{"title":"Universality of stable multi-cluster periodic solutions in a population model of the cell cycle with negative feedback.","authors":"Kiattisak Prathom, Todd R Young","doi":"10.1080/17513758.2021.1971781","DOIUrl":null,"url":null,"abstract":"<p><p>We study a population model where cells in one part of the cell cycle may affect the progress of cells in another part. If the influence, or feedback, from one part to another is negative, simulations of the model almost always result in multiple temporal clusters formed by groups of cells. We study regions in parameter space where periodic '<i>k</i>-cyclic' solutions are stable. The regions of stability coincide with sub-triangles on which certain events occur in a fixed order. For boundary sub-triangles with order '<math><mrow><mi>r</mi><mi>s</mi><mn>1</mn></mrow></math>', we prove that the <i>k</i>-cyclic periodic solution is asymptotically stable if the index of the sub-triangle is relatively prime with respect to the number of clusters <i>k</i> and neutrally stable otherwise. For negative linear feedback, we prove that the interior of the parameter set is covered by stable sub-triangles, i.e. a stable <i>k</i>-cyclic solution always exists for some <i>k</i>. We observe numerically that the result also holds for many forms of nonlinear feedback, but may break down in extreme cases.</p>","PeriodicalId":48809,"journal":{"name":"Journal of Biological Dynamics","volume":"15 1","pages":"455-522"},"PeriodicalIF":1.8000,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Biological Dynamics","FirstCategoryId":"99","ListUrlMain":"https://doi.org/10.1080/17513758.2021.1971781","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"ECOLOGY","Score":null,"Total":0}
引用次数: 0
Abstract
We study a population model where cells in one part of the cell cycle may affect the progress of cells in another part. If the influence, or feedback, from one part to another is negative, simulations of the model almost always result in multiple temporal clusters formed by groups of cells. We study regions in parameter space where periodic 'k-cyclic' solutions are stable. The regions of stability coincide with sub-triangles on which certain events occur in a fixed order. For boundary sub-triangles with order '', we prove that the k-cyclic periodic solution is asymptotically stable if the index of the sub-triangle is relatively prime with respect to the number of clusters k and neutrally stable otherwise. For negative linear feedback, we prove that the interior of the parameter set is covered by stable sub-triangles, i.e. a stable k-cyclic solution always exists for some k. We observe numerically that the result also holds for many forms of nonlinear feedback, but may break down in extreme cases.
期刊介绍:
Journal of Biological Dynamics, an open access journal, publishes state of the art papers dealing with the analysis of dynamic models that arise from biological processes. The Journal focuses on dynamic phenomena at scales ranging from the level of individual organisms to that of populations, communities, and ecosystems in the fields of ecology and evolutionary biology, population dynamics, epidemiology, immunology, neuroscience, environmental science, and animal behavior. Papers in other areas are acceptable at the editors’ discretion. In addition to papers that analyze original mathematical models and develop new theories and analytic methods, the Journal welcomes papers that connect mathematical modeling and analysis to experimental and observational data. The Journal also publishes short notes, expository and review articles, book reviews and a section on open problems.