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 clusteringis 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 .
Cooperation is ubiquitous across all levels of biological systems ranging from microbial communities to human societies. It, however, seemingly contradicts the evolutionary theory, since cooperators are exploited by free-riders and thus are disfavored by natural selection. Many studies based on evolutionary game theory have tried to solve the puzzle and figure out the reason why cooperation exists and how it emerges. Network reciprocity is one of the mechanisms to promote cooperation, where nodes refer to individuals and links refer to social relationships. The spatial arrangement of mutant individuals, which refers to the clustering of mutants, plays a key role in network reciprocity. Besides, many other mechanisms supporting cooperation suggest that the clustering of mutants plays an important role in the expansion of mutants. However, the clustering of mutants and the game dynamics are typically coupled. It is still unclear how the clustering of mutants alone alters the evolutionary dynamics. To this end , we employ a minimal model with frequency independent fitness on a circle. It disentangles the clustering of mutants from game dynamics. The distance between two mutants on the circle is adopted as a natural indicator for the clustering of mutants or assortment. We find that 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 mutants are, the greater the chance of invasion is. It gives rise to the non-monotonic effect of clustering, which is counterintuitive. On the other hand, we find that less assortative mutants speed up fixation. Our model shows that the clustering of mutants plays a non-trivial role in fixation, which has emerged even if the game interaction is absent .
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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
Cooperation is ubiquitous across all levels of biological systems ranging from microbial communities to human societies. It, however, seemingly contradicts the evolutionary theory, since cooperators are exploited by free-riders and thus are disfavored by natural selection. Many studies based on evolutionary game theory have tried to solve the puzzle and figure out the reason why cooperation exists and how it emerges. Network reciprocity is one of the mechanisms to promote cooperation, where nodes refer to individuals and links refer to social relationships. The spatial arrangement of mutant individuals, which refers to the clustering of mutants, plays a key role in network reciprocity. Besides, many other mechanisms supporting cooperation suggest that the clustering of mutants plays an important role in
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rules indicate that clustering plays a heavy role in the expansion of mutants.
expansion of mutants.
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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
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of mutants and the game dynamics are typically coupled. It is still unclear how the clustering of mutants alone alters the evolutionary dynamics. To this end
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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
with frequency independent fitness on a circle. It disentangles the clustering of mutants from game dynamics. The distance between two mutants on the circle is adopted as a natural indicator for the clustering of mutants or assortment. We find that
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mutants
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the
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The non-monotonical affection of clusteringis 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
It gives rise to the non-monotonic effect of clustering, which is counterintuitive. On the other hand, we find that less assortative mutants speed up fixation. Our model shows that the clustering of mutants plays a non-trivial role in fixation, which has emerged even if the game interaction is absent