Sisyphus and his rock: Quasi-random walk inspired by the motion of a ball transported by a dung beetle on combined terrain.

The majority of biologically inspired dynamic problems are essentially defined by the complexity of the contact surface where such motion takes place. From a statistical point of view, such a surface in many biological problems is typically a combination of a universal scale invariant (fractal) component and a well-defined component having a characteristic scale. If the biological object, here a dung ball, or its parts have a size comparable to the dimensions of the surface peculiarities, one can expect a strong influence on the motion. To avoid competition for the same food resource, some dung-feeding insect species form a dung ball and roll it away from the dung pile. In order to quickly escape competition, dung beetles seem to strictly follow an initial bearing. On flat terrain, they manage to roll a dung ball along a nearly perfect straight path. However, on a more realistic terrain, which normally includes both components mentioned above, the motion is more complex. In this study, we numerically model the ball transportation on terrain with different scales of surface profile. A strong correlation is observed between effective ball transportation (time, distance, work) and the ratio of the size of the ball relative to the size of the terrain roughness. Surface irregularities, with a characteristic size comparable to the ball diameter, are negatively correlated to the efficiency of ball transportation. In addition a strong correlation is found between the quasi random noise, numerically simulating the activity of a dung beetle trying to escape from a valley in which it is trapped, and the success in ball transportation.