{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T19:18:22Z","timestamp":1768418302077,"version":"3.49.0"},"reference-count":56,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,3,26]],"date-time":"2022-03-26T00:00:00Z","timestamp":1648252800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,3,26]],"date-time":"2022-03-26T00:00:00Z","timestamp":1648252800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No. 11871473"],"award-info":[{"award-number":["No. 11871473"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["No. ZR2019MA010"],"award-info":[{"award-number":["No. ZR2019MA010"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["No. ZR2019MA006"],"award-info":[{"award-number":["No. ZR2019MA006"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2022,6]]},"DOI":"10.1007\/s00332-022-09789-7","type":"journal-article","created":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T01:03:41Z","timestamp":1648515821000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Stationary Distribution, Extinction and Probability Density Function of a Stochastic Vegetation\u2013Water Model in Arid Ecosystems"],"prefix":"10.1007","volume":"32","author":[{"given":"Baoquan","family":"Zhou","sequence":"first","affiliation":[]},{"given":"Bingtao","family":"Han","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7026-3127","authenticated-orcid":false,"given":"Daqing","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Tasawar","family":"Hayat","sequence":"additional","affiliation":[]},{"given":"Ahmed","family":"Alsaedi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,3,26]]},"reference":[{"issue":"4302","key":"9789_CR1","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1126\/science.197.4302.463","volume":"197","author":"JR Beddington","year":"1977","unstructured":"Beddington, J.R., May, R.M.: Harvesting natural populations in a randomly fluctuating environment. Science 197(4302), 463\u2013465 (1977)","journal-title":"Science"},{"key":"9789_CR2","first-page":"221","volume":"305","author":"Y Cai","year":"2017","unstructured":"Cai, Y., Kang, Y., Wang, W.: A stochastic SIRS epidemic model with nonlinear incidence rate. Appl. Math. Comput. 305, 221\u2013240 (2017)","journal-title":"Appl. Math. Comput."},{"key":"9789_CR3","doi-asserted-by":"crossref","first-page":"125666","DOI":"10.1016\/j.cam.2020.113215","volume":"390","author":"Z Chen","year":"2021","unstructured":"Chen, Z., Wua, Y., Feng, G., Qian, Z., Sun, G.: Effects of global warming on pattern dynamics of vegetation: Wuwei in China as a case. Appl. Math. Comput. 390, 125666 (2021)","journal-title":"Appl. Math. Comput."},{"key":"9789_CR4","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1038\/35041539","volume":"408","author":"PM Cox","year":"2000","unstructured":"Cox, P.M., Betts, R.A., Jones, C.D., et al.: Acceleration of global warming due to carbon-cycle feedbacks in a coupled climate model. Nature 408, 184\u2013187 (2000)","journal-title":"Nature"},{"issue":"1","key":"9789_CR5","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1038\/nclimate1633","volume":"3","author":"A Dai","year":"2013","unstructured":"Dai, A.: Increasing drought under global warming in observations and models. Nat. Clim. Change 3(1), 52\u201358 (2013)","journal-title":"Nat. Clim. Change"},{"key":"9789_CR6","doi-asserted-by":"publisher","first-page":"1013","DOI":"10.1002\/mma.6809","volume":"44","author":"M Gao","year":"2021","unstructured":"Gao, M., Jiang, D., Hayat, T., Alsaedi, A.: Stationary distribution and extinction for a food chain chemostat model with random peturbation. Math. Meth. Appl. Sci. 44, 1013\u20131028 (2021)","journal-title":"Math. Meth. Appl. Sci."