A method for classifying and comparing non-linear trajectories of ecological variables

Abstract Temporal dynamics in ecological variables are usually assessed using linear trends or smoothing methods. Those trends qualitatively summarise the increase or decrease in the variable of interest over a given time period. Yet, linear trends do not capture changes in the direction or in the rate of change of indices such as population trajectories, that may typically occur when conditions improve or worsen following conservation actions or environmental disturbances. In a similar way, non-linear methods while aiming to fully characterise population trajectories, fail to end up with a standard classification. Here, we propose and test a simple method to classify the trajectory of a given ecological variable according to its trend and acceleration. Our method is based on fitting a second order polynomial that is used to describe a trajectory according to its direction, velocity, and curvature (accelerated or decelerated). We apply this method to the temporal dynamics of bird populations monitored by the French Breeding Bird Survey as a case study. Using data for more than 100 species monitored from 1989 to 2017 in more than 2000 sites, we show that one quarter of the studied species have dynamics that can be better described by our polynomial approach than typically-used linear analysis. We also show how it can be used to analyse indicators constructed with multi-species indices. Our method can be applied to any type of ecological variable either to classify trajectories of ecological variables in time or trajectories of ecological variables across pressure gradients. Overall, our results suggest a more systematic investigation of non-linear trajectories when analysing the dynamics of ecological variables.