Bressloff P C
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA.
Proc Math Phys Eng Sci. 2020 Oct;476(2242):20200475. doi: 10.1098/rspa.2020.0475. Epub 2020 Oct 14.
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 at a random sequence of times. Whenever the particle is captured by a target, it delivers a packet of resources and then returns to , 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 small interior targets. We show that slow resetting increases the total number of resources across all targets provided that , where is the Neumann Green's function and is the location of the -th target. This implies that can be optimized by varying . We also show that the -th target has a competitive advantage if .
我们开发了一个通用框架，用于分析在多个独立搜索与捕获事件下目标群体中资源的分布情况。每个事件都涉及单个粒子执行随机搜索，该搜索会在随机时间序列重置到固定位置。每当粒子被一个目标捕获时，它会输送一包资源，然后返回 ，在那里它会重新装载货物并开始新一轮的搜索与捕获。利用更新理论，我们确定每个目标中的平均资源数量，作为相应无重置搜索过程的分裂概率和无条件平均首次通过时间的函数。然后，我们使用渐近偏微分方程方法来确定在具有小内部目标的有界二维域中，重置对扩散搜索产生的资源分布的影响。我们表明，只要 ，其中 是诺伊曼格林函数， 是第 个目标的位置，缓慢重置会增加所有目标中的资源总数。这意味着可以通过改变 来优化 。我们还表明，如果 ，第 个目标具有竞争优势。