},{"key":"9789_CR7","volume-title":"Introduction to Stochastic Differential Equations","author":"TC Gard","year":"1988","unstructured":"Gard, T.C.: Introduction to Stochastic Differential Equations. Marcel Dekker INC, New York (1988)"},{"key":"9789_CR8","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02377-8","volume-title":"Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences","author":"CW Gardiner","year":"1983","unstructured":"Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin (1983)"},{"key":"9789_CR9","doi-asserted-by":"publisher","first-page":"680","DOI":"10.1016\/j.jtbi.2006.08.006","volume":"244","author":"E Gilad","year":"2007","unstructured":"Gilad, E., von Hardenberg, J., Provenzale, A., et al.: A mathematical model of plants as ecosystems engineers. J. Theor. Biol. 244, 680\u2013691 (2007)","journal-title":"J. Theor. Biol."},{"key":"9789_CR10","doi-asserted-by":"publisher","first-page":"420","DOI":"10.1016\/j.ecolmodel.2006.10.005","volume":"201","author":"V Guttal","year":"2007","unstructured":"Guttal, V., Jayaprakash, C.: Impact of noise on bistable ecological systems. Ecol. Model. 201, 420\u20138 (2007)","journal-title":"Ecol. Model."},{"key":"9789_CR11","doi-asserted-by":"publisher","first-page":"450","DOI":"10.1111\/j.1461-0248.2008.01160.x","volume":"11","author":"V Guttal","year":"2008","unstructured":"Guttal, V., Jayaprakash, C.: Changing skewness: an early warning signal of regime shifts in ecosystems. Ecol. Lett. 11, 450\u201360 (2008)","journal-title":"Ecol. Lett."},{"key":"9789_CR12","doi-asserted-by":"publisher","first-page":"96","DOI":"10.1016\/j.physa.2014.04.015","volume":"408","author":"Q Han","year":"2014","unstructured":"Han, Q., Yang, T., Zeng, C., et al.: Impact of time delays on stochastic resonance in an ecological system describing vegetation. Phys. A 408, 96\u2013105 (2014)","journal-title":"Phys. A"},{"key":"9789_CR13","doi-asserted-by":"publisher","first-page":"110238","DOI":"10.1016\/j.chaos.2020.110238","volume":"140","author":"B Han","year":"2020","unstructured":"Han, B., Jiang, D., Hayat, T., et al.: Stationary distribution and extinction of a stochastic staged progression AIDS model with staged treatment and second-order perturbation. Chaos Soliton Fract. 140, 110238 (2020)","journal-title":"Chaos Soliton Fract."},{"issue":"5","key":"9789_CR14","first-page":"110519","volume":"142","author":"B Han","year":"2020","unstructured":"Han, B., Jiang, D., Zhou, B., et al.: Stationary distribution and probability density function of a stochastic SIRSI epidemic model with saturation incidence rate and logistic growth. Chaos Soliton Fract. 142(5), 110519 (2020)","journal-title":"Chaos Soliton Fract."},{"issue":"6232","key":"9789_CR15","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1126\/science.aaa1788","volume":"348","author":"Y Hautier","year":"2015","unstructured":"Hautier, Y., Tilman, D., Isbell, F., et al.: Anthropogenic environmental changes affect ecosystem stability via biodiversity. Science 348(6232), 336\u2013340 (2015)","journal-title":"Science"},{"key":"9789_CR16","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1137\/S0036144500378302","volume":"43","author":"DJ Higham","year":"2001","unstructured":"Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525\u2013546 (2001)","journal-title":"SIAM Rev."},{"key":"9789_CR17","doi-asserted-by":"publisher","first-page":"1131","DOI":"10.1007\/s00382-015-2636-8","volume":"46","author":"J Huang","year":"2016","unstructured":"Huang, J., Ji, M., Xie, Y., et al.: Global semi-arid climate change over last 60 years. Clim. Dyn. 46, 1131\u20131150 (2016)","journal-title":"Clim. Dyn."