Optimal noise in a stochastic model for local search

Abstract

We develop a prototypical stochastic model for a local search around a given home. The stochastic dynamic model is motivated by experimental findings of the motion of a fruit fly around a given spot of food but will generally describe the local search behavior. The local search consists of a sequence of two epochs. In the first the searcher explores new space around the home, whereas it returns to the home during the second epoch. In the proposed two-dimensional model both tasks are described by the same stochastic dynamics. The searcher moves with constant speed and its angular dynamics is driven by a symmetric α-stable noise source. The latter stands for the uncertainty to decide the new direction of motion. The main ingredient of the model is the nonlinear interaction dynamics of the searcher with its home. In order to determine the new heading direction, the searcher has to know the actual angles of its position to the home and of the heading vector. A bound state to the home is realized by a permanent switch of a repulsive and attractive forcing of the heading direction from the position direction corresponding to search and return epochs. Our investigation elucidates the analytic tractability of the deterministic and stochastic dynamics. Noise transforms the conservative deterministic dynamics into a dissipative one of the moments. The noise enables a faster finding of a target distinct from the home with optimal intensity. This optimal situation is related to the noise-dependent relaxation time. It is uniquely defined for all α and distinguishes between the stochastic dynamics before and after its value. For times large compared to this, we derive the corresponding Smoluchowski equation and find diffusive spreading of the searcher in the space. We report on the qualitative agreement with the experimentally observed spatial distribution, noisy oscillatory return times, and spatial autocorrelation function of the fruit fly. However, as a result of its simplicity, the model aims to reproduce the local search behavior of other units during their exploration of surrounding space and their quasiperiodic return to a home.

Analysis of aligning active local searchers orbiting around their common home position

“…As solution for the weakly coupled ensemble, we obtain the same stationary marginal density as it was derived for a single independent particle [26,27]. There we have found…”

“…Remarkably, the two equations for r, z define without noise σ = 0 a deterministic oscillatory motion in the (r, z) space. Stationary fixed points of the dynamics are saddles or centers as it was reported in [26]. Trajectories perform an oscillatory loops with alternating search and return epochs.…”

“…dynamics [26]. With alignment present this is not the case as the position angle enters the z i dynamics through the neighborhood Ω i .…”

“…The main ingredient of the kinematics of the searcher is the dynamics of the heading direction standing for the decision making step of the searcher to choose a new direction. In [26,27], we assumedφ…”

“…In [26] we also derived the Smoluchowski-equation for the marginal spatial density ρ(x, y, t). Its stationary solution was found to be a exponential function…”