We present a simple paradigm for detection of an immobile target by a space-time coupled random walker with a finite lifetime. The motion of the walker is characterized by linear displacements at a fixed speed and exponentially distributed duration,
interrupted by random changes in the direction of motion and resumption of motion in the new direction with the same speed. We call these walkers mortal creepers. A mortal creeper may die at any time during its motion according to an exponential decay law characterized by a finite mean death rate $omega_m$. While still alive, the creeper has a finite mean frequency $omega$ of change of the direction of motion. In particular, we consider the efficiency of the target search process, characterized by the probability that the creeper will eventually detect the target. Analytic results confirmed by numerical results show that there is an $omega_m$-dependent optimal frequency $omega=omega_{opt}$ that maximizes the probability of eventual target detection. We work primarily in one-dimensional ($d=1$) domains and examine the role of initial conditions and of finite domain sizes. Numerical results in $d=2$ domains confirm the existence of an optimal frequency of change of direction, thereby suggesting that the observed effects are robust to changes in dimensionality. In the $d=1$ case, explicit expressions for the probability of target detection in the long time limit are given. In the case of an infinite domain, we compute the detection probability for arbitrary times and study its early- and late-time behavior. We further consider the survival probability of the target in the presence of many independent creepers beginning their motion at the same location and at the same time. We also consider a version of the standard target problem in which many creepers start at random locations at the same time.
This chapter is a contribution in the Handbook of Applications of Chaos Theory ed. by Prof. Christos H Skiadas. The chapter is organized as follows. First we study the statistical properties of combs and explain how to reduce the effect of teeth on t
he movement along the backbone as a waiting time distribution between consecutive jumps. Second, we justify an employment of a comb-like structure as a paradigm for further exploration of a spiny dendrite. In particular, we show how a comb-like structure can sustain the phenomenon of the anomalous diffusion, reaction-diffusion and Levy walks. Finally, we illustrate how the same models can be also useful to deal with the mechanism of ta translocation wave / translocation waves of CaMKII and its propagation failure. We also present a brief introduction to the fractional integro-differentiation in appendix at the end of the chapter.
