Target detection by space-time coupled walkers with finite lifetime

In biology and ecology, target search processes where the lifetime of the searchers is shorter or comparable to the typical time needed to reach the target by diffusion are a common occurrence (e.g., oocyte fertilization by sperm). Here, we present a simple paradigm for detection of an immobile target by a Continuous Time Random Walker with a finite lifetime, which we alternatively call a mortal creeper. The motion of the creeper is characterized by ballistic 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. The mortal creeper may die at any time during its motion according to an exponential decay law characterized by a finite mean death rate. While still alive, the creeper has a finite mean frequency of changes in direction. Given a random uniform initial condition for the velocity direction, the question is how to maximize the efficiency of the search process, characterized by the probability that the creeper will eventually detect the target upon hitting it for the first time. Analytic results and simulations for the 1d problem show that in a proper parameter regime there exists an optimal non-zero frequency of reorientation maximizing the probability of eventual target detection. A non-zero optimal frequency is also seen in simulations for a 2d periodic system, thereby suggesting that the observed optimization effect is robust with respect to changes in dimensionality. Finally, we consider the case of more than one creeper searching for the immobile target.