kinetic theory [31] and follows earlier ideasin [37]. The mean square speed v2 of individual C. typicus wasemployed to define a ‘single-organism temperature’ accountingfor individual activity, considering that single specimens maymodify their motion activity as a function of physiological trig-gers. The cumulative distribution functions of the individualecological temperatures were then used to evaluate differencesin motion behaviour associated with crowding conditions experi-enced by C. typicus. The Kolmogorov–Smirnov test was used todistinguish pairs of empirical cumulative distributions.
3. ResultsThe behaviour of Centropages typicus (figure 1) is characterizedby three motion states, namely swimming, sinkingand jumping. In the typical swimming habit, the body ofC. typicus females and males is slightly oblique, describing ahelical path. At variable frequency, the copepods alternate
such behaviour to sinking, i.e. a passive downward dis-placement without any movement of the cephalothoracicappendages. Only rarely do C. typicus females and males dis-playquick (less than0.1 s) and long (greater than10 mm) jumps.
3.1. Mean square displacementFigure 2a shows the 3D MSD as a function of time for allrecorded experiments. We use a bi-logarithmic plot to have aclear overview of the motion over the different investigatedtime scales. On this scale, the absence of a clear-cut trendupon changing the concentration, or sharp differences betweenfemales andmales, is clearly noticeable. Conversely, a commonbehaviour emerges from all datasets investigated. At shorttimes, Δt≤ 1 s, the MSD increase is compatible with ballisticdiffusion, ⟨Δr2(Δt)⟩∝ Δt2. At intermediate time, Δt > 1 s, asmooth crossover takes place, where the MSDs show a cross-over to a Fickian regime, ⟨Δr2(Δt)⟩ = 6 DLΔt, with DL beingthe long-time diffusion coefficient. Fickian diffusion is in factattained at times close to the overall observation time, Δt≃10 s. Overall, this behaviour indicates C. typicus females andmales perform straight and sudden steps at short times (onthe timescale of the ballistic regime) and subsequent steps arepoorly correlated, giving rise to a random-walk-like motionon longer timescales (corresponding to the Fickian regime).
To get more details on changes induced by individualdensity and sex, figure 2b shows the same data as figure 2ain linear scale. This implies that we are now focusing onlyon the last decade in time (≃1–30 s), corresponding to theattaining of the Fickian regime. Within this visualization,some spreading among different datasets is detectable. Inorder to readily capture this variation, we measured the dif-fusion coefficient DL, through a long-time Fickian fit to thedata. The obtained estimation of DL is shown in the inset offigure 2b as a function of the individual density, for bothfemales and males. The diffusion coefficient lies in therange 4–7 mm2 s−1 and does not show a clear, monotonictrend at increasing densities. Conversely, data for bothfemales and males display a moderate minimum at the inter-mediate investigated density of 50 ind. l−1 a factor of about
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Figure 2. (a) Bi-logarithmic and (b) linear plot of MSD as a function of time for females (full symbols) and males (empty symbols), at different individual densities,as indicated. The lines in (a) are guides to the eyes, indicative of ballistic diffusion (dashed), ⟨Δr2(Δt)⟩∝ t2, and Fickian diffusion (solid), ⟨Δr2(Δt)⟩∝ Δt,respectively. The lines in (b) are long-time fits ⟨Δr2(Δt)⟩ = 6DLΔt for females (solid) and males (dashed), with the diffusion coefficient DL being the fittingparameter. Inset: DL as a function of the individual density for both females (full symbols) and males (empty symbols). The symbol size is similar to theerrors on DL, as obtained from the fits.
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Figure 3. Displacement distribution Gs(l, Δt) of females (full symbols) and males (empty symbols) as function of the displacement length l, and at different timesΔt, as indicated. The individual density is 10, 50 and 100 ind. l−1 in (a–c), respectively.
