This document is the accepted manuscript version of the following article: Giometto, A., Rinaldo, A., Carrara, F., & Altermatt, F. (2014). Emerging predictable features of replicated biological invasion fronts. Proceedings of the National Academy of Sciences of the United States of America PNAS, 111(1), 297-301. https://doi.org/10.1073/pnas.1321167110

Emerging predictable features of replicated biological invasion fronts Andrea Giometto ∗ †, Andrea Rinaldo ∗ ‡ Francesco Carrara ∗ , and Florian Altermatt † ∗Laboratory of Ecohydrology, School of Architecture, Civil and Environmental Engineering, Ecole´ Polytechnique F´ed´eralede Lausanne,CH-1015 Lausanne, Switzerland,†Department of Aquatic , Eawag: Swiss Federal Institute of Aquatic Science and Technology, CH-8600 D¨ubendorf, Switzerland, and ‡Dipartimento di Ingegneria Civile, Edile ed Ambientale, Universit`adi Padova, I-35131 Padova, Italy Submitted to Proceedings of the National Academy of Sciences of the United States of America Biological dispersal shapes ’ distribution and affects their co- logical range expansions is fundamental to the study of inva- existence. The spread of organisms governs the dynamics of invasive sive species dynamics [1–10], shifts in species ranges due to species, the spread of pathogens and the shifts in species ranges due climate or environmental change [11–14] and, in general, the to climate or environmental change. Despite its relevance for fun- spatial distribution of species [3, 15–18]. Dispersal is the key damental ecological processes, however, replicated experimentation agent that brings favorable genotypes or highly competitive on biological dispersal is lacking and current assessments point at inherent limitations to predictability, even in the simplest ecologi- species into new ranges much faster than any other ecological cal settings. In contrast, we show, by replicated experimentation or evolutionary process [1, 19]. Understanding the potential on the spread of the ciliate Tetrahymena sp. in linear landscapes, and realized dispersal is thus key to ecology in general [20]. that information on local unconstrained movement and reproduc- When organisms’ spread occurs on the timescale of multiple tion allows to predict reliably the existence and speed of traveling generations, it is the byproduct of processes that take place at waves of invasion at the macroscopic scale. Furthermore, a theoret- finer spatial and temporal scales, that are the local movement ical approach introducing demographic stochasticity in the Fisher- and reproduction of individuals [5,10]. The main difficulty in Kolmogorov framework of reaction-diffusion processes captures the causally understanding dispersal is thus to upscale processes observed fluctuations in range expansions. Therefore, predictability that happen at the short-term individual level to long-term of the key features of biological dispersal overcomes the inherent biological stochasticity. Our results establish a causal link from the and broad-scale patterns [5, 20–22]. Furthermore, short-term individual level to the long-term, broad-scale population the large fluctuations observed in range expansions have been patterns and may be generalized, possibly providing a general pre- claimed to reflect an intrinsic lack of predictability of the dictive framework for biological invasions in natural environments. phenomenon [23]. Whether the variability observed in na- ture or in experimental ensembles might be accounted for by Fisher wave | | microcosm | limits to prediction | coloniza- systematic differences between landscapes or by demographic tion stochasticity affecting basic vital rates of the organisms in- volved is an open research question [10, 20, 23, 24]. Modeling of biological dispersal established the theoret- Significance Statement ical framework of reaction-diffusion processes [1–3, 25–27], Biological dispersal is a key driver of several fundamental which now finds common application in dispersal ecology processes in nature, crucially controlling the distribution of [5,15,24,28–32] and in other fields [19,25,27,33–36]. The clas- species and affecting their coexistence. Despite its relevance sical prediction of reaction-diffusion models [1,2,26,27] is the for important ecological processes, however, the subject suf- propagation of an invading wavefront traveling undeformed fers an acknowledged lack of experimentation and current as- at a constant speed (Fig. 1E). Such models have been widely sessments point at inherent limitation to predictability even adopted by ecologists to describe the spread of organisms in in the simplest ecological settings. We show, by combin- a variety of comparative studies [5, 10, 28] and to control the ing replicated experimentation on the spread of the ciliate dynamics of invasive species [3, 4, 6]. The extensive use of Tetrahymena sp. to a theoretical approach based on stochas- these models and the good fit to observational data favored tic differential equations, that information on local uncon- their common endorsement as a paradigm for biological dis- strained movement and reproduction of organisms (including persal [6]. However, current assessments [23] point at inher- demographic stochasticity) allows to predict reliably both the ent limitations to the predictability of the phenomenon, due propagation speed and range of variability of invasion fronts to its intrinsic stochasticity. Therefore, single realizations of over multiple generations. a dispersal event (as those addressed in comparative studies) might deviate significantly from the mean of the process, mak- hat is the source of variance in the spread rates of bio- ing replicated experimentation necessary to allow hypothesis Wlogical invasions? The search for processes that affect testing, identification of causal relationships and to potentially biological dispersal and sources of variability observed in eco- falsify the models’ assumptions.

