1. Despite the popular use of diffusion models to predict the spatial spread of populations over time, we currently know little about how diffusion rates change with the state of the environment or the internal condition of individuals. To address this gap in our understanding, we measured rates of spread for many populations of the rotifer Brachionus calyciflorus in a suite of well-replicated experiments.
2. In one set of experiments, we manipulated food availability and population density along a continuous range of densities. In a second set, we manipulated the internal state of entire populations via food deprivation and exposure to predator kairomones.
3. Across replicate populations, diffusion rates were positively correlated with conspecific density. Diffusion rates were negatively correlated with food availability, especially when conspecific density was high. Diffusion rates of food-deprived populations or those exposed to predation risk were lower than controls.
4. Our results provide direct experimental evidence that rates of population spread are conditional on population density, food availability, body condition and predation risk.
Despite much theoretical precedence, few empirical studies have progressed beyond simple representations of diffusion because it is extremely difficult to measure variation in diffusion rates under complex field conditions with free-ranging individuals. Here, we take a novel empirical approach to measure diffusion under controlled experimental conditions to test how diffusion rates are influenced by continuous variation in external environmental conditions and also the internal state of individuals. We use simple, one-dimensional, microcosms to test state-dependent diffusion in over 200 populations of the rotifer (Brachionus calyciflorus). Using small-scale, but highly replicated, experiments we were able to measure the degree of sensitivity of the diffusion process to consumer population density, resource abundance, and levels of hunger and predation risk experienced by the diffusing individuals, and evaluate whether interactions exist among either external (environmental) or internal (organismal) state variables.
Resource-dependent movement processes logically dictate that consumers spend more foraging time in locations with abundant resources than in those with lower resource density. Everything else being equal, populations of consumers can achieve this by reducing diffusion rates when resources are locally abundant and increasing diffusion rates when resources are scarce, as assumed in numerous theoretical models (Karieva & Odell 1987; Morris & Karieva 1991; Grunbaum 1998; Wilson & Richards 2000; Dwyer & Morris 2006; Avgar, Kuefler & Fryxell 2011). Population density–dependent processes may be expected to either suppress or inflate diffusion rates depending on the ecological circumstances. If rotifers move away from conspecifics to reduce competition for locally available resources, then diffusion rates may increase with population density (Shigesada, Kawasaki & Teramoto 1979; Turchin 1989). On the other hand, if rotifers are sensitive to predation risk, they may move towards conspecifics and thus diffuse less at higher densities (Courchamp, Clutton-Brock & Grenfell 1999). We tested these predictions by estimating rotifer diffusion coefficients across a range of experimentally manipulated population and resource densities.
Our experimental system consisted of a single resource species, the freshwater green algae Monoraphidium minutum, and a single consumer species, the planktonic rotifer Brachionus calyciflorus. These rotifers were originally isolated from natural population in Germany by the Institute for Freshwater Ecology and Inland Fisheries and donated to us by Prof. G. Fussmann (McGill University). This strain has been cultured for more than 10 years in laboratory populations, and we maintained cultures for a minimum of 6 months in our laboratory prior to conducting experiments, making no attempt to isolate particular clonal strains. Adults of this rotifer species self-propel via a band of ciliary cells that serve to capture food and generate movement (Clement 1987; Wallace et al. 2006). Given the extremely narrow perceptual field of this species (Salt 1987) and lack of prior observations of directed movement, we assume that the movement of any individual in the population approximates an uncorrelated, unbiased movement which is well described (at the population level) by the simple Fokker-Planck diffusion model. Our experiments reflect primarily the behaviours, and not demographic dynamics, of these populations as the average generational time of this species under similar environmental conditions is approximately 5 days (Guo, Snell & Yang 2011), whereas our experiments were conducted over a span of 8 h.
