Target competition for resources under diffusive search-and-capture

In this paper we use asymptotic analysis to determine the steady-state mean number of resources in each of $N$ small interior targets within a three-dimensional bounded domain. The accumulation of resources is based on multiple rounds of search-and-capture events; whenever a searcher finds a target it delivers a resource packet to the target, after which it escapes and returns to its initial position (resetting after capture). The searcher is then resupplied with cargo and a new search process is initiated after a random delay. Assuming that the accumulation of resources is counterbalanced by degradation, one can derive general expressions for the moments of the resource distribution. We use this to show that the mean number of resources in a target is proportional to its effective "shape capacitance." We then extend the analysis to the case of diffusive search with stochastic resetting before capture, where the position of the searcher is reset to its initial position at a random sequence of times that is statistically independent of the ongoing search process, in contrast to the sequence of resetting times after capture.