Reserved for Publication Footnotes

www.pnas.org/cgi/doi/10.1073/pnas.0709640104 PNAS Issue Date Volume Issue Number 1–7 A B C

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Fig. 1. Schematic representation of the experiment. (A), Linear landscape. (B), Individuals of the ciliate Tetrahymena sp. move and reproduce within the landscape. (C ), Examples of reconstructed trajectories of individuals (see Video S1). (D), Individuals are introduced at one end of a linear landscape and are observed to reproduce and disperse within the landscape (not to scale). (E), Illustrative representation of density profiles along the landscape at subsequent times. A wavefront is argued to propagate undeformed at a constant speed v according to the Fisher-Kolmogorov equation.

Methods). Density profiles are shown in Fig. 2, in six repli- Here, we provide replicated and controlled experimental cated dispersal events (panels A-F ). Organisms introduced at support to the theory of reaction-diffusion processes for mod- one end of the landscape rapidly formed an advancing front eling biological dispersal [25–27] in a generalized context that that propagated at a remarkably constant speed (Fig. 3). reproduces the observed fluctuations. Firstly, we experimen- The front position at each time was calculated as the first oc- tally substantiate the Fisher-Kolmogorov prediction [1, 2] on currence, starting from the end of the landscape, of a fixed the existence and the mean speed of traveling wavefronts by value of the density (Fig. 2). As for traveling waves predicted measuring the individual components of the process. Sec- by the Fisher-Kolmogorov equation, the mean front speed in ondly, we manipulate the inclusion of demographic - our experiment is notably constant for different choices of the ity in the model to reproduce the observed variability in range reference density value (Fig. 3C ). expansions. We move from the Fisher-Kolmogorov equation The species’ traits r, K and D were measured in inde- (Materials and Methods) to describe the spread of organisms pendent experiments. In the local growth experiment, a low- in a linear landscape [1, 2, 26, 27]. The equation couples a density population of Tetrahymena sp. was introduced evenly logistic term describing the reproduction of individuals with across the landscape and its density was measured locally at growth rate r [T −1] and carrying capacity K [L−1] and a different times. Recorded density measurements were fitted diffusion term accounting for local movement, epitomized by to the logistic growth model, which gave the estimates for r the diffusion coefficient D [L2T −1]. These species’ traits de- and K (SI). In the local unimpeded movement experiment, fine the characteristic scales of the dispersal process. In this we computed the mean square displacement (SI) of individu- framework, a population initially located at one end of a linear als’ trajectories [39–41] to estimate the diffusion coefficient D landscape is predicted to form a wavefront of√ colonization in- in density-independent conditions (Materials and Methods). vading empty space at a constant speed v = 2 rD [1,2,26,27], The growth and movement measurements were performed in which we measured in our dispersal experiment (Fig. 1D and the same linear landscape settings as in the dispersal exper- SI). iment and therefore are assumed to accurately describe the In the experiments, we used the freshwater ciliate Tetrahy- dynamics at the front of the traveling wave in the dispersal mena sp. (Materials and Methods) because of its short gen- events. √ eration time [18] and its history as a model system in ecol- The comparison of the predicted front speed v = 2 rD ogy [18, 37, 38]. The experimental setup consisted of linear to the wavefront speed measured in the dispersal experiment, landscapes, filled with a nutrient medium, kept in constant vo, yields a compelling agreement. The observed speed in the environmental conditions and of suitable size to meet the as- dispersal experiment was vo = 52.0 ± 1.8 cm/day (mean±SE) sumptions about the relevant dispersal timescales (Materials (SI), which we compare to the predicted one v = 51.9 ± 1.1 and Methods). Replicated dispersal events were conducted cm/day (mean±SE). The two velocities are compatible within by introducing an ensemble of individuals at one end of the one standard error. A t-test between the√ replicated observed landscape and measuring density profiles throughout the sys- speeds and bootstrap estimates of v = 2 rD gives a p-value tem at different times, through image analysis (Materials and of p = 0.96 (t = 0.05, df = 9). Thus, the null hypothesis that

