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 location xr at a random sequence of times. Whenever the particle is captured by a target, it delivers a packet of resources and then returns to xr, 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 with N small interior targets. We show that slow resetting increases the total number of resources Mtot across all targets provided that ∑j=1NG(xr,xj)<0, where G is the Neumann Green’s function and xj is the location of the j-th target. This implies that Mtot can be optimized by varying r. We also show that the k-th target has a competitive advantage if ∑j=1NG(xr,xj)>NG(xr,xk).

中文翻译：

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具有随机重置的多个搜索和捕获事件下的目标资源竞争

我们开发了一个通用框架，用于分析多个独立搜索和捕获事件下目标群体中的资源分布。每个事件都涉及执行随机搜索的单个粒子，该搜索以随机的时间序列重置到固定位置 xr。每当粒子被目标捕获时，它就会传送一个资源包，然后返回到 xr，在那里重新装载货物并开始新一轮的搜索和捕获。使用更新理论，我们将每个目标中的平均资源数量确定为相应搜索过程的分裂概率和无条件平均首次通过时间的函数，无需重置。然后，我们使用渐近 PDE 方法来确定重置对具有 N 个小内部目标的有界二维域中扩散搜索生成的资源分布的影响。我们表明，如果 ∑j=1NG(xr,xj)<0，慢重置会增加所有目标的资源总数 Mtot，其中 G 是诺依曼格林函数，xj 是第 j 个目标的位置。这意味着可以通过改变 r 来优化 Mtot。我们还表明，如果 ∑j=1NG(xr,xj)>NG(xr,xk)，则第 k 个目标具有竞争优势。