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Decomposition formula and stationary measures for stochastic Lotka-Volterra system with applications to turbulent convection
Chen, Lifeng1; Dong, Zhao2; Jiang, Jifa1; Niu, Lei3; Zhai, Jianliang4
2019-05-01
Source PublicationJOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
ISSN0021-7824
Volume125Pages:43-93
AbstractMotivated by the remarkable works of Busse and his collaborators in the 1980s on turbulent convection in a rotating layer, we explore the long-run behavior of stochastic Lotka-Volterra (LV) systems both in pull-back trajectories and in stationary measures. A decomposition formula is established to describe the relationship between the solutions of stochastic and deterministic LV systems and the stochastic logistic equation. By virtue of this formula, it can be verified that every pull-back omega limit set is an omega limit set of the deterministic LV system multiplied by the random equilibrium of the stochastic logistic equation. The formula is also used to derive the existence of a stationary measure, its support and ergodicity. We prove the tightness of stationary measures and that their weak limits are invariant with respect to the corresponding deterministic system and supported on the Birkhoff center. The developed theory is successfully utilized to completely classify three dimensional competitive stochastic LV systems into 37 classes. The expected occupation measures weakly converge to a strongly mixing measure and all stationary measures are obtained for each class except class 27 c). Among them there are two classes possessing a continuum of random closed orbits and strongly mixing measures supported on the cone surfaces, which weakly converge to the Haar measures of periodic orbits as the noise intensity vanishes. The class 27 c) is an exception, almost every pull-back trajectory cyclically oscillates around the boundary of the stochastic carrying simplex characterized by three unstable stationary solutions. The limit of the expected occupation measures is neither unique nor ergodic. These are consistent with symptoms of turbulence. (C) 2019 Elsevier Masson SAS. All rights reserved.
KeywordStochastic Lotka-Volterra system Stationary measure Ergodicity Support Stochastically cyclical oscillation Turbulence
DOI10.1016/j.matpur.2019.02.013
Language英语
Funding ProjectNational Natural Science Foundation of China (NSFC)[11771295] ; National Natural Science Foundation of China (NSFC)[11371252] ; National Natural Science Foundation of China (NSFC)[11271356] ; National Natural Science Foundation of China (NSFC)[11371041] ; National Natural Science Foundation of China (NSFC)[11431014] ; National Natural Science Foundation of China (NSFC)[11671372] ; National Natural Science Foundation of China (NSFC)[11721101] ; Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, CAS ; Fundamental Research Funds for the Central Universities[WK0010450002] ; Shanghai Gaofeng Project for University Academic Program Development
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000466257300002
PublisherELSEVIER SCIENCE BV
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Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/34601
Collection应用数学研究所
Corresponding AuthorJiang, Jifa
Affiliation1.Shanghai Normal Univ, Math & Sci Coll, Shanghai 200234, Peoples R China
2.Chinese Acad Sci, RCSDS, Acad Math & Syst Sci, Beijing 100190, Peoples R China
3.Univ Helsinki, Dept Math & Stat, FIN-00014 Helsinki, Finland
4.Chinese Acad Sci, Wu Wen Tsun Key Lab Math, USTC, Hefei 230026, Anhui, Peoples R China
Recommended Citation
GB/T 7714
Chen, Lifeng,Dong, Zhao,Jiang, Jifa,et al. Decomposition formula and stationary measures for stochastic Lotka-Volterra system with applications to turbulent convection[J]. JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES,2019,125:43-93.
APA Chen, Lifeng,Dong, Zhao,Jiang, Jifa,Niu, Lei,&Zhai, Jianliang.(2019).Decomposition formula and stationary measures for stochastic Lotka-Volterra system with applications to turbulent convection.JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES,125,43-93.
MLA Chen, Lifeng,et al."Decomposition formula and stationary measures for stochastic Lotka-Volterra system with applications to turbulent convection".JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES 125(2019):43-93.
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