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0.75 smaller than the average diffusion coefficient. This resultcalls for future experimental campaigns to sample the inter-mediate density range through many other experiments, soas to clarify whether the observed minimum originatesfrom a smooth trend or simply from statistical noise. Wealso note that at the highest density, i.e. 100 ind. l−1, themale diffusion coefficient is a factor of about 1.2 larger thanthe female one. This suggests that, on average, males movefaster than females, consistently with previous experimentsthat also measured the diffusion coefficient for the samespecies [38].
With regards to the spatial isotropy of the motion, theresults drawn for the 3D MSD are found to be qualitativelysimilar to those obtained analysing the horizontal and verti-cal components separately.
3.2. Distribution of individual displacementsFigure 3 provides a broad overview of the distribution oforganism displacements, Gs(l, Δt). The three panels refer tothe different investigated crowding conditions and reportGs(l, Δt) both for males and females. The selected time Δtfalls within the ballistic regime (Δt = 0.67 s), the ballistic–Fickian crossover (Δt = 2 s), and the Fickian regime (Δt = 10 s),
respectively. A distinctive feature emerging from figure 3is the clear evidence of fat (i.e. fatter than Gaussian),exponential-like tails at the intermediate time.
At shorter time, the distribution is instead more similar toa Gaussian. A hint to restore the Gaussian property seems tobe present also at the longest time in figure 3, as expected fora standard random walk. Overall, figure 3 confirms that nomajor differences exist between the groups of males andfemales. Slight deviations in the tails are observed only atthe intermediate time Δt = 2 s, and densities 50 and100 ind. l−1, where the characteristic decay length of malesis a factor about 1.75 and 1.3, respectively, larger than thefemale one (as obtained through exponential fits).
To better observe this aspect as well as the exponentialnature of the tails at intermediate time, figure 4 shows adirect comparison between the distribution at the differentdensities and a time Δt = 1 s, corresponding to the overallduration ballistic regime (i.e. right at the edge between bal-listic regime and ballistic–Fickian crossover). By doing so,the distributions of single ballistic steps are monitored.Accordingly, the exponential tails appearing in figure 4reveal the heterogeneous distribution of these steps.Remarkably, datasets corresponding to different densitiesnearly overlap both for females (figure 4a) and males
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Figure 4. The displacement distribution Gs(l, Δt) for females (a) and males (b), as a function of the displacement length l, at an intermediate time Δt = 1 s, andfor the indicated densities. The solid lines are guides to the eyes, indicative of an exponential decay, Gs(l, Δt)∝ e−l/ξ, with ξ = 2 mm.
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Figure 5. Boxplots of the D3D values for Centropages typicus females (red) and males (sky blue) under the different crowding conditions analysed. On the right ofeach box, the number of samples for each group is reported, while on the left the 25th, 50th and 75th percentiles are indicated.
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(figure 4b), indicating that the decay length is essentiallyconstant at different crowding conditions.
Concerning the spatial isotropy of the displacement distri-bution, the only noticeable difference is a local maximum inGs(l,Δt) at short time (Δt= 0.67 s, 1 s and 2 s in figures 3 and 4) andrelatively small lengths (e.g. ≃2.5 mm in figure 4), which isclearly detectable in three dimensions, but absent in the distri-bution of the horizontal displacements. This local maximum isreadily ascribed to the gravity-induced drift in the vertical direc-tion. As time progresses, such a passive displacement isprogressively overwhelmed by the ‘active’ motion, with theconsequent disappearing of the local maximum.
