{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:17:23Z","timestamp":1760235443908,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,23]],"date-time":"2021-08-23T00:00:00Z","timestamp":1629676800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Because the two-dimensional coupled ecosystem has perfect symmetry, the dynamical behavior of symmetric dynamical system is discussed. The analysis of the dynamical behavior of a two-dimensional coupled ecosystem with stochastic parameters is explored in this paper. Firstly, a two-dimensional coupled ecosystem with stochastic parameters is established, it is transformed into a deterministic equivalent system by orthogonal polynomial approximation. Then, analysis of the dynamical behaviour of equivalently deterministic coupled ecosystems is performed using stability theory. At last, we analyzed the dynamical behaviour of non-trivial points by means of the mathematics analysis method and found the influence of random parameters on asymptotic stability in coupled ecosystem is prominent. The dynamical behaviour analysis results were verified by numerical simulation.<\/jats:p>","DOI":"10.3390\/sym13081547","type":"journal-article","created":{"date-parts":[[2021,8,23]],"date-time":"2021-08-23T23:19:33Z","timestamp":1629760773000},"page":"1547","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Analysis of the Dynamical Behaviour of a Two-Dimensional Coupled Ecosystem with Stochastic Parameters"],"prefix":"10.3390","volume":"13","author":[{"given":"Xuefen","family":"Li","sequence":"first","affiliation":[{"name":"College of History and Ethnic Culture, Guizhou University, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fangfang","family":"Shen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Guangxi Normal University, Guiling 541004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,23]]},"reference":[{"key":"ref_1","first-page":"1083","article-title":"Looking to the Future of Mathematical Ecology and Ecological Modelling","volume":"20","author":"Li","year":"2000","journal-title":"Acta Ecol. Sinica"},{"key":"ref_2","first-page":"1","article-title":"Stability Analysis of Enterprises Competition Based on Ecological Model","volume":"46","author":"Liu","year":"2016","journal-title":"Math. Pract. Theory"},{"key":"ref_3","first-page":"577","article-title":"Stability and Hopf Bifurcation of a Kind of Pinus Koraiensis Ecological System with Time Delay","volume":"29","author":"Chen","year":"2014","journal-title":"J. Biomath."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"120508","DOI":"10.7498\/aps.62.120508","article-title":"Bifurcation Analysis of Complex Behavior in the Logistic Map via Periodic Impulsive Force","volume":"62","author":"Jiang","year":"2013","journal-title":"Acta Phys. Sin."},{"key":"ref_5","first-page":"925","article-title":"Study on the Complicated Dynamlcal Behaviors of Nonlinear Ecosystem","volume":"9","author":"Zan","year":"1988","journal-title":"Appl. Math. Mech."},{"key":"ref_6","first-page":"276","article-title":"Stability Analysis of a Stochastic Predator-Prey Model with Harrison Function Response","volume":"14","author":"Niu","year":"2016","journal-title":"J. Dyn. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1038\/261459a0","article-title":"Simple Mathematical Models with Very Complicated Dynamics","volume":"261","author":"May","year":"1976","journal-title":"Nature"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1038\/35098000","article-title":"Catastrophic Shifts in Ecosystems","volume":"413","author":"Scheffer","year":"2001","journal-title":