{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,4]],"date-time":"2026-04-04T11:05:27Z","timestamp":1775300727734,"version":"3.50.1"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"9","license":[{"start":{"date-parts":[[2018,12,7]],"date-time":"2018-12-07T00:00:00Z","timestamp":1544140800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100007129","name":"Shandong Province Natural Science Foundation","doi-asserted-by":"crossref","award":["ZR2018MA005"],"award-info":[{"award-number":["ZR2018MA005"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100007129","name":"Shandong Province Natural Science Foundation","doi-asserted-by":"crossref","award":["ZR2018MA020"],"award-info":[{"award-number":["ZR2018MA020"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100007129","name":"Shandong Province Natural Science Foundation","doi-asserted-by":"crossref","award":["ZR2017MA008"],"award-info":[{"award-number":["ZR2017MA008"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Key Project of Science and Technology of Weihai","award":["2014DXGJMS08"],"award-info":[{"award-number":["2014DXGJMS08"]}]},{"name":"Innovation Technology Funding Project in Harbin Institute of Technology","award":["HIT.NSRIF.201703"],"award-info":[{"award-number":["HIT.NSRIF.201703"]}]},{"name":"Project of Shandong Province Higher Educational Science and Technology Program of China","award":["J18KA218"],"award-info":[{"award-number":["J18KA218"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Comput & Applic"],"published-print":{"date-parts":[[2020,5]]},"DOI":"10.1007\/s00521-018-3918-y","type":"journal-article","created":{"date-parts":[[2018,12,7]],"date-time":"2018-12-07T03:09:49Z","timestamp":1544152189000},"page":"5001-5013","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Synchronized stationary distribution of stochastic multi-group models with dispersal"],"prefix":"10.1007","volume":"32","author":[{"given":"Yan","family":"Liu","sequence":"first","affiliation":[]},{"given":"Anran","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Wenxue","family":"Li","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,12,7]]},"reference":[{"key":"3918_CR1","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.chaos.2014.09.009","volume":"69","author":"Y Wang","year":"2014","unstructured":"Wang Y, Gao J (2014) Global dynamics of multi-group SEI animal disease models with indirect transmission. Chaos Solitons Fractals 69:81\u201389","journal-title":"Chaos Solitons Fractals"},{"key":"3918_CR2","doi-asserted-by":"crossref","first-page":"2044","DOI":"10.1109\/TCYB.2017.2681119","volume":"47","author":"H Li","year":"2017","unstructured":"Li H, Huang C, Chen G, Liao X, Huang T (2017) Distributed consensus optimization in multiagent networks with time-varying directed topologies and quantized communication. IEEE Trans Cybern 47:2044\u20132057","journal-title":"IEEE Trans Cybern"},{"key":"3918_CR3","doi-asserted-by":"crossref","first-page":"1434","DOI":"10.1016\/j.nonrwa.2012.10.007","volume":"14","author":"Q Yang","year":"2013","unstructured":"Yang Q, Mao X (2013) Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations. Nonlinear Anal Real World Appl 14:1434\u20131456","journal-title":"Nonlinear Anal Real World Appl"},{"key":"3918_CR4","doi-asserted-by":"crossref","first-page":"1339","DOI":"10.1002\/rnc.3631","volume":"27","author":"W Zhu","year":"2017","unstructured":"Zhu W, Li H, Jiang Z (2017) Consensus of multi-agent systems with time-varying topology: an event-based dynamic feedback scheme. Int J Robust Nonlinear Control 27:1339\u20131350","journal-title":"Int J Robust Nonlinear Control"},{"key":"3918_CR5","doi-asserted-by":"crossref","first-page":"2762","DOI":"10.1002\/mma.4196","volume":"40","author":"X Feng","year":"2017","unstructured":"Feng X, Wang K, Zhang F, Teng Z (2017) Threshold dynamics of a nonlinear multi-group epidemic model with two infinite distributed delays. Math Methods Appl Sci 40:2762\u20132771","journal-title":"Math Methods Appl