3.3. Fractal dimensions of swimming tracksThe characterization of the tracks in terms of their D3D revealsseveral intriguing aspects of the motion behaviour of
C. typicus. All tracks display an intermittent and recursivealternation in the magnitude of the instantaneous 3D dis-placements at different temporal scales, which constitutes anunderlining signature of a fractal process (not shown). As evi-dent from figure 5, both females and males describemoderately convoluted tracks, with meanD3D values between1.15 and 1.21 (table 1). Remarkably, this range of values fallsin between the fractal dimensions of ballistic motion andrandom walk (1 and 2, respectively), consistent with theMSD results; consequently, the estimated D3D can be con-sidered as characteristic of a mixed, ballistic/random-walk-like motion. The non-parametric Kruskal–Wallis analysis ofvariance results in p = 0.64, indicating overall statistical simi-larity among the D3D recorded both by each sex in the threedifferent conditions, and by females and males at the samelevel of crowding, as further confirmed by the boxplots(figure 5). The R2 and RMSE returned values close to one
Table 1. D3D, R2 and RMSE values for Centropages typicus females and
males under the different crowding conditions tested. All values arestatistically similar.
crowding condition D3D R2 RMSE
C10 ind: l�1 1.20 ± 0.09 0.98 ± 0.02 0.13 ± 0.06
C50 ind: l�1 1.17 ± 0.09 0.97 ± 0.02 0.16 ± 0.05
C100 ind: l�1 1.18 ± 0.10 0.98 ± 0.01 0.12 ± 0.05
F10 ind: l�1 1.15 ± 0.07 0.98 ± 0.01 0.13 ± 0.04
F50 ind: l�1 1.21 ± 0.07 0.96 ± 0.02 0.18 ± 0.03
F100 ind: l�1 1.16 ± 0.10 0.97 ± 0.01 0.15 ± 0.04
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and zero, respectively, in allD3D estimates (table 1), thus ensur-ing robustness of the method.
In 2D, for each crowding condition, the Dxy, Dxz and Dyz
of a given sex are always statistically similar (Kruskal–Wallis analysis of variance p > 0.01) (table 2). This indicatesthat both females and males move isotropically in theirenvironment. This outcome is backed up by the analysis ofthe boxplots (not shown). The confidence in the fractaldimension estimations is ensured by R2 close to 1 and bylow RMSE values (table 2). Additionally, the analysis of theinterquartile ranges from the boxplots points to a comparableinter-individual variability in the fractal dimension, thusmirroring crowding unbiased behaviour in both sexes.
3.4. Ecological temperatureThe cumulative distribution functions of the ecological temp-erature are shown in figure 6 and in electronic supplementarymaterial, figure S1. The Kolmogorov–Smirnov test leads tothe rejection of the null hypothesis that the data are drawnfrom the same distribution in all cases (electronic supple-mentary material, table SI). The deviation from the overallbehaviour shown by the males at their lowest density iswithin the range of variability of the ecological temperatureamong the treatments. As a matter of fact, the averageecological temperatures of the different experiments are con-stant within the standard deviation (figure 7). For eachdensity, the comparison between the sexes is shown inelectronic supplementary material, figure S1.
4. DiscussionInvestigating the modalities by which individual organismsintegrate exogenous information from their environment aswell endogenous stimuli, and use them accordingly to modu-late their behaviour, is central to relate movement ecology tocommunity theory and biodiversity research [39]. Individualmovement in animal kingdom displays a wide gamut of pat-terns and typologies [40,41]. It can be modulated by triggersworking at individual, population, community or ecosystemlevels, and in a feedback loop, variations in movement canimpact on each of these hierarchically organized levels ofaggregation [40]. Animal movement is generally quanti-tatively characterized by analysing the trajectory patterns[40,42]. Such quantification permits the comparison of differ-ent behaviours and the emergence of peculiar strategies oradaptations to specific conditions. The results in the present
study reveal new features of the behavioural adaptationsof Centropages typicus, a planktonic copepod common in tropi-cal to temperate waters of the Northern Hemisphere. Adultfemales and males were exposed to three crowding conditions(10, 50 and 100 ind. l−1), and their swimming trajectories wereanalysed in terms of mean square displacement, displacementdistribution, fractal dimension and ecological temperature.
Behaviour can hint at possible stress in organisms that mayimpact on their physiological state [43]; consequently, a changein one of these parameters may reflect a change in the others.The complexity in spatial and temporal patterns of C. typicusswimming motion can therefore be used as a proxy ofbehavioural stress [27,44], unveiling important underpinningbiological and ecological responses. The integrated outcomeemerging from the application of the above-listed descriptorsexcludes any appreciable effect of crowding on the behaviour-al metrics tested. Inter-individual variations in movement canbe linked to peculiar traits in the species investigated [40].Independently of the treatment considered, both C. typicusfemales and males manifest coherent motion patterns, indica-tive of negligible inter-individual changes. This suggeststhat the responses recorded in this study do not depend onthe internal state of the single individuals, but rather can beconsidered as population-level adaptations.
4.1. Homeostatic mechanismsThe scenario emerging from our results suggests thata homeostatic mechanism may be at play, preserving theswimming performance typical of dilute conditions in spiteof the increasing individual density. Here, we adopt theterm homeostatic as commonly used in a broad context—including generic models of dynamical systems [45], brainplasticity [46], artificial intelligence [47], networks [48]—toindicate a category of dynamical processes where a systemcontinuously self-organizes in order to maintain a targetedstationary condition.
Indeed, a simple possibility to understand our results isthat the organisms tend to avoid clustering and dynamicallymaintain a spatial distribution that closely maximizes theiraverage distance λ, making the effect of crowding negligible,at least over the investigated density range. Indeed, maximiz-ing λ closely corresponds to minimizing the encounter rate,the local food consumption and, of course, any other typeof interaction among different organisms. As a matter offact, homogeneous spatial distributions of individuals domaximize the average inter-organism distance.
It is therefore interesting to compare the values of λ forspatially homogeneous distributions at the different densitieswith the characteristic length scales that can be identified fromthe MSD. Assuming a spatially homogeneous distribution ofindividuals, the average distance among nearest neighboursis given by λ = (V/N )1/3. This corresponds to λ = 46.4, 27.1,21.5 mm for the experiments at densities 10, 50, 100 ind. l−1,respectively. On the other hand, the typical, closely density-independent length of the ballistic step is ξb≃ 4 mm, as esti-mated by evaluating ⟨Δr2(Δt)⟩1/2 at the end of the ballisticbehaviour Δt = 1 s. Similarly, the root MSD at Δt = 25 s, atime-span corresponding to the typical trajectory duration,provides an estimation of the overall distance, ξmax, travelledby an organism during the whole observation time. Forinstance, ξmax≃ 28 mm, at the highest density. Accordingly,the average inter-organism distance λ is always significantly
Table 2. Isotropic fractal behaviour of Centropages typicus females and males. For each crowding condition and two-dimensional projection (xy, xz and yzplanes), the values for D2D, R
2 and RMSE are reported. All values are statistically similar.
crowdingcondition xy xz yz parameter
C10 ind: l�1 1.11 ± 0.07 1.20 ± 0.09 1.21 ± 0.08 D2D0.99 ± 0.01 0.99 ± 0.01 0.99 ± 0.01 R2
0.09 ± 0.02 0.09 ± 0.03 0.09 ± 0.03 RMSE
C50 ind: l�1 1.18 ± 0.12 1.25 ± 0.13 1.26 ± 0.12 D2D0.98 ± 0.01 0.99 ± 0.01 0.99 ± 0.01 R2
0.12 ± 0.04 0.11 ± 0.03 0.11 ± 0.03 RMSE
C100 ind: l�1 1.15 ± 0.11 1.23 ± 0.08 1.24 ± 0.12 D2D0.99 ± 0.01 0.99 ± 0.01 0.99 ± 0.01 R2
0.09 ± 0.04 0.09 ± 0.02 0.10 ± 0.03 RMSE
F10 ind: l�1 1.11 ± 0.07 1.22 ± 0.07 1.23 ± 0.09 D2D0.99 ± 0.01 0.99 ± 0.07 0.99 ± 0.01 R2
0.10 ± 0.03 0.10 ± 0.03 0.09 ± 0.02 RMSE
F50 ind: l�1 1.27 ± 0.11 1.27 ± 0.10 1.30 ± 0.11 D2D0.98 ± 0.01 0.99 ± 0.01 0.98 ± 0.01 R2
0.12 ± 0.03 0.11 ± 0.02 0.13 ± 0.03 RMSE
F100 ind: l�1 1.16 ± 0.14 1.21 ± 0.10 1.24 ± 0.11 D2D0.98 ± 0.01 0.99 ± 0.01 0.98 ± 0.01 R2
0.11 ± 0.03 0.10 ± 0.02 0.11 ± 0.03 RMSE
20 40 60 800
0.2
0.4
0.6
0.8
1.0
10 20 30 40 50 60 700
0.2
0.4
0.6
0.8
1.0
density 10density 50
density 100
density 10density 50
density 100
v2 (mm2 s–2) v2 (mm2 s–2)
(a) (b)
Figure 6. Cumulative distribution functions of ecological temperature of ensembles of C. typicus females (a) and males (b) at different densities.