2 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author the mean difference is 0 is not rejected at the 5 percent level experiments, at scales that were orders of magnitude smaller and there is no indication that the two means are different. As than in the dispersal events, the agreement between the two the measurements of r and D were performed in independent estimates of the front velocity is deemed remarkable.

1500 A Time (day) B Time (day) 3.6 3.7 2.6 1000 2.7 1.9 2.0 1.5 1.6 1.0 500 1.0 0.5 0.5 0.0 0.0 0 1500 C Time (day) D Time (day) 3.7 3.8 Experiment 2.7 1000 2.8 2.0 2.0 1.6 1.6 1.0 1.0 500 0.5 0.5 0.0 0.0 0 1500 E Time (day) F Time (day) 3.9 4.0 2.8 1000 2.8 2.0 2.0 Density (individuals/cm) 1.6 1.6 1.0 500 1.0 0.5 0.5 0.0 0.0 0

1500 G Time (day) H Time (day) model Stochastic 3.5 3.5 3.0 3.0 1000 2.5 2.5 2.0 2.0 1.5 1.5 500 1.0 1.0 0.5 0.5 0.0 0.0 0 0 50 100 150 200 0 50 100 150 200 Distance from origin (cm)

Fig. 2. Density profiles in the dispersal experiment and in the stochastic model. (A-F), Density profiles of six replicated experimentally measured dispersal events, at different times. Legends link each color to the corresponding measuring time. Black dots are the estimates of the front position at each time point. Organisms were introduced at the origin and subsequently colonized the whole landscape in 4 days (∼ 20 generations). (G-H ), Two dispersal events simulated according to the generalized model equation, with initial conditions as at the second experimental time point. Data are binned in 5cm intervals, typical length scale of the process.

accordance with simulations (Fig. 3A). In particular, most Although the Fisher-Kolmogorov equation correctly pre- experimental data are within the 95% confidence interval for dicts the mean speed of the experimentally observed invad- the simulated front position and the observed range variability ing wavefront, its deterministic formulation prevents it to re- is well captured by our stochastic model (Fig. 3B). Accord- produce the variability that is inherent to biological disper- ingly, the estimate for the front speed and its variability in sal [23]. In particular, it cannot reproduce the fluctuations the experiment are in good agreement with simulations (SI). in range expansion between different replicates of our dis- Demographic stochasticity can therefore explain the observed persal experiment (Fig. 3A). We propose a generalization variability in range expansions. of the Fisher-Kolmogorov equation (Materials and Methods) We suggest that measuring and suitably interpreting pro- accounting for demographic stochasticity that is able to cap- cesses acting locally allows to accurately predict the features ture the observed variability. The strength of demographic of global invasions. The deterministic Fisher-Kolmogorov stochasticity is embedded in an additional species’ trait σ equation correctly predicts the mean speed of invasion, but [T −1/2]. In this stochastic framework, the demographic pa- cannot capture the observed variability. Instead, inferring rameters r, K and σ were estimated from the local growth effective growth rates and the magnitude of demographic experiment with a maximum likelihood approach (Materials stochasticity allows to characterize and predict both the mean and Methods), while the estimate of the diffusion coefficient and the variability in range expansions, which is of great in- D was left unchanged (SI). We then used these local indepen- terest even for practical purposes, such as the delineation of dent estimates to numerically integrate the generalized model worst-case scenarios for the spread of invasive species. Thus, equation [42,43] with initial conditions as in the dispersal ex- at least in the simple ecological setting investigated here, pre- periment and found that the measured front positions are in dictability remains notwithstanding biological fluctuations, owing to the stochastic treatment devised. Our approach al-

