Fractal characterization of three-dimensional zooplankton swimming trajectories

Several natural processes, such as swimming and flying, occur in a three-dimensional reference frame and thus describe three-dimensional trajectories. Analysing such tracks is no easy task. To this aim, here we propose a new technique, based on the box-counting approach, by which to estimate the three-dimensional fractal dimension (D3D) of spatially-evolving routes. This technique has been preliminary tested on two sets of random walks (pure and correlated). Then, our approach has been utilised to characterise the swimming trajectories performed by the freshwater cladoceran Daphnia pulex under two light conditions. logically different tracks attain statistically different D3D values, smoother paths having lower values than more tortuous ones; therefore, our proposed methodology proves to be fruitful in direct comparisons. Moreover D3D can highlight bizarre situations, in which differently convoluted tracks are characterised by the same degree of contortion.