females males
density 10
density 50
density 100
v2 (m
m2
s–2)
0
10
20
30
40
50
60
70
Figure 7. The averages and standard deviations of the ecological temperatures for both sexes.
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larger than the ballistic step and comparable with the overalltravelled distance, even at the largest individual density.
Further exploiting this framework, it is possible to evaluatethe timescales, τo, on which the regions explored by nearestneighbour individuals start overlapping. This is obtained byconsidering that, by definition, ⟨Δr2(τo)⟩
1/2 = λ/2 and that, onthe length scale λ, the diffusion is closely Fickian-like,⟨Δr2(τo)⟩ = (6DLτo), which results in the relation (6DLτo)
1/2 =λ/2. By doing so, we obtain τo = λ2/24DL ≃ 3.5 s. This timecan be interpreted as a very lower boundary for the waitingtime needed for an individual to meet one of its neighbours.Overall, the just discussed computations demonstrate that ahomogeneous spatial distribution ensures negligible inter-actions among individuals and, in turn, poor crowdingeffects over the whole range of investigated densities.
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4.2. Behavioural adaptations in C. typicus motionThe occurrence of the mixed ballistic–diffusive motion in theMSD behaviour is emerging as a quite universal property inswimming zooplankton [35], as also recently reported for adifferent calanoid species, Clausocalanus furcatus [28]. Thecharacteristic time and length scales for C. furcatus are alsostrikingly similar to those recorded for C. typicus, pointingto possible evolutionary mechanisms in copepod movement.
The ballistic-to-diffusive crossover in the MSD seems tobe a quite universal property in swimming zooplankton, asreported for other species [28,35] but the shape of this cross-over may be species-dependent: smooth in C. typicus andalong a plateau in C. furcatus, at least in similar prey-richenvironmental conditions. This difference is likely attribu-table to the swimming behaviour of the two copepods.C. typicus is a cruise swimmer, moving continuously andsmoothly in the fluid [49]. The smooth transition hereshown is compliant with that reported for C. typicus nauplii[35], for adult Temora longicornis [35] and Acartia tonsa [43].On the other hand, C. furcatus has a unique motion patternwith repetitive looping [34,50] developing over scales compa-tible with the plateau in the MSD. The results of the presentinvestigation thus confirm the ability of MSD function to dis-criminate distinct behaviour and point to species-specific(and consequently motion-specific) emergent properties.
As concerns the displacement distributions, we note thatthe observed presence of exponential tails is an intriguing fea-ture. It has been previously reported for a variety of materialand biological systems and is typically ascribed to spatial cor-relations among displacements of different particles/organisms or to a distribution of single particle/organismdiffusion coefficients [1,51–53]. Testing whether similarmechanisms were at play also in the present experiments isan interesting problem that calls for further investigation.