Sci"},{"key":"3918_CR6","doi-asserted-by":"crossref","first-page":"1475","DOI":"10.1002\/mma.4068","volume":"40","author":"J Xu","year":"2017","unstructured":"Xu J, Geng Y, Zhou Y (2017) Global stability of a multi-group model with distributed delay and vaccination. Math Methods Appl Sci 40:1475\u20131486","journal-title":"Math Methods Appl Sci"},{"key":"3918_CR7","doi-asserted-by":"crossref","first-page":"2980","DOI":"10.1016\/j.jfranklin.2016.06.001","volume":"353","author":"Y Guo","year":"2016","unstructured":"Guo Y, Ding X, Li Y (2016) Stochastic stability for pantograph multi-group models with dispersal and stochastic perturbation. J Frankl Inst Eng Appl Math 353:2980\u20132998","journal-title":"J Frankl Inst Eng Appl Math"},{"key":"3918_CR8","doi-asserted-by":"crossref","first-page":"259","DOI":"10.3934\/dcdsb.2015.20.259","volume":"20","author":"C Zhang","year":"2015","unstructured":"Zhang C, Li W, Wang K (2015) Graph-theoretic approach to stability of multi-group models with dispersal. Discrete Contin Dyn Syst Ser B 20:259\u2013280","journal-title":"Discrete Contin Dyn Syst Ser B"},{"key":"3918_CR9","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.amc.2017.10.034","volume":"321","author":"W Chen","year":"2018","unstructured":"Chen W, Zhang B, Ma Q (2018) Decay-rate-dependent conditions for exponential stability of stochastic neutral systems with Markovian jumping parameters. Appl Math Comput 321:93\u2013105","journal-title":"Appl Math Comput"},{"key":"3918_CR10","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.neucom.2017.07.026","volume":"272","author":"J Feng","year":"2018","unstructured":"Feng J, Xu C (2018) A graph-theoretic approach to exponential stability of stochastic complex networks with time-varying delays. Neurocomputing 272:453\u2013460","journal-title":"Neurocomputing"},{"key":"3918_CR11","doi-asserted-by":"crossref","first-page":"043114","DOI":"10.1063\/1.5007139","volume":"28","author":"P Wang","year":"2018","unstructured":"Wang P, Jin W, Su H (2018) Synchronization of coupled stochastic complex-valued dynamical networks with time-varying delays via aperiodically intermittent adaptive control. Chaos 28:043114","journal-title":"Chaos"},{"key":"3918_CR12","doi-asserted-by":"crossref","first-page":"1303","DOI":"10.1017\/S030821051500089X","volume":"146","author":"X Zong","year":"2016","unstructured":"Zong X, Wu F, Huang C (2016) The moment exponential stability criterion of nonlinear hybrid stochastic differential equations and its discrete approximations. Proc R Soc Edinb Sect A Math 146:1303\u20131328","journal-title":"Proc R Soc Edinb Sect A Math"},{"key":"3918_CR13","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1088\/0951-7715\/22\/4\/011","volume":"22","author":"J Lu","year":"2009","unstructured":"Lu J, Ho D, Wu L (2009) Exponential stabilization of switched stochastic dynamical networks. Nonlinearity 22:889\u2013911","journal-title":"Nonlinearity"},{"key":"3918_CR14","doi-asserted-by":"crossref","first-page":"1974","DOI":"10.1016\/j.nonrwa.2010.12.013","volume":"12","author":"J Lu","year":"2011","unstructured":"Lu J, Ho D (2011) Stabilization of complex dynamical networks with noise disturbance under performance constraint. Nonlinear Anal Real World Appl 12:1974\u20131984","journal-title":"Nonlinear Anal Real World Appl"},{"key":"3918_CR15","doi-asserted-by":"crossref","first-page":"738","DOI":"10.1007\/s11424-010-8039-3","volume":"23","author":"Y Zhang","year":"2010","unstructured":"Zhang Y, Sun J (2010) Stability of impulsive linear hybrid systems with time delay. J Syst Sci Complex 23:738\u2013747","journal-title":"J Syst Sci Complex"},{"key":"3918_CR16","doi-asserted-by":"crossref","first-page":"2178","DOI":"10.1049\/iet-cta.2013.0444","volume":"7","author":"Y Zhang","year":"2013","unstructured":"Zhang Y (2013) Stability of discrete-time delay Markovian jump systems with stochastic non-linearity and impulses. IET Control Theory Appl 7:2178\u20132187","journal-title":"IET Control Theory Appl"},{"key":"3918_CR17","doi-asserted-by":"crossref","first-page":"2141","DOI":"10.1007\/s00521-015-2013-x","volume":"27","author":"E