Clustering mutants favor and disfavor fixation
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Exploration of the emerging patterns of mutants in a finite wild-type group on spatial structure is an attracting topic. The mutation is assumed to appear with a low probability. Many studies have investigated the expansion and invasion chances for a few mutants under different games on spatial structures. It is shown that different games could lead to different evolving patterns. There are rules generalized for when the process favors the mutants, where many of the rules indicate that clustering plays a heavy role in the expansion of mutants. However, on the one hand, clustering hasn't been considered as an independent variable and is ignored in some previous analysis, where the patterns of the games might be affected by clustering and thus might not be well compared. On the other hand, many simulations for exploring the clustering effects are analytically intractable due to the great complexity of calculation and the difficulty to find proper categorizations on complex spatial structures. In order to explore the solely and fundamental effect of the clustering, we employ a minimal model where no game is involved and a simple cycle is adopted. In this way, we disentangle the clustering and the games. The applied process reflects very nontrivial effects of the clustering. Our results show the assortment is an amplifier of the selection for the connected mutants compared with the separated ones. Nevertheless, as mutants are separated, the more dispersed they are, the greater chance of invasion is. The non-monotonical affection of clustering is quite unintuitive. And what it affects under the minimal model reveals the great potential effects and also the great complexity of clustering when games are involved. We also notice an uncoordination between the fixation probability and fixation time for mutants.
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