Fractal dimension has been used to characterize theswimming motion of several copepod species and hasproved to be a robust descriptor of track convolution[50,54]. It is important to underline that the adoption of afractal approach has sometimes been criticized [55,56], someworks from the literature lacking the necessary formal robust-ness and preliminary verification of the applicability of fractalframework. In the present work, we have verified the self-similar signature, which is maintained over three to fourorders of magnitude, thus guaranteeing the safe applicabilityof this descriptor. The estimation of D3D reveals the mainten-ance of the same level of track convolution independent of
the crowding, while the parallel calculation of Dxy, Dxz andDyz points to an isotropic motion, without any directionalpreference. Numerical simulations [57–59] reveal the closerelationship between D3D and the encounter probabilitywith prey, predators and mates. As such, a given D3D rep-resents a behavioural trade off between maximization of‘positive’ encounters (with prey and mates) and minimizationof ‘negative’ ones (with predators). The description of trackswith statistically similar degree of convolution (both in 3Dand in 2D) suggests that the motion of C. typicus is optimizedalso in terms of potential disturbing effects due to crowding,and that the level of convolution set by the tracks represents adynamic adaptation to a variable environment. The resultshere presented also confirm the impossibility of retrieving3D information from 2D projections, as previously indicatedby [29] and [16]. This latter evidence supports the adoption ofspecifically 3D computational tools to evaluate the full degreeof convolution of space-occupying tracks. Overall, the fractalcharacterization carried out in the present study backs up thesuitability of this descriptor as a non-invasive assessment ofthe health of copepods [44]. The same approach has beenused to check for any anisotropy in the 2D motion of C. typi-cus. For each crowding condition, the Dxy, Dxz and Dyz of agiven sex are always statistically similar (Kruskal–Wallisanalysis of variance p > 0.01). This indicates that both femalesand males move isotropically in their environment, and con-sequently their motion does not display any directionalpreference but the same degree of convolution is maintainedover the three Cartesian planes.
In [31], the average squared velocitywas used as ameasurefor the activity level in a population of the freshwatercladoceran Daphnia pulicaria infected by Vibrio cholerae. Theinfected D. pulicaria showed a considerably higher level ofactivity (or, equivalently, higher ecological temperature),which would make them more easily detectable to predatorssuch as fish [60], verifying the suitability of the descriptor tohighlight behavioural differences. In the present study, weapply for the first time the ecological temperature to trajec-tories described by a marine copepod, extending the rangeof applications of this metric. The results indicate that theaverage ecological temperature does not show a clear trendon crowding for C. typicus, and is overall compatible witha constant value within the statistical uncertainty. Hence,the outcomes of ecological temperature do support theaforementioned picture of a homeostatic mechanism at play.
4.3. Adapting to crowding: ecological implicationsThe results gathered in this study specifically refer toC. typicus, and consequently cannot be automatically general-ized, as species-specific responses may differ. Nonetheless, itmight be realistic to expect that similar levels of crowding inother pelagic calanoids of similar size and cruising behaviour(e.g. Pseudocalanus elongatus, Eurytemora affinis, Temora styli-fera and Temora longicornis [61]) may lead to similar results.The experiments presented here were run over short-timewindows; therefore the behaviour recorded must be con-sidered as an immediate adaptation of C. typicus to anacute stress. Future efforts will be devoted to the investigationof any time-dependent behavioural acclimation, as done in[43]. It should also be considered that the highest crowdinglevels used in this study correspond to the lower end rangeof abundances often tested in the literature on other species,
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e.g. Acartia steueri [15], A. tonsa [19,43,62] and E. affinis [63].Future investigations will therefore need to include alsocrowding levels above the ones here tested.