Footline Author PNAS Issue Date Volume Issue Number 3 Experimental setup 250 A Experiments were performed in linear landscapes (Fig. 1A) filled with a nutrient medium and of the three species above mentioned. The linear landscapes were 2m long, 5mm wide and 3mm deep, respectively 200  105, 350 and 200 times the size of the study organism [38]. Landscapes  consisted of channels drilled on a plexiglass sheet, a second sheet was used   150  as lid and a gasket was introduced to avoid water spillage (Fig. 1A). At one end of the landscapes, an opening was placed for the introduction of  ciliates. The plexiglass sheets were sterilized with a 70% alcohol solu-  o 100  tion and gaskets were autoclaved at 120 C before filling the landscape  with medium. As plexiglass is transparent, the experimental units could be  placed under the objective of a stereomicroscope, to record pictures (for Front position (cm)  50  counting of individuals) or videos (to track ciliates). Individuals were ob- served to distribute mainly at the bottom of the landscape, whose length   was three orders of magnitude larger than its width (w) and depth (d) and  0   two orders of magnitude larger than the typical length scale of the process 0 1 2 3 4 (pD/r ' 5cm). The population was thus assumed to be confidently Time (day) well-mixed within the cross section after a time ∼ w2/D, which in our 30 B 80 C case is of the order of a minute after introduction of the ciliates in the  60 landscape. 20  40 Experimental protocol 10  We performed three independent and complementing experiments, specif-  20  ically: i) A dispersal experiment was carried out to study the possible ex-  istence and the propagation of traveling invasion wavefronts in replicated 95% range width (cm) 0 Mean speed (cm/day) 0 0 1 2 3 4 0 100 200 300 400 500 dispersal events; ii) A growth experiment was run to obtain estimates of the Time (day) Reference density (ind/cm) demographic species’ traits, that are r and K in the deterministic frame- work of Eq. 1 and r, K and σ in the stochastic framework of Eq. 2; iii) A Fig. 3. Range expansion in the dispersal experiment and in the stochastic local movement experiment was performed to study the local unimpeded model. (A), Front position of the expanding population in six replicated dis- movement of Tetrahymena sp. over a short timescale (in a time window persal events, colors identify replicas as in Fig. 2. The dark and light grey t  r−1), in order to estimate the diffusion coefficient D for our study shadings are respectively the 95% and 99% confidence intervals computed species, independently from the dispersal and growth experiments. by numerically integrating the generalized model equation, with initial con- ditions as at the second experimental time point, in 1020 iterations. The black curve is the mean front position in the stochastic integrations. (B), Dispersal experiment - We performed six replicated dispersal events in The increase in range variability between replicates in the dispersal experi- the linear landscapes. After filling the landscapes with medium and bac- ment (blue diamonds) is well described by the stochastic model (red line). teria, a small ensemble of Tetrahymena sp. was introduced at the origin. (C), Mean front speed for different choices of the reference density value Subsequently, the density of Tetrahymena sp. was measured at 1cm in- at which we estimated the front position in the experiment, error bars are tervals, five times in the first 48h and twice in the last 48h. The whole smaller than symbols. experiment lasted for about 20 generations of the study species.