The resource, M. minutum, was batch-cultured on COMBO medium (Kilham et al. 1998). New cultures were inoculated biweekly, as experimental volume demanded, and old cultures were immediately discarded if contamination was detected. The consumer, B. calyciflorus, was likewise propagated and maintained in batches on M. minutum in COMBO. We used only asexual females in experiments based on our visual observations that there were no males or resting-egg-bearing females in the cultures used for experiments. New cultures were inoculated in 250-mL Erlenmeyer flasks weekly and, upon initial inoculation, contained algal and rotifer densities of approximately 2 × 106 mL−1 and 2 mL−1, respectively. As a source of predator kairomones for risk-conditioned experiments, we maintained cultures of the rotifer Asplanchna brightwelli (a natural predator of B. calyciflorus) in COMBO medium with the algae Cryptomonas erosa as a food source. These A. brightwelli were originally isolated in 2000 from a natural population in a pond system near Munich, Germany (J. J. Gilbert pers. comm.). We propagated batch cultures of A. brightwelli approximately biweekly in 250-mL Erlenmeyer flasks, making no attempt to isolate particular clonal strains. Media conditioned with this predator have a demonstrated effect on the consumer’s defensive spine formation (Gilbert & Waage 1967) and growth rates (Guo, Snell & Yang 2011) but have not previously been demonstrated to influence movement behaviours. All cultures were housed in a 20 °C incubator with 24-h light.
To estimate diffusion coefficients, we measured changes in rotifer distribution within capillary tubes over time. Parallel rows of 20 glass capillary tubes were supported on a levelled 1 × 1 m platform. The clear plastic material and architecture of the platform eliminated shadows on the tubes, which were suspended in slots cut in the platform. Each tube was 1 m long, with a 2 mm outside diameter and 1 mm inside diameter. Individual tubes were spaced 3 cm apart. The entire apparatus was positioned on a laboratory bench beneath laboratory ceiling lights such that all experiments were conducted under constant temperature and lighting conditions. For each trial, we injected rotifers from the same stock culture into five replicate tubes filled with an algal solution. A different tube was censused every 2 h by breaking it into 10 equal segments and counting the rotifers in each segment. Each set of five tubes was analysed as a single independent trial, yielding one set of time-sequenced snapshots of rotifer redistribution along a one-dimensional axis.
External (Environmental) Factors
We conducted experiments to assess the influence of resource and conspecific density on rates of rotifer redistribution, spanning a wide range of resource (0·3–4 × 106 algae cells mL−1) and consumer (20–600 rotifers mL−1) densities. Densities of both algae and rotifers were manipulated to achieve a continuous, evenly distributed range of experimental densities rather than prescribed bins of different density treatments. To manipulate algal density, we diluted algal solutions with fresh medium to meet a specific density (±5%), measured using a Partec CyFlow® flow cytometer (Partec; Munster, Germany). Tubes were prefilled with these solutions, taking care to avoid injecting any air pockets. Algal concentrations were measured only once at the onset of every trial. To manipulate rotifer density, we concentrated rotifers into a dense solution by gently filtering an entire population through a small funnel capped with a 20-μm Nitex filter, which allowed for the passage of algal solution while trapping rotifers in the funnel. This rotifer concentrate was then diluted with fresh media to meet a specific rotifer density. We used new randomly assigned populations of rotifers each day. Newly inoculated cultures or very old cultures were discarded. For this first set of experiments, we counted rotifers in 460 tubes to calculate the numbers of individuals at certain distances after certain time intervals. The absolute densities of individual rotifers varied from tube to tube, ranging between 20 individuals per tube and 580 per tube across all trials, with a mean density of 174 (±13 SE) individuals per tube. At the lower extreme, these densities overlap with those observed in laboratory-reared populations, and at the upper extreme, we used the highest densities we could achieve experimentally to increase our power to detect treatment responses. The rotifer densities we tested are higher than those commonly observed in natural populations, so in future studies it remains to be tested whether density-dependent responses can be found under field conditions or whether the responses shown here are only sensitive to these experimental densities. The absolute number of rotifers in any single 10-cm tube segment varied according to the absolute tube density and the rate of spread. The range of absolute densities in any single tube segment for a given time intervals varied between 0 per section (e.g. in the section furthest from the injection site at t = 0) and 279 per section (e.g. in the section nearest to the injection site at t = 0). For standardization, rotifer counts in each segment were divided by the overall population size (i.e. the sum of counts from all segments).