Zhu","year":"2016","unstructured":"Zhu E, Yin G, Yuan Q (2016) Stability in distribution of stochastic delay recurrent neural networks with Markovian switching. Neural Comput Appl 27:2141\u20132151","journal-title":"Neural Comput Appl"},{"key":"3918_CR18","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1016\/j.cnsns.2017.10.012","volume":"57","author":"S Ma","year":"2018","unstructured":"Ma S, Kang Y (2018) Exponential synchronization of delayed neutral-type neural networks with L\u00e9vy noise under non-Lipschitz condition. Commun Nonlinear Sci Numer Simul 57:372\u2013387","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"3918_CR19","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1016\/j.apm.2017.09.035","volume":"54","author":"Y Xiao","year":"2018","unstructured":"Xiao Y, Tang S, Sun Z, Song X (2018) Positive role of multiplication noise in attaining complete synchronization on large complex networks of dynamical systems. Appl Math Model 54:803\u2013816","journal-title":"Appl Math Model"},{"key":"3918_CR20","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1002\/rnc.3867","volume":"1","author":"Y Wu","year":"2018","unstructured":"Wu Y, Yan S, Fan M, Li W (2018) Stabilization of stochastic coupled systems with Markovian switching via feedback control based on discrete-time state observations. Int J Robust Nonlinear Control 1:247\u2013265","journal-title":"Int J Robust Nonlinear Control"},{"key":"3918_CR21","doi-asserted-by":"crossref","first-page":"3146","DOI":"10.1016\/j.cnsns.2013.03.021","volume":"18","author":"J Wang","year":"2013","unstructured":"Wang J, Feng J, Xu C, Zhao Y (2013) Exponential synchronization of stochastic perturbed complex networks with time-varying delays via periodically intermittent pinning. Commun Nonlinear Sci Numer Simul 18:3146\u20133157","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"3918_CR22","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.amc.2016.06.030","volume":"291","author":"J Feng","year":"2016","unstructured":"Feng J, Yang P, Zhao Y (2016) Cluster synchronization for nonlinearly time-varying delayed coupling complex networks with stochastic perturbation via periodically intermittent pinning control. Appl Math Comput 291:52\u201368","journal-title":"Appl Math Comput"},{"key":"3918_CR23","doi-asserted-by":"crossref","first-page":"1275","DOI":"10.1007\/s00521-014-1608-y","volume":"25","author":"W Zhang","year":"2014","unstructured":"Zhang W, Li C, Huang T, Qi J (2014) Global exponential synchronization for coupled switched delayed recurrent neural networks with stochastic perturbation and impulsive effects. Neural Comput Appl 25:1275\u20131283","journal-title":"Neural Comput Appl"},{"key":"3918_CR24","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1007\/s00521-012-0871-z","volume":"22","author":"Q Gan","year":"2013","unstructured":"Gan Q (2013) Synchronization of unknown chaotic neural networks with stochastic perturbation and time delay in the leakage term based on adaptive control and parameter identification. Neural Comput Appl 22:1095\u20131104","journal-title":"Neural Comput Appl"},{"key":"3918_CR25","doi-asserted-by":"crossref","first-page":"016218","DOI":"10.1103\/PhysRevE.64.016218","volume":"64","author":"J Acebr\u00f3n","year":"2001","unstructured":"Acebr\u00f3n J, Perales A, Spigler R (2001) Bifurcations and global stability of synchronized stationary states in the Kuramoto model for oscillator populations. Phys Rev E 64:016218","journal-title":"Phys Rev E"},{"key":"3918_CR26","unstructured":"Hasminskii R (1980) Stochastic stability of differential equations. In: Mechanics and analysis. Sijthoff and Noordhoff, Netherlands"},{"key":"3918_CR27","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1007\/s10884-015-9454-x","volume":"27","author":"W Huang","year":"2015","unstructured":"Huang W, Ji M, Liu Z, Yi Y (2015) Steady states of Fokker\u2013Planck equations: I. Existence. J Dyn Differ Equ 27:721\u2013742","journal-title":"J Dyn Differ Equ"},{"key":"3918_CR28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jde.2009.09.003","volume":"248","author":"M Li","year":"2010","unstructured":"Li M, Shuai Z (2010) Global-stability problem for coupled systems of differential equations on networks. J Differ Equ 248:1\u201320","journal-title":"J Differ Equ"},{"key":"3918_CR29","doi-asserted-by":"crossref","first-page":"2793","DOI":"10.1090\/S0002-9939-08-09341-6","volume":"136","author":"H Guo","year":"2008","unstructured":"Guo H, Li M, Shuai Z (2008) A graph-theoretic approach to the method of global Lyapunov functions. Proc Am Math Soc 136:2793\u20132802","journal-title":"Proc Am Math Soc"},{"key":"3918_CR30","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1007\/s10884-016-9566-y","volume":"30","author":"Y Liu","year":"2018","unstructured":"Liu Y, Li W, Feng J (2018) Graph-theoretical method to the existence of stationary distribution of stochastic coupled systems. J Dyn Differ Equ 30:667\u2013685","journal-title":"J Dyn Differ Equ"},{"key":"3918_CR31","doi-asserted-by":"crossref","first-page":"1285","DOI":"10.1007\/s11538-015-0086-4","volume":"77","author":"Y Zhao","year":"2015","unstructured":"Zhao Y, Yuan S, Ma J (2015) Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment. Bull Math Biol 77:1285\u20131326","journal-title":"Bull Math Biol"},{"key":"3918_CR32","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1080\/10739140600809702","volume":"34","author":"Y He","year":"2006","unstructured":"He Y, Zhang D, Zhang H (2006) A novel atomic force microscope with high stability and scan speed. Instrum Sci Technol 34:547\u2013554","journal-title":"Instrum Sci Technol"},{"key":"3918_CR33","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.cnsns.2016.01.013","volume":"37","author":"Y Zhao","year":"2016","unstructured":"Zhao Y, Yuan S, Zhang T (2016) The stationary distribution and ergodicity of a stochastic phytoplankton allelopathy model under regime switching. Commun Nonlinear Sci Numer Simul 37:131\u2013142","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"3918_CR34","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.probengmech.2012.03.002","volume":"30","author":"C Zhu","year":"2012","unstructured":"Zhu C, Zhu W, Yang Y (2012) Design of feedback control of a nonlinear stochastic system for targeting a pre-specified stationary probability distribution. Probab Eng Mech 30:20\u201326","journal-title":"Probab Eng Mech"},{"key":"3918_CR35","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1214\/EJP.v15-756","volume":"15","author":"M Kinnally","year":"2010","unstructured":"Kinnally M, Williams R (2010) On existence and uniqueness of stationary distributions for stochastic delay differential equations with positivity constraints. Electron J Probab 15:409\u2013451","journal-title":"Electron J Probab"},{"key":"3918_CR36","volume-title":"Introduction to graph theory","author":"D West","year":"1996","unstructured":"West D (1996) Introduction to graph theory. Prentice Hall, Upper Saddle River"},{"key":"3918_CR37","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/0025-5564(76)90125-5","volume":"28","author":"A Lajmanovich","year":"1976","unstructured":"Lajmanovich A, Yorke J (1976) A deterministic model for gonorrhea in a nonhomogeneous population. Math Biosci 28:221\u2013236","journal-title":"Math Biosci"},{"key":"3918_CR38","doi-asserted-by":"crossref","first-page":"1292","DOI":"10.1016\/j.jmaa.2016.09.043","volume":"446","author":"M Li","year":"2017","unstructured":"Li M, Jin Z, Sun G, Zhang J (2017) Modeling direct and indirect disease transmission using multi-group model. J Math Anal Appl 446:1292\u20131309","journal-title":"J Math Anal Appl"},{"key":"3918_CR39","volume-title":"Stochastic differential equations and applications","author":"X Mao","year":"2007","unstructured":"Mao X (2007) Stochastic differential equations and applications. Horwood Publishing, Chichester"},{"key":"3918_CR40","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1007\/s10884-015-9470-x","volume":"27","author":"W Huang","year":"2015","unstructured":"Huang W, Ji