lows to make predictions on spread without the need to in- Local growth experiment - We performed five replicated growth mea- troduce all details on the movement behavior, biology or any surements in the linear landscapes, in order to measure the demographic other information, as all these details are averaged in the tran- species’ traits, in the same environmental conditions as in the dispersal ex- sition from small to large scales. Such details can be synthe- periment, but independently from it. A low density culture of Tetrahymena sp. was introduced in the whole landscape and its density was measured sized in three parameters describing the density-independent by taking pictures and counting individuals, covering a region of 7cm yet stochastic behavior of individuals riding the invasion wave. along the landscape. Density measurements were performed at several The parsimony of our model allows generalization to organ- time points for each of the five replicates, in a time window of 3 days. isms with different biology (e.g., growth rates and diffusion co- efficients are available for several species in the literature [6]) and supports the view that our protocol may possibly pro- Local movement experiment - We performed four additional, replicated vide a general predictive framework for biological invasions in dispersal events in the linear landscapes, initialized in the same way as in natural environments. the dispersal experiment, in order to measure the diffusion coefficient of Tetrahymena sp. The diffusion coefficient D is the proportionality con- stant that links the mean square displacement of organisms’ trajectories Materials and Methods to time [39, 41] (SI). Macroscopically, it relates the local flux to the den- sity of individuals, under the assumption of steady state [41]. To es- timate the diffusion coefficient we recorded several videos of individuals Study species moving at the front of the traveling wave (at low density), reconstructed The species used in this study is Tetrahymena sp. (Fig. 1B), a freshwater their trajectories [39, 40] and computed their mean square displacement ciliate, purchased at Carolina Biological Supply (Burlington, NC, USA). In- hx2(t)i = h[x(t) − x(0)]2i. dividuals of Tetrahymena sp. have typical linear size (equivalent diameter) Video recording. We recorded videos of Tetrahymena sp. at the front of of 14µm [38]. Freshwater bacteria of the species Serratia fonticola, Bre- the traveling wave in four replicated dispersal events, at various times over viacillus brevis and Bacillus subtilis were used as a food resource for ciliates, 4 days. The area covered in each video was of 24mm in the direction of which were kept in a medium made of sterilized spring water and proto- the landscape and 5mm orthogonal to it. Each video lasted for 12min. zoan pellets (Carolina Biological Supply) at a density of 0.45g/L. The Trajectories reconstruction. For each recorded video we extracted individ- experimental units were kept under constant fluorescent light for the whole o uals’ spatial coordinates in each frame and used the MOSAIC plugin for duration of the study, at a constant temperature of 22 C. Overall, exper- the software ImageJ to reconstruct trajectories [40]. The goodness of the imental protocols are well established [18, 38, 44–46] and the contribution tracking was checked on several trajectories by direct comparison with the of laboratory experiments on protists to the understanding of population videos. Examples of reconstructed trajectories can be seen in Fig. 1C or and dynamics proved noteworthy [44]. in Video S1. Diffusion coefficient estimate. For each video, the square displacement of