We initiated each trial by injecting 0·07 mL of rotifers in solution (the volume of a 10-cm tube segment is 0·0785 mL) into one end of five algal-filled capillary tubes and immediately capped both ends of the tubes to prevent drainage. We processed one tube immediately at the beginning of each trial to determine the initial distribution of rotifers and subsequently sampled a new tube every 2 h to measure how the distribution had changed over time. Processing involved breakage of the tubes at 10-cm increments. We did this by scoring the tube with a diamond-edged glass cutter, snapping sections apart and flushing the contents of each section into glass well trays while diluting the samples with 0·5 mL of fresh media. Rotifers in each well were counted using a dissecting microscope. When rotifer densities were markedly high (>30 mL−1), we subdivided the contents of each well into several new wells to facilitate more accurate counting. This procedure yielded five single snapshots of rotifer spatial distribution over the course of 8 h for each experimental trial.
Internal (Organismal State-Conditioning) Factors
We conducted a second set of experiments to assess the influence of food deprivation and risk exposure on diffusion rates. For these experiments, we preconditioned rotifers via one of four treatments for 15 h prior to conducting trials but otherwise followed the protocol described above. State-conditioning treatments included a control (i.e. no treatment), food deprivation, risk exposure and the latter combined. Food deprivation was achieved by filtering rotifers from a stock population into a new flask containing the same volume of fresh media, but with no algae present, and returning to the incubator for 15–16 h. Risk exposure was achieved by injecting 10 mL of media laden with predator kairomones into a population before returning to the incubator for 15–16 h. We extracted the predator-conditioned media from our cultures of A. brightwelli (density >5 mL−1) and filtered samples through glass fibre paper to ensure no actual predators or their algae food (C. erosa) physically entered the B. calyciflorus stock, but only kairomones. The algae C. erosa do not exude any chemical cue perceived by B. calyciflorus that we are aware of based on a review of the literature. We varied the state-conditioning treatments daily, holding the algae and rotifer densities roughly constant at levels approximately mid-range of levels used in the first set of experiments (i.e. at 2·06 × 106 algae mL−1 ± 4·2 × 104 SE and 115 rotifers mL−1 ± 4·6 SE, respectively).
For the resource- and density-influenced diffusion rates, we analysed 92 independent trials, for which we measured redistributions at 2-h time intervals. For the predator- and starvation-dependent diffusion rates, we analysed 120 independent trails (30 independent trials for each treatment), for which we measured redistributions at 2-h time intervals. Each independent trial generated a time series of population proportions at different distances from the initial point of origin (Fig. 1a). Using a maximum likelihood approach, we fitted a one-dimensional Fokker-Plank diffusion model (Appendix S1, Supporting Information for Matlab code) to these distributions (Fig. 1b) to estimate the most likely diffusion coefficient (Fig. 1d). The discretized approximation to the Fokker-Plank diffusion model generally fits our experimental data quite well (Fig. 1c) – of 212 trials, none of the R2 values were <0·5 and most exceeded 0·8.
For experiments that manipulated a continuous range of external factors, variation in diffusion rates across trials was analysed using a set of nested linear regression models with resource density, consumer density and their interaction as potential explanatory variables. A likelihood ratio test was then used to determine the best model. For the second set of experiments that manipulated only discrete organismal state variables across constant environmental conditions, we used anova to assess the influence of food deprivation and predator exposure on estimated diffusion coefficients. All statistical analyses were performed with R version 2·2·0 software (R Core Development Team, http://www.r-project.org).
