{"title":"具有密度依赖速度的细胞迁移非局部动力学模型的稳定性。","authors":"Nadia Loy, Luigi Preziosi","doi":"10.1093/imammb/dqaa013","DOIUrl":null,"url":null,"abstract":"<p><p>The aim of this article is to study the stability of a non-local kinetic model proposed by Loy & Preziosi (2020a) in which the cell speed is affected by the cell population density non-locally measured and weighted according to a sensing kernel in the direction of polarization and motion. We perform the analysis in a $d$-dimensional setting. We study the dispersion relation in the one-dimensional case and we show that the stability depends on two dimensionless parameters: the first one represents the stiffness of the system related to the cell turning rate, to the mean speed at equilibrium and to the sensing radius, while the second one relates to the derivative of the mean speed with respect to the density evaluated at the equilibrium. It is proved that for Dirac delta sensing kernels centered at a finite distance, corresponding to sensing limited to a given distance from the cell center, the homogeneous configuration is linearly unstable to short waves. On the other hand, for a uniform sensing kernel, corresponding to uniformly weighting the information collected up to a given distance, the most unstable wavelength is identified and consistently matches the numerical solution of the kinetic equation.</p>","PeriodicalId":49863,"journal":{"name":"Mathematical Medicine and Biology-A Journal of the Ima","volume":"38 1","pages":"83-105"},"PeriodicalIF":0.8000,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imammb/dqaa013","citationCount":"0","resultStr":"{\"title\":\"Stability of a non-local kinetic model for cell migration with density-dependent speed.\",\"authors\":\"Nadia Loy, Luigi Preziosi\",\"doi\":\"10.1093/imammb/dqaa013\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>The aim of this article is to study the stability of a non-local kinetic model proposed by Loy & Preziosi (2020a) in which the cell speed is affected by the cell population density non-locally measured and weighted according to a sensing kernel in the direction of polarization and motion. We perform the analysis in a $d$-dimensional setting. We study the dispersion relation in the one-dimensional case and we show that the stability depends on two dimensionless parameters: the first one represents the stiffness of the system related to the cell turning rate, to the mean speed at equilibrium and to the sensing radius, while the second one relates to the derivative of the mean speed with respect to the density evaluated at the equilibrium. It is proved that for Dirac delta sensing kernels centered at a finite distance, corresponding to sensing limited to a given distance from the cell center, the homogeneous configuration is linearly unstable to short waves. On the other hand, for a uniform sensing kernel, corresponding to uniformly weighting the information collected up to a given distance, the most unstable wavelength is identified and consistently matches the numerical solution of the kinetic equation.</p>\",\"PeriodicalId\":49863,\"journal\":{\"name\":\"Mathematical Medicine and Biology-A Journal of the Ima\",\"volume\":\"38 1\",\"pages\":\"83-105\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2021-03-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1093/imammb/dqaa013\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Medicine and Biology-A Journal of the Ima\",\"FirstCategoryId\":\"99\",\"ListUrlMain\":\"https://doi.org/10.1093/imammb/dqaa013\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"BIOLOGY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Medicine and Biology-A Journal of the Ima","FirstCategoryId":"99","ListUrlMain":"https://doi.org/10.1093/imammb/dqaa013","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"BIOLOGY","Score":null,"Total":0}
Stability of a non-local kinetic model for cell migration with density-dependent speed.
The aim of this article is to study the stability of a non-local kinetic model proposed by Loy & Preziosi (2020a) in which the cell speed is affected by the cell population density non-locally measured and weighted according to a sensing kernel in the direction of polarization and motion. We perform the analysis in a $d$-dimensional setting. We study the dispersion relation in the one-dimensional case and we show that the stability depends on two dimensionless parameters: the first one represents the stiffness of the system related to the cell turning rate, to the mean speed at equilibrium and to the sensing radius, while the second one relates to the derivative of the mean speed with respect to the density evaluated at the equilibrium. It is proved that for Dirac delta sensing kernels centered at a finite distance, corresponding to sensing limited to a given distance from the cell center, the homogeneous configuration is linearly unstable to short waves. On the other hand, for a uniform sensing kernel, corresponding to uniformly weighting the information collected up to a given distance, the most unstable wavelength is identified and consistently matches the numerical solution of the kinetic equation.
期刊介绍:
Formerly the IMA Journal of Mathematics Applied in Medicine and Biology.
Mathematical Medicine and Biology publishes original articles with a significant mathematical content addressing topics in medicine and biology. Papers exploiting modern developments in applied mathematics are particularly welcome. The biomedical relevance of mathematical models should be demonstrated clearly and validation by comparison against experiment is strongly encouraged.
The journal welcomes contributions relevant to any area of the life sciences including:
-biomechanics-
biophysics-
cell biology-
developmental biology-
ecology and the environment-
epidemiology-
immunology-
infectious diseases-
neuroscience-
pharmacology-
physiology-
population biology