},{"key":"9789_CR18","first-page":"619","volume":"14","author":"N Ikeda","year":"1977","unstructured":"Ikeda, N., Watanade, S.: A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. 14, 619\u2013633 (1977)","journal-title":"Osaka J. Math."},{"key":"9789_CR19","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1016\/j.jde.2005.06.017","volume":"217","author":"L Imhof","year":"2005","unstructured":"Imhof, L., Walcher, S.: Exclusion and persistence in deterministic and stochastic chemostat models. J. Differ. Equ. 217, 26\u201353 (2005)","journal-title":"J. Differ. Equ."},{"issue":"1","key":"9789_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/S0036141096303359","volume":"29","author":"R Jordan","year":"1998","unstructured":"Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker\u2013Planck equation. SIAM J. Math. Anal. 29(1), 1\u201317 (1998)","journal-title":"SIAM J. Math. Anal."},{"key":"9789_CR21","doi-asserted-by":"publisher","first-page":"332","DOI":"10.1016\/j.tpb.2008.09.004","volume":"74","author":"S Kefi","year":"2008","unstructured":"Kefi, S., Rietkerka, M., Katul, G.G.: Vegetation pattern shift as a result of rising atmospheric $$\\text{ CO}_2$$ in arid ecosystems. Theor. Popu. Biol. 74, 332\u2013344 (2008)","journal-title":"Theor. Popu. Biol."},{"key":"9789_CR22","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1007\/s12080-009-0067-z","volume":"3","author":"S Kefi","year":"2010","unstructured":"Kefi, S., Eppinga, M.B., de Ruiter, P.C., Rietkerk, M.: Bistability and regular spatial patterns in arid ecosystems. Theor. Ecol. 3, 257\u2013269 (2010)","journal-title":"Theor. Ecol."},{"key":"9789_CR23","volume-title":"Stochastic Stability of Differential Equations","author":"R Khasminskii","year":"2011","unstructured":"Khasminskii, R.: Stochastic Stability of Differential Equations. Springer, Berlin (2011)"},{"key":"9789_CR24","first-page":"97","volume":"21","author":"X Li","year":"2004","unstructured":"Li, X., Liu, X., Ma, Z.: Analysis on the drought characteristics in the main arid regions in the world since recent hundred-odd years. Arid Zone Res. 21, 97\u2013103 (2004)","journal-title":"Arid Zone Res."},{"issue":"4","key":"9789_CR25","doi-asserted-by":"publisher","first-page":"1079","DOI":"10.1007\/s10884-014-9408-8","volume":"26","author":"Y Lin","year":"2014","unstructured":"Lin, Y., Jiang, D.: Threshold behavior in a stochastic SIS epidemic model with standard incidence. J. Dyn. Differ. Equ. 26(4), 1079\u20131094 (2014)","journal-title":"J. Dyn. Differ. Equ."},{"key":"9789_CR26","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1080\/17442508008833146","volume":"3","author":"R Lipster","year":"1980","unstructured":"Lipster, R.: A strong law of large numbers for local martingales. Stochastics 3, 217\u2013228 (1980)","journal-title":"Stochastics"},{"key":"9789_CR27","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1007\/s00332-017-9413-2","volume":"28","author":"Q Liu","year":"2018","unstructured":"Liu, Q., Jiang, D.: Periodic solution and stationary distribution of stochastic predator-prey models with higher-order perturbation. J. Nonlinear Sci. 28, 423\u2013442 (2018)","journal-title":"J. Nonlinear Sci."},{"key":"9789_CR28","doi-asserted-by":"publisher","first-page":"1151","DOI":"10.1007\/s00332-018-9444-3","volume":"28","author":"Q Liu","year":"2018","unstructured":"Liu, Q., Jiang, D., Hayat, T., et al.: Dynamics of a stochastic predator-prey model with stage structure for predator and Holling Type II functional response. J. Nonlinear Sci. 28, 1151\u20131187 (2018)","journal-title":"J. Nonlinear Sci."