For external factors, likelihood ratio tests show that the full model (rotifer diffusion rate as a function of rotifer density, resource density and an interaction term) explained significantly more variation in the data than any of the nested models (full model vs. density-dependent model: χ2 = 46·1, P < 0·001; full model vs. resource-dependent model: χ2 = 86·0, P < 0·001; full model vs. density- and resource-dependent additive model: χ2 = 33·4, P < 0·001). The full model, including resource density, consumer density and their interaction, explained 63% (adjusted R2) of the variability in observed diffusion rates (F3,88 = 51·7, P < 0·001). Rotifer diffusion rates increased with consumer density but decreased with resource density (Fig. 2). A negative interaction between these two factors resulted in a stronger influence of resource density when consumers were most dense and a likewise stronger influence of consumer density when resources were scarce. In other words, rotifer movement rates were highest at high competitor densities, but low resource densities (Fig. 2).
We found significant variability in diffusion coefficients between state-conditioning treatments. Movement rates were significantly lower for food-deprived populations relative to controls (F1,117 = 16·5, P < 0·001) and marginally lower for risk-exposed populations relative to controls (F1,117 = 3·4, P = 0·07; Fig. 3). However, we did not find statistically significant evidence of an interaction between starvation and predation risk on diffusion rates (F1,117 = 2·1, P = 0·15; Fig. 3).
In this study, we have directly tested whether rates of population spread are sensitive to environmental conditions and the internal state of individuals in consumer populations. Our results demonstrate clearly that biological diffusion is conditional upon consumer density, resource density, body condition and predation risk. There is a substantial empirical literature on organismal movement, particularly natal dispersal (Holyoak et al. 2008). Proximate mechanisms associated with individual dispersal probabilities or distance travelled by individual dispersers, for example, include factors such as age, body size, sex, personality type, local population density, resource density and degree of relatedness to other members in a population (Johnson & Gaines 1990; Clobert et al. 2001; Lambin, Aars & Piertnet 2001; Bowler & Benton 2005; Mattheysen 2005; Clobert et al. 2009;. Our experimental approach extends this growing literature by measuring the rate of population spread (i.e. the diffusion coefficient) under controlled experimental conditions. This is a necessary first step to parameterizing more complex continuous models of diffusive spread under heterogeneous environmental conditions (Patlak 1953; Gurney & Nisbet 1975; Karieva & Odell 1987; Grunbaum 1998; Moorcroft & Lewis 2006).
Our finding that diffusion rates decrease with resource abundance adds an Eulerian perspective to the ample evidence that individual consumers travel faster through resource-poor areas and linger in resource-rich ones (Barraquand & Benhamou 2008; Avgar, Kuefler & Fryxell 2011 and references therein). Theory suggests that a negative relationship between diffusion rates and resource abundance would be expected if consumers adaptively respond to high resource density by taking shorter steps or making more frequent turns (Okubo 1986; Karieva & Odell 1987; Morris & Karieva 1991; Moorcroft & Lewis 2006). Numerous empirical studies have demonstrated that at the level of individuals such behavioural responses can serve to increase time spent in favourable patches (Grunbaum 1998; Viet 1999; Wilson & Richards 2000; Klaassen, Nolet & Bankert 2006; Kuefler & Haddad 2006; Fryxell et al. 2008; Dias, Granadeiro & Palmeirim 2009). Recent theoretical work has demonstrated that such patterns may simply reflect the mechanical truncation of step lengths between resource encounters and thus do not require any decisive behavioural response (Avgar, Kuefler & Fryxell 2011). Regardless of the mechanisms involved, our work demonstrates empirically that a common pattern of resource-dependent movement is translated, at the population level, to resource-dependent diffusion and may thus play a crucial role in determining population redistribution patterns. Our experimental objective was to assess population responses to conditions that remained relatively constant through time. In our study, however, it is possible that resource densities within capillary tubes changed over time owing to algal growth or rotifer consumption. Such changes are likely monotonic (as demonstrated by Fussmann, Weithoff & Yoshida 2005) and should thus have no qualitative effect on the overall form of the population response surface we observed (Fig. 2).
