Anomalous diffusions of the composite processes: Generalized Lévy walk with jumps or rests

Composite processes appear in a wide field, such as biology, ecology and natural science, so it is necessary to establish corresponding models to describe them. This manuscript builds up two two-mode random walk models which are the generalized Lévy walk with jumps (GLWJ) model and the generalized Lévy walk with rests (GLWR) model. The GLW processes in these two models will be interrupted by jumps and rest events, respectively, and move at a new velocity which is coupled with the motion time. The motion time and waiting time densities follow power-law forms and the jump density follows a Lévy form. We discuss the diffusive behaviors by analytically calculating the mean square displacement (MSD) and numerically simulating the probability density function (PDF). The results reveal their MSDs in both models exhibit crossover phenomena during the evolution processes. Meanwhile, the diffusion type of the GLWJ can be determined by either the largest exponent term or the largest pre-coefficient term. However, since the GLWR does not have random fluctuations from another mechanism that competes with the GLW. Without considering the scales of two time density functions, its diffusion type is determined only by the largest exponent team. The studies of these two models provide theoretical guidances for practical observations of some complex processes such as intermittent search strategies, human activity patterns in cities and cold atom fluctuations.