The integrative approach applied in this study supportsbuilding a more complete description of the swimming per-formance of C. typicus, thus gaining a comprehensive pictureof any potential effect of crowding on their swimming fitness.It is worth underlining that, over similar ranges of densities,also the calanoid copepod E. affinis did not show significantvariations in movement descriptors [16]. At even muchhigher values (up to 80 ind. ml−1), the cyclopoid Oithona davi-sae showed metabolic rates in proportion to the temperaturealthough the motor activity parameters were unaffected byshort-term crowding [64,65]. The density levels adopted inthe present experiments could consequently be usedin observational and behavioural studies or in rearing con-ditions without expected consequences on animal health ormotion patterns.
The time span of the experiments also allowed excludingany relevant oxygen depletion in the set-up. Considering anindicative oxygen consumption rate of 0.157 μl O2 h
−1 cop−1
for C. typicus [66], at the highest copepod density used inour tests (100 ind. l−1) undersaturation might be criticalafter several hours. Running short-time experiments thusexcludes that the behaviour observed might have beeninfluenced by hypoxia.
The maintenance of similar values for all the descriptorsused in this work, independently of the crowding conditiontested, suggests an overall optimality in the swimming per-formance of C. typicus. Even at the highest density tested,none of the descriptors used showed significant variations.This indicates that, over the range of crowding tested, the typi-cal distance between the individuals is always large enough todetermine appreciable changes in the descriptors. Such an evi-dence can be explained considering the distance betweenindividuals. Assuming a perceptive radius of 1mm centredon C. typicus body, each individual would have a perceptivevolume of 4.19 mm3. An interaction between two organismswould be effective only when their perceptive volumeswould overlap. At the highest density tested, however, thetotal volume occupied by the individuals would representless than 0.1% of that of the experimental aquarium (1 l).Such a small ratio ensures that encounters are rare, providedthat the spatial distribution is homogeneous (no clustering),
as suggested in §4.1. Individual behaviour can be modulatedby triggers acting at different scales [40]. When both organ-isms involved in the interaction are moving, the shape of theencounter zone may impact the effective encounter rate [67].The assumption of a spherical encounter zone for C. typicus,however, can be considered a valid oversimplification atleast in the framework of the process-oriented model hereoutlined.
Fine-scale movement details may not only impact theoverall fitness of single organisms, but their consequencesmay also scale up to population, community and ecosystemlevels [40,41]. Individual behaviour can respond differently,with different outcomes at community level [68], but it canalso impact on the environment itself [69]. Based on this evi-dence, it is important to pay attention to variations inindividual movement, and include such details in predictivemodels to properly interpret ecological dynamics [40]. Alongthis line, the study here presented underlines the critical needto resolve individual-scale behaviour in copepods to aptlyunderstand the functioning of pelagic ecosystems underdifferent conditions.
Data accessibility. Data are provided as electronic supplementarymaterial.Authors’ contributions. G.B., M.R.d.A. and M.G.M. conceived the exper-iments. G.B. carried out the laboratory observations. M.U., P.H.and R.P. planned the analysis. M.U. performed the analysis of thefractal dimension. P.H. performed the analysis of the ecological temp-erature. R.P. performed the analysis of MSD and of the displacementdistribution. R.P. conceived the idea of the homeostatic mechanism.All authors interpreted the results. M.U., P.H. and R.P. drafted themanuscript. All authors revised the manuscript and gave finalapproval for publication and agree to be held accountable for thework performed therein.Competing interests. We declare we have no competing interests.
Funding. The experiments were performed in the framework of G.B.’sPhD thesis supported by the Stazione Zoologica Anton Dohrn.M.U.’s visit to the University of Wisconsin–Milwaukee in January2017 was supported by the Simons Foundation grant ‘Collaborationon Mathematical Biology’ (award no. 278436) to P.H. This workwas financially supported by Stazione Zoologica Anton Dohrn PhDprogramme and internal grant funds, and by Simons Foundationgrant ‘Collaboration on Mathematical Biology’ (award no. 278436).The funding agencies had no role in study design, data collectionand analysis, decision to publish, or preparation of the manuscript.Acknowledgements. The authors thank two anonymous reviewers forvaluable comments on the manuscript.
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