Our finding that diffusion rates increase with conspecific density is in agreement with speculations in the literature about the adaptive value of conspecific repulsion promoting more rapid diffusion as densities increase (Shigesada, Kawasaki & Teramoto 1979). This might suggest a pre-emptive adaptive response to competition for resources that become more rapidly depleted as local competitor densities increase. As with the response to resource density, the explicit mechanisms underlying these observed patterns may be informed by more detailed studies of individual movement. In a related paper, we describe a set of experiments demonstrating that rotifers vary their turning frequencies and swimming velocities. Interestingly, these two different movement mechanisms, turning regulation and velocity regulation, are each influenced uniquely by densities of either resources or conspecific competitors, respectively (D. Kuefler, T. Avgar & J.M. Fryxell, unpublished data). These different mechanisms may reflect different selection regimes operating on these two determinants of population spread rates: escaping competition through increased movement when conspecific density is high and enhancing intake through decreased movement when resource density is high.
The results from our state-conditioning treatments suggest that nutritional stress and predation risk also influenced patterns of movement in rotifers. Reduced diffusion rates exhibited by starved populations of B. calyciflorus may reflect a metabolic deficit owing to starvation superimposed upon a relatively heavy cost of swimming (Epp & Lewis 1984; Charoy & Clement 1993; Charoy 1995). The lower diffusion rates of rotifer populations exposed to predation risk may be attributable to a reduction in swimming velocity to minimize predator encounters (Charoy & Clement 1993; Preston, Cecchine & Snell 1999) or a retraction of the swimming appendage (i.e. the corona) to reduce detectable vibrations (Wallace et al. 2006). Such suppressive effects of both starvation and predation on local diffusion rates could have significant ecological implications for food web interactions, by altering the spatial structure of populations.
Biological diffusion has received a great deal of theoretical attention (Turchin 1998; Okubo & Levin 2001), but far less empirical study. Empirical studies have begun to examine how individual movement characteristics respond to external drivers (Haddad 1999; Fortin et al. 2005; Morales et al. 2005; Klaassen, Nolet & Bankert 2006; Dalziel, Morales & Fryxell 2008; Kuefler et al. 2010). Extending observations of individuals to derive patterns of population spread is hampered, however, by the challenging task of scaling up individual animal movements to a landscape scale, particularly for ecological state variables that vary over time and space (Morales & Ellner 2002; Moorcroft & Lewis 2006). Other studies have modelled changes in the spatial distribution of a single population over time in relation to resource abundance, rates of energy gain or predation risk (Karieva & Odell 1987; Morales & Ellner 2002; Fryxell, Wilmshurst & Sinclair 2004; Holdo, Holt & Fryxell 2009). Such field studies are often challenging to interpret, because of complex patterns of spatial heterogeneity in key environmental variables. Single realizations of a spatial process yield limited insight into the degree of variability in behavioural responses and are rarely suitable for testing for interactions among state variables. While physiological or motivational states have long been thought to influence the movement behaviour of individuals in an energy-seeking vs. risk-avoiding framework (Sih 1980; Mangel & Clark 1986; McNamara & Houston 1986), our understanding of how internal state affects patterns of population redistribution is often obscured by other aspects of individual variation. We suggest that experimental studies may help in meeting these substantial challenges.
Predictions of population spread in rotifers were improved by considering diffusion as a function of both population density and resource density, tempered by predation risk and body condition experienced in the past by individuals. The implementation of such processes in diffusion models could conceivably improve our ability to predict how populations spread over time and space. Such models would be conditioned on past resource levels experienced by dispersers, local resource densities as the population spreads across space, as well as gradients in density of competing consumers and predators. In sum, our experimental results suggest a richer array of state-dependent models of diffusion are worthy of deeper consideration.
This work was supported by an NSERC Discovery Grant (to J. M. F.), an NSERC Vanier Fellowship (to T. A.) and an OGS fellowship (to D. K.). We thank Prof. Marcel Holyoak for comments on an earlier version of this manuscript. We thank Prof. Gregor Fussmann and his laboratory for providing rotifer stock cultures and advice on culture care. Finally, we acknowledge the fastidious laboratory assistance of Allegra Fryxell, Laurence Gillespie and Regina Kruse.