M, Liu Z, Yi Y (2015) Steady states of Fokker\u2013Planck equations: II. Non-existence. J Dyn Differ Equ 27:743\u2013762","journal-title":"J Dyn Differ Equ"},{"key":"3918_CR41","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/s10884-015-9476-4","volume":"28","author":"W Huang","year":"2016","unstructured":"Huang W, Ji M, Liu Z, Yi Y (2016) Steady states of Fokker\u2013Planck equations: III. Degenerate diffusion. J Dyn Differ Equ 28:127\u2013141","journal-title":"J Dyn Differ Equ"},{"key":"3918_CR42","doi-asserted-by":"crossref","first-page":"1507","DOI":"10.3934\/dcdsb.2013.18.1507","volume":"18","author":"X Zou","year":"2013","unstructured":"Zou X, Fan D, Wang K (2013) Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete Contin Dyn Syst Ser B 18:1507\u20131519","journal-title":"Discrete Contin Dyn Syst Ser B"},{"key":"3918_CR43","doi-asserted-by":"crossref","first-page":"6119","DOI":"10.1016\/j.jfranklin.2017.07.012","volume":"354","author":"X Zou","year":"2017","unstructured":"Zou X, Lv J (2017) A new idea on almost sure permanence and uniform boundedness for a stochastic predator-prey model. J Frankl Inst Eng Appl Math 354:6119\u20136137","journal-title":"J Frankl Inst Eng Appl Math"},{"key":"3918_CR44","doi-asserted-by":"crossref","first-page":"2220","DOI":"10.1016\/j.physa.2017.11.137","volume":"492","author":"W Guo","year":"2018","unstructured":"Guo W, Cai Y, Zhang Q, Wang W (2018) Stochastic persistence and stationary distribution in an SIS epidemic model with media coverage. Physica A 492:2220\u20132236","journal-title":"Physica A"},{"key":"3918_CR45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3934\/nhm.2017001","volume":"12","author":"S Ha","year":"2017","unstructured":"Ha S, Lee J, Li Z (2017) Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators. Netw Heterog Media 12:1\u201324","journal-title":"Netw Heterog Media"},{"key":"3918_CR46","doi-asserted-by":"crossref","first-page":"3226","DOI":"10.1109\/TAC.2015.2418400","volume":"60","author":"L T\u00f4rres","year":"2015","unstructured":"T\u00f4rres L, Hespanha J, Moehlis J (2015) Synchronization of identical oscillators coupled through a symmetric network with dynamics: a constructive approach with applications to parallel operation of inverters. IEEE Trans Autom Control 60:3226\u20133241","journal-title":"IEEE Trans Autom Control"},{"key":"3918_CR47","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1063\/1.1505811","volume":"13","author":"I Heisler","year":"2003","unstructured":"Heisler I, Braun T, Zhang Y (2003) Experimental investigation of partial synchronization in coupled chaotic oscillators. Chaos 13:185\u2013194","journal-title":"Chaos"},{"key":"3918_CR48","doi-asserted-by":"crossref","first-page":"013110","DOI":"10.1063\/1.3677253","volume":"22","author":"Y Xiao","year":"2012","unstructured":"Xiao Y, Tang S, Xu S (2012) Theoretical analysis of multiplicative-noise-induced complete synchronization in global coupled dynamical network. Chaos 22:013110","journal-title":"Chaos"}],"container-title":["Neural Computing and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-018-3918-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00521-018-3918-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-018-3918-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,4]],"date-time":"2026-04-04T10:07:26Z","timestamp":1775297246000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00521-018-3918-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,12,7]]},"references-count":48,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2020,5]]}},"alternative-id":["3918"],"URL":"https:\/\/doi.org\/10.1007\/s00521-018-3918-y","relation":{},"ISSN":["0941-0643","1433-3058"],"issn-type":[{"value":"0941-0643","type":"print"},{"value":"1433-3058","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,12,7]]},"assertion":[{"value":"31 March 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 November 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 December 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}