Emerging predictable features of replicated biological invasion fronts

Biological dispersal shapes species’ distribution and affects their coexistence. The spread of organisms governs the dynamics of invasive species, the spread of pathogens, and the shifts in species ranges due to climate or environmental change. Despite its relevance for fundamental ecological processes, however, replicated experimentation on biological dispersal is lacking, and current assessments point at inherent limitations to predictability, even in the simplest ecological settings. In contrast, we show, by replicated experimentation on the spread of the ciliate Tetrahymena sp. in linear landscapes, that information on local unconstrained movement and reproduction allows us to predict reliably the existence and speed of traveling waves of invasion at the macroscopic scale. Furthermore, a theoretical approach introducing demographic stochasticity in the Fisher–Kolmogorov framework of reaction–diffusion processes captures the observed fluctuations in range expansions. Therefore, predictability of the key features of biological dispersal overcomes the inherent biological stochasticity. Our results establish a causal link from the short-term individual level to the long-term, broad-scale population patterns and may be generalized, possibly providing a general predictive framework for biological invasions in natural environments.What is the source of variance in the spread rates of biological invasions? The search for processes that affect biological dispersal and sources of variability observed in ecological range expansions is fundamental to the study of invasive species dynamics (1–10), shifts in species ranges due to climate or environmental change (11–13), and, in general, the spatial distribution of species (3, 14–16). Dispersal is the key agent that brings favorable genotypes or highly competitive species into new ranges much faster than any other ecological or evolutionary process (1, 17). Understanding the potential and realized dispersal is thus key to ecology in general (18). When organisms’ spread occurs on the timescale of multiple generations, it is the byproduct of processes that take place at finer spatial and temporal scales that are the local movement and reproduction of individuals (5, 10). The main difficulty in causally understanding dispersal is thus to upscale processes that happen at the short-term individual level to long-term and broad-scale population patterns (5, 18–20). Furthermore, the large fluctuations observed in range expansions have been claimed to reflect an intrinsic lack of predictability of the phenomenon (21). Whether the variability observed in nature or in experimental ensembles might be accounted for by systematic differences between landscapes or by demographic stochasticity affecting basic vital rates of the organisms involved is an open research question (10, 18, 21, 22).Modeling of biological dispersal established the theoretical framework of reaction–diffusion processes (1–3, 23–25), which now finds common application in dispersal ecology (5, 14, 22, 26–30) and in other fields (17, 23, 25, 31–36). Reaction–diffusion models have also been applied to model human colonization processes (31), such as the Neolithic transition in Europe (25, 37, 38). The classical prediction of reaction–diffusion models (1, 2, 24, 25) is the propagation of an invading wavefront traveling undeformed at a constant speed (Fig. 1E). Such models have been widely adopted by ecologists to describe the spread of organisms in a variety of comparative studies (5, 10, 26) and to control the dynamics of invasive species (3, 4, 6). The extensive use of these models and the good fit to observational data favored their common endorsement as a paradigm for biological dispersal (6). However, current assessments (21) point at inherent limitations to the predictability of the phenomenon, due to its intrinsic stochasticity. Therefore, single realizations of a dispersal event (as those addressed in comparative studies) might deviate significantly from the mean of the process, making replicated experimentation necessary to allow hypothesis testing, identification of causal relationships, and to potentially falsify the models’ assumptions (39).Open in a separate windowFig. 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 (Movie 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.Here, we provide replicated and controlled experimental support to the theory of reaction–diffusion processes for modeling biological dispersal (23–25) in a generalized context that reproduces the observed fluctuations. Firstly, we experimentally substantiate the Fisher–Kolmogorov prediction (1, 2) on the existence and the mean speed of traveling wavefronts by measuring the individual components of the process. Secondly, we manipulate the inclusion of demographic stochasticity in the model to reproduce the observed variability in range expansions. We move from the Fisher–Kolmogorov equation (Materials and Methods) to describe the spread of organisms in a linear landscape (1, 2, 24, 25). The equation couples a logistic term describing the reproduction of individuals with growth rate r and carrying capacity K and a diffusion term accounting for local movement, epitomized by the diffusion coefficient D. These species’ traits define the characteristic scales of the dispersal process. In this framework, a population initially located at one end of a linear landscape is predicted to form a wavefront of colonization invading empty space at a constant speed (1, 2, 24, 25), which we measured in our dispersal experiment (Fig. 1D and SI Text).