},{"key":"9789_CR29","first-page":"255","volume":"342","author":"C Liu","year":"2019","unstructured":"Liu, C., Li, L., Wang, Z., Wang, R.: Pattern transitions in a vegetation system with cross-diffusion. Appl. Math. Comput. 342, 255\u2013262 (2019)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"9789_CR30","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1007\/s00332-019-09576-x","volume":"30","author":"Q Liu","year":"2020","unstructured":"Liu, Q., Jiang, D., Hayat, T., et al.: Stationary distribution and extinction of a stochastic HIV-1 infection model with distributed delay and logistic growth. J. Nonlinear. Sci. 30(1), 369\u2013395 (2020)","journal-title":"J. Nonlinear. Sci."},{"issue":"3","key":"9789_CR31","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1016\/j.camwa.2019.07.022","volume":"79","author":"G Liu","year":"2020","unstructured":"Liu, G., Qi, H., Chang, Z., Meng, X.: Asymptotic stability of a stochastic May mutualism system. Comput. Math. Appl. 79(3), 735\u2013745 (2020)","journal-title":"Comput. Math. Appl."},{"key":"9789_CR32","unstructured":"London, N., Unep, N.: United nations convention to combat desertification in countries experiencing serious drought and\/or desertification, particularly in Africa. Document A\/AC. 241\/27, 12. 09. 1994 with Annexes, New York, NY (1994)"},{"key":"9789_CR33","unstructured":"Ma, Z., Zhou, Y., Li, C.: Qualitative and Stability Methods for Ordinary Differential Equations. Science Press, Beijing (2015). (in Chinese)"},{"key":"9789_CR34","volume-title":"Stochastic Differential Equations and Applications","author":"X Mao","year":"1997","unstructured":"Mao, X.: Stochastic Differential Equations and Applications. Horwood Publishing, Chichester (1997)"},{"key":"9789_CR35","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1016\/j.nonrwa.2012.07.012","volume":"14","author":"K Marinov","year":"2013","unstructured":"Marinov, K., Wang, T., Yang, Y.: On a vegetation pattern formation model governed by a nonlinear parabolic system. Nonlinear Anal. Real World Appl. 14, 507\u2013525 (2013)","journal-title":"Nonlinear Anal. Real World Appl."},{"issue":"5628","key":"9789_CR36","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1038\/269471a0","volume":"269","author":"RM May","year":"1977","unstructured":"May, R.M.: Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature 269(5628), 471\u2013477 (1977)","journal-title":"Nature"},{"key":"9789_CR37","doi-asserted-by":"publisher","first-page":"814","DOI":"10.1137\/19M1246973","volume":"80","author":"DH Nguyen","year":"2020","unstructured":"Nguyen, D.H., Yin, G., Zhu, C.: Long-term analysis of a stochastic SIRS model with general incidence rates. SIAM J. Appl. Math. 80, 814\u2013838 (2020)","journal-title":"SIAM J. Appl. Math."},{"issue":"9","key":"9789_CR38","doi-asserted-by":"publisher","first-page":"6043","DOI":"10.1002\/mma.6346","volume":"43","author":"S Pan","year":"2020","unstructured":"Pan, S., Zhang, Q., Anke, M.: Near-optimal control of a stochastic vegetation\u2013water system with reaction diffusion. Math. Meth. Appl. Sci. 43(9), 6043\u20136061 (2020)","journal-title":"Math. Meth. Appl. Sci."},{"key":"9789_CR39","doi-asserted-by":"publisher","first-page":"107589","DOI":"10.1016\/j.aml.2021.107589","volume":"123","author":"S Pan","year":"2022","unstructured":"Pan, S., Zhang, Q., Anke, M.: Stationary distribution of a stochastic vegetation\u2013water system with reaction\u2013diffusion. Appl. Math. Lett. 123, 107589 (2022)","journal-title":"Appl. Math. Lett."