Abstract
For describing irregular movement of animals in random search as well as patterns of human travel during daily activity an original model of Brownian nonlinear motion is proposed. The wandering particle is represented as a point in the extended phase space comprising its position, velocity, and, in addition, the acceleration. The acceleration is assumed to be governed by a certain random process with stochastic self-acceleration. The acceleration dynamics is described by a nonlinear stochastic differential equation of the Hänggi-Klimontovich type. Its regular component represents the preference of the particle moving with a certain fixed velocity. The stochastic component with the noise intensity growing with the acceleration is related to the active behavior of the wandering particle in changing the motion direction. The model is studied numerically. The obtained results allow us to state that the developed model generates motion trajectories that can be treated as Lévy random walks. The latter statement can be regarded as the main original point of the present work demonstrating a new approach to modeling the random searching based on continuous Markovian processes.
