The Correlated Random Walk and the Rise of Movement Ecology

Abstract

Understanding why, how, and when animals move is essential to many areas of ecology and related fields. Indeed, key aspects of animal behavior, population genetics, biological control, predator–prey dynamics, ecosystem engineering, and conservation biology all hinge upon knowing what critters are moving from where to where in a landscape, including information on how quickly, how regularly, and by what route they travel. The complexities involved in such processes have spawned tremendous efforts in both field research (where goals include measuring and characterizing such movements) and theoretical research (where goals include exploring the nature and potential consequences of movement). Ecologists today routinely receive some training in both empirical and theoretical research, and in the role of statistical analyses and model fitting as a way of linking the two perspectives. However, that has not always been the case. Spatial questions in ecology were long an area where the gulf between theory and reality was particularly wide. In part, this was due to the additional mathematical challenges of spatial models, but it also due to the perhaps greater technological challenges of measuring and contextualizing animal movements. More than 30 years ago, an early, sturdy bridge between field data and spatial ecological theory was built when the article “Analyzing insect movement as a correlated random walk” was published in Oecologia. This paper, which represented a collaboration between ecologist Peter Kareiva and mathematician Nanako Shigesada, is a milestone along the Paper Trail because it marks a critical link between the abstract world of ecological theory and the hands-on way in which ecologists actually collect data on individual animals. Even now, this paper, which has been cited almost 500 times, continues to attract interest as a key nexus linking the realms of the rumpled shirts and the muddy boots. Kareiva and Shigesada's paper helped transform the quantitative study of animal movement from a purely theoretical venture into an integrative science, where theory and data are merged to generate new understanding. Combining clear prose and instructive equations, Kareiva and Shigesada (1983) was one of the first papers in ecology to provide a concrete, tractable linkage between spatial ecological models and features that could be readily observed—and quantified—by field biologists. Kareiva and Shigesada (1983) introduced a generalized two-dimensional correlated random walk (CRW) model to ecology, and demonstrated how it could be parameterized by decomposing an individual animal's movement path into a series of movement steps and turning angles. The CRW was a clear advance in spatial ecology because it dealt with an obvious discrepancy between previously used, simple (uncorrelated) random walks and empirical reality—namely that moving animals very frequently exhibit directional persistence. The key to their approach was to write the model in terms of the moments of the step length and turn angle distributions, which is important for two reasons. First, these moments do not require complex statistical methods to estimate, and can instead be calculated from directly from movement path data via simple paper-and-pencil formulas. Second, by focusing on the statistics of the step length and turn angle distributions instead of making particular distributional assumptions, Kareiva and Shigesada ensured that their model would apply to a wide range of ecological scenarios. The mean step length, mean squared step length, and mean cosine of turn angles are now standard statics used to summarize movement paths and parameterize movement models. Kareiva and Shigesada (1983) established the mean squared displacement (MSD, also called the net squared displacement) as a standard yardstick for judging the appropriateness of the CRW for particular data sets. The centerpiece of their paper was a closed-form expression for the expected value of the MSD under the CRW model in terms of the moments of the step length and turn angle distributions. Comparing the observed MSD to that predicted by the fitted CRW model allows users to gauge how well the CRW describes their data. The combination of biological plausibility, mathematical tractability, and a clear connection between model and data established the CRW as a simple, yet nontrivial null model against which real animal movements could be compared. The authors were clear that, while the CRW is more realistic than a simple random walk, it is still a radical simplification of real animal movement. Importantly, they noted the specific ways in which real animal movements deviate from the CRW may reveal important biological insights into the underlying movement process. For example, an observed MSD that increases consistently faster than model predictions suggests more directed movement than can be captured with a CRW. When coupled with bootstrap confidence intervals reflecting parameter uncertainty (Turchin 1998), Kareiva and Shigesada's (1983) approach provides an unambiguous gauge of the degree to which a more complicated and biologically realistic movement model is justified. In our opinion, this is one of the most important and lasting contributions of their work. Because diffusion models can be derived from the CRW via the diffusion approximation, the approach initiated by Kareiva and Shigesada (1983) helped to provide a pathway between the quantities field ecologists' can observe (movement paths of individuals), and the population-level diffusion rates employed by theoreticians (Turchin 1998). For example, Turchin (1991) showed how habitat-specific individual movement behaviors could be estimated via CRW methods and then translated into the spatial distribution of the population via a diffusion approximation. Other areas of spatial ecology have subsequently emulated this individual-level to population-level upscaling approach, including moment equations for spatial population dynamics, spatially explicit metapopulation models, and individual-based spatial population models. Collectively, these frameworks have fundamentally changed the character of spatial ecology by forging links between observable, individual-level phenomenon and their population- or community-level consequences. The CRW approach has steadily grown and developed over the years to become the workhorse of modern movement ecology. Many exciting advances in spatial analyses of animal movement can trace an intellectual ancestry to Kareiva and Shigesada (1983). For example, composite random walk models allow movement behavior to vary in space and time, and can help contextualize movement by linking behavioral changes to environmental covariates (Benhamou 2014). Behavioral change point analyses (Gurarie et al. 2009) take a similar approach, but are based on a continuous-space analog of the discrete CRW. Mechanistic home range models (Moorcroft and Lewis 2013) are based on CRWs and allow home ranges of individuals (or groups) to arise naturally from realistic movement behavior and interactions between the individual and its environment and conspecifics. These approaches extend Kareiva and Shigesada's ideas in important directions by allowing movement behavior to depend on context. While great strides have been made in building biological realism into CRW-based models, it is now clear that this framework is nearing its limits. For instance, many animals exhibit movement behaviors that repeat at regular intervals (e.g., daily, seasonally). Animals may also use memory to navigate, or may avoid recently exploited areas when foraging. All of these biological realities and many others violate the Markovian assumption under which Kareiva and Shigesada (1983) derived their results. Memory, avoidance, repetition and other biological complexities introduce long-term autocorrelations into the movement paths of individuals, which the CRW and other Markovian movement models cannot accommodate or utilize. A new frontier of movement ecology is to relax the first-order Markovian assumption such that movement models can use the information contained in long-term autocorrelations to identify critical behaviors (Fleming et al. 2014a, b). Doing so will allow ecology to go beyond purely random movement, and to begin incorporating real biological mechanisms into movement models. It is remarkable to note that Kareiva and Shigesada saw this frontier on the horizon over 30 years ago. That it remains an open challenge for movement ecology is testament both to the enduring contributions of their paper, and to the inherent difficulty in taking the next major step beyond their pioneering work.