4 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author each trajectory in the direction parallel to the landscape was computed at describing demographic stochasticity in a population [43] and needs extra- all time points and then averaged across trajectories. Precisely, for each care in simulations [42, 48]. In particular, standard stochastic integration 2 2 trajectory i we computed the quantity xi (t) = [Xi(t) − Xi(0)] , where schemes fail to preserve the positivity of ρ. We adopted a recently devel- Xi(t) is the 1-dimensional coordinate of organism i at time t in the direc- oped split-step method [42] to numerically integrate Eq. 2. This method tion parallel to the landscape and Xi(0) is its initial position. The mean allows to perform the integration with relatively large spatial and temporal 2 steps maintaining numerical accuracy. square displacement in a video was then computed as the mean of xi (t) 2 1 P 2 Data from the growth experiment were fitted to the equation: across all trajectories, that is, hx (t)i = N i xi (t) (where N is the total number of trajectories). A typical measurement of hx2(t)i is shown dρ h ρ i σ √ = rρ 1 − + √ ρ η, [3] in Fig. S1. As shown in the figure, there exists an initial correlated phase, dt K which we discuss in the SI. To estimate the diffusion coefficient from the l mean square displacement we fitted the measured hx2(t)i to the func- where ρ = ρ(t) is the local density, η = η(t) is a gaussian, zero-mean 2 h −t/τ i tion hx (t)i = 2Dt − 2Dτ 1 − e (SI) with the two parameters white noise (i.e., with correlations hη(t)η(t0)i = δ(t − t0)), σ > 0 is D (diffusion coefficient) and τ (correlation time). The total number of constant and l is the size of the region over which densities were measured recorded videos was 28, that is, 7 for each replica. (SI). Eq. 3 describes the time-evolution of the density in a well-mixed patch of length l, see SI. The likelihood function for Eq. 3 can be written as: n Mathematical models Y Deterministic framework - The Fisher-Kolmogorov equation [1,2] reads: L(θ) = P [ρ(tj ), tj |ρ(tj−1), tj−1; θ] , [4] j=2 ∂ρ ∂2ρ h ρ i = D + rρ 1 − , [1] ∂t ∂x2 K where n is the total number of observation in the growth time series, θ = (r, K, σ) is the vector of demographic parameters and P (ρ, t|ρ0, t0; θ) where ρ = ρ(x, t) is the density of organisms, r the species’ growth rate, is the transitional probability density of having a density of individuals D the diffusion coefficient and K the carrying capacity. Eq. 1 is known ρ at time t, given that the density at time t0 was ρ0 (for a given θ). to foster the development of undeformed traveling waves of the density The transitional probability density P (ρ, t|ρ0, t0; θ) satisfies the Fokker- profile. Mathematically, this implies that ρ(x, t) = ρ(x − vt), where Planck equation associated to Eq. 3 (SI), which was solved numerically v is the speed of the advancing wave. Fisher√ [1] proved that traveling for all observed transitions and choices of parameters (SI), adopting the wave solutions can only exist with speed v ≥ 2 rD and Kolmogorov [2] implicit Crank-Nicholson scheme. The best fit parameters were those that demonstrated that, with suitable initial conditions, the speed of the wave- maximized the likelihood function Eq. 4 (SI). front is the lower bound. Stochastic framework - The stochastic model equation reads: ACKNOWLEDGMENTS. We thank Amos Maritan and Enrico Bertuzzo 2 for invaluable discussions, Janick Cardinale for support with the MOSAIC ∂ρ ∂ ρ h ρ i √ software and Regula Illi for the ciliates’ picture. The authors gratefully = D + rρ 1 − + σ ρ η, [2] ∂t ∂x2 K acknowledge the support provided by the discretionary funds of Eawag: Swiss Federal Institute of Aquatic Science and Technology; the Euro- where η = η(x, t) is a gaussian, zero-mean white noise (i.e., with corre- pean Research Council advanced grant program through the project River lations hη(x, t)η(x0, t0)i = δ(x − x0)δ(t − t0), where δ is the Dirac’s networks as ecological corridors for biodiversity, and water- borne disease (RINEC-227612); Swiss National Science Foundation projects delta distribution) and σ > 0 is constant. We adopt the Itˆo’sstochastic 200021 124930/1 and 31003A 135622. calculus [47], as appropriate in this case. Note, in fact, that the choice of the Stratonovich framework would make no sense here, as the noise term Author contributions: A.G., A.R. and F.A. designed the study. A.G. and would have a constant non-zero mean [24, 47], which would allow an ex- F.C. performed the experiments. A.G. performed the analysis. A.R., F.A. tinct population to possibly escape the zero-density absorbing state. The and F.C. assisted with the analysis, A.G. developed the theoretical model, square-root multiplicative noise term in Eq. 2 is commonly interpreted as A.G., A.R. and F.A. prepared the manuscript.

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