{"title":"Solution landscape of reaction-diffusion systems reveals a nonlinear mechanism and spatial robustness of pattern formation","authors":"Shuonan Wu, Bing Yu, Yuhai Tu, Lei Zhang","doi":"arxiv-2408.10095","DOIUrl":null,"url":null,"abstract":"Spontaneous pattern formation in homogeneous systems is ubiquitous in nature.\nAlthough Turing demonstrated that spatial patterns can emerge in\nreaction-diffusion (RD) systems when the homogeneous state becomes linearly\nunstable, it remains unclear whether the Turing mechanism is the only route for\npattern formation. Here, we develop an efficient algorithm to systematically\nmap the solution landscape to find all steady-state solutions. By applying our\nmethod to generic RD models, we find that stable spatial patterns can emerge\nvia saddle-node bifurcations before the onset of Turing instability.\nFurthermore, by using a generalized action in functional space based on large\ndeviation theory, our method is extended to evaluate stability of spatial\npatterns against noise. Applying this general approach in a three-species RD\nmodel, we show that though formation of Turing patterns only requires two\nchemical species, the third species is critical for stabilizing patterns\nagainst strong intrinsic noise in small biochemical systems.","PeriodicalId":501040,"journal":{"name":"arXiv - PHYS - Biological Physics","volume":"280 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - PHYS - Biological Physics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.10095","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Spontaneous pattern formation in homogeneous systems is ubiquitous in nature.
Although Turing demonstrated that spatial patterns can emerge in
reaction-diffusion (RD) systems when the homogeneous state becomes linearly
unstable, it remains unclear whether the Turing mechanism is the only route for
pattern formation. Here, we develop an efficient algorithm to systematically
map the solution landscape to find all steady-state solutions. By applying our
method to generic RD models, we find that stable spatial patterns can emerge
via saddle-node bifurcations before the onset of Turing instability.
Furthermore, by using a generalized action in functional space based on large
deviation theory, our method is extended to evaluate stability of spatial
patterns against noise. Applying this general approach in a three-species RD
model, we show that though formation of Turing patterns only requires two
chemical species, the third species is critical for stabilizing patterns
against strong intrinsic noise in small biochemical systems.