},{"key":"9789_CR40","doi-asserted-by":"publisher","first-page":"1793","DOI":"10.1137\/0149110","volume":"49","author":"H Roozen","year":"1989","unstructured":"Roozen, H.: An asymptotic solution to a two-dimensional exit problem arising in population dynamics. SIAM J. Appl. Math. 49, 1793 (1989)","journal-title":"SIAM J. Appl. Math."},{"key":"9789_CR41","doi-asserted-by":"publisher","first-page":"1717","DOI":"10.5194\/hess-11-1717-2007","volume":"11","author":"PM Saco","year":"2007","unstructured":"Saco, P.M., Willgoose, G.R., Hancock, G.R.: Eco-geomorphology of banded vegetation patterns in arid and semi-arid regions. Hydrol. Earth Syst. Sci. 11, 1717\u20131730 (2007)","journal-title":"Hydrol. Earth Syst. Sci."},{"issue":"3","key":"9789_CR42","doi-asserted-by":"publisher","first-page":"038101","DOI":"10.1103\/PhysRevLett.90.038101","volume":"90","author":"NM Shnerb","year":"2003","unstructured":"Shnerb, N.M., Sarah, P., Lavee, H., Solomon, S.: Reactive glass and vegetation patterns. Phys. Rev. Lett. 90(3), 038101 (2003)","journal-title":"Phys. Rev. Lett."},{"issue":"4","key":"9789_CR43","doi-asserted-by":"publisher","first-page":"2207","DOI":"10.1007\/s11071-013-0935-3","volume":"73","author":"G Sun","year":"2013","unstructured":"Sun, G., Li, L., Zhang, Z.: Spatial dynamics of a vegetation model in an arid flat environment. Nonlinear Dyn. 73(4), 2207\u20132219 (2013)","journal-title":"Nonlinear Dyn."},{"key":"9789_CR44","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.chaos.2014.06.003","volume":"66","author":"F Wang","year":"2014","unstructured":"Wang, F., Wang, X., Zhang, S., Ding, C.: On pulse vaccine strategy in a periodic stochastic SIR epidemic model. Chaos Soliton Fract. 66, 127\u2013135 (2014)","journal-title":"Chaos Soliton Fract."},{"issue":"3","key":"9789_CR45","first-page":"541","volume":"40","author":"D Yang","year":"2016","unstructured":"Yang, D., Liu, Y., Cai, Z., et al.: The spatial and temporal distribution of carbon dioxide over China based on GOSAT observations. Chin. J. Atmos. Sci. 40(3), 541\u2013550 (2016)","journal-title":"Chin. J. Atmos. Sci."},{"key":"9789_CR46","first-page":"125833","volume":"394","author":"X Zhang","year":"2021","unstructured":"Zhang, X., Yuan, R.: A stochastic chemostat model with mean-reverting Ornstein\u2013Uhlenbeck process and Monod\u2013Haldane response function. Appl. Math. Comput. 394, 125833 (2021)","journal-title":"Appl. Math. Comput."},{"key":"9789_CR47","doi-asserted-by":"publisher","first-page":"106217","DOI":"10.1016\/j.aml.2020.106217","volume":"103","author":"H Zhang","year":"2020","unstructured":"Zhang, H., Zhang, T.: The stationary distribution of a microorganism flocculation model with stochastic perturbation. Appl. Math. Lett. 103, 106217 (2020)","journal-title":"Appl. Math. Lett."},{"key":"9789_CR48","doi-asserted-by":"publisher","first-page":"258","DOI":"10.1016\/j.cnsns.2019.05.001","volume":"77","author":"H Zhang","year":"2019","unstructured":"Zhang, H., Xu, W., Lei, Y., Qiao, Y.: Early warning and basin stability in a stochastic vegetation\u2013water dynamical system. Commun. Nonlinear. Sci. Numer. Simul. 77, 258\u2013270 (2019)","journal-title":"Commun. Nonlinear. Sci. Numer. Simul."},{"issue":"1","key":"9789_CR49","doi-asserted-by":"publisher","first-page":"516","DOI":"10.1016\/j.jfranklin.2020.10.035","volume":"358","author":"H Zhang","year":"2020","unstructured":"Zhang, H., Liu, X., Xu, W.: Threshold dynamics and pulse control of a stochastic ecosystem with switching parameters. J. Frankl. Inst. 358(1), 516\u2013532 (2020)","journal-title":"J. Frankl. Inst."