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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 Publication JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES ISSN 0021-7824 Volume 125Pages:43-93 Abstract Motivated 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. Keyword Stochastic Lotka-Volterra system Stationary measure Ergodicity Support Stochastically cyclical oscillation Turbulence DOI 10.1016/j.matpur.2019.02.013 Language 英语 Funding Project National 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 Area Mathematics WOS Subject Mathematics, Applied ; Mathematics WOS ID WOS:000466257300002 Publisher ELSEVIER SCIENCE BV Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/34601 Collection 应用数学研究所 Corresponding Author Jiang, Jifa Affiliation 1.Shanghai Normal Univ, Math & Sci Coll, Shanghai 200234, Peoples R China2.Chinese Acad Sci, RCSDS, Acad Math & Syst Sci, Beijing 100190, Peoples R China3.Univ Helsinki, Dept Math & Stat, FIN-00014 Helsinki, Finland4.Chinese Acad Sci, Wu Wen Tsun Key Lab Math, USTC, Hefei 230026, Anhui, Peoples R China Recommended CitationGB/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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