Target competition for resources under multiple search-and-capture events with stochastic resetting

We develop a general framework for analysing the distribution of resources in a population of targets under multiple independent search-and-capture events. Each event involves a single particle executing a stochastic search that resets to a fixed locationx r at a random sequence of times. Whenever the particle is captured by a target, it delivers a packet of resources and then returns tox r , where it is reloaded with cargo and a new round of search and capture begins. Using renewal theory, we determine the mean number of resources in each target as a function of the splitting probabilities and unconditional mean first passage times of the corresponding search process without resetting. We then use asymptotic PDE methods to determine the effects of resetting on the distribution of resources generated by diffusive search in a bounded two-dimensional domain withN small interior targets. We show that slow resetting increases the total number of resourcesM tot across all targets provided that∑ j = 1 N G ( x r , x j ) < 0 , whereG is the Neumann Green’s function andx j is the location of thej -th target. This implies thatM tot can be optimized by varyingr . We also show that thek -th target has a competitive advantage if∑ j = 1 N G ( x r , x j ) > N G ( x r , x k ) .