},{"key":"9789_CR50","doi-asserted-by":"publisher","first-page":"124809","DOI":"10.1016\/j.physa.2020.124809","volume":"556","author":"H Zhang","year":"2020","unstructured":"Zhang, H., Xu, W., Han, P., Qiao, Y.: Stochastic dynamic balance of a bi-stable vegetation model with pulse control. Phys. A 556, 124809 (2020)","journal-title":"Phys. A"},{"key":"9789_CR51","doi-asserted-by":"publisher","first-page":"109767","DOI":"10.1016\/j.chaos.2020.109767","volume":"135","author":"H Zhang","year":"2020","unstructured":"Zhang, H., Xu, W., Guo, Q., et al.: First escape probability and mean first exit time for a time-delayed ecosystem driven by non-Gaussian colored noise. Chaos Soliton Fract. 135, 109767 (2020)","journal-title":"Chaos Soliton Fract."},{"issue":"7","key":"9789_CR52","doi-asserted-by":"publisher","first-page":"1285","DOI":"10.1007\/s11538-015-0086-4","volume":"77","author":"Y Zhao","year":"2015","unstructured":"Zhao, Y., Yuan, S., Ma, J.: Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment. Bull. Math. Biol. 77(7), 1285\u20131326 (2015)","journal-title":"Bull. Math. Biol."},{"key":"9789_CR53","doi-asserted-by":"publisher","first-page":"110601","DOI":"10.1016\/j.chaos.2020.110601","volume":"143","author":"B Zhou","year":"2021","unstructured":"Zhou, B., Jiang, D., Dai, Y., et al.: Stationary distribution and probability density function of a stochastic SVIS epidemic model with standard incidence and vacinnation. Chaos Soliton Fract. 143, 110601 (2021)","journal-title":"Chaos Soliton Fract."},{"key":"9789_CR54","first-page":"126388","volume":"410","author":"B Zhou","year":"2021","unstructured":"Zhou, B., Han, B., Jiang, D., et al.: Ergodic stationary distribution and extinction of a hybrid stochastic SEQIHR epidemic model with media coverage, quarantine strategies and pre-existing immunity under discrete Markov switching. Appl. Math. Comput. 410, 126388 (2021)","journal-title":"Appl. Math. Comput."},{"key":"9789_CR55","doi-asserted-by":"publisher","first-page":"1155","DOI":"10.1137\/060649343","volume":"46","author":"C Zhu","year":"2007","unstructured":"Zhu, C., Yin, G.: Asymptotic properties of hybrid diffusion systems. SIAM J. Control. Optim. 46, 1155\u20131179 (2007)","journal-title":"SIAM J. Control. Optim."},{"key":"9789_CR56","first-page":"93","volume":"330","author":"L Zu","year":"2018","unstructured":"Zu, L., Jiang, D., O\u2019Regan, D., et al.: Ergodic property of a Lotka\u2013Volterra predator-prey model with white noise higher order perturbation under regime switching. Appl. Math. Comput. 330, 93\u2013102 (2018)","journal-title":"Appl. Math. Comput."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-022-09789-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-022-09789-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-022-09789-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,12]],"date-time":"2022-11-12T09:10:07Z","timestamp":1668244207000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-022-09789-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,26]]},"references-count":56,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,6]]}},"alternative-id":["9789"],"URL":"https:\/\/doi.org\/10.1007\/s00332-022-09789-7","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,26]]},"assertion":[{"value":"8 March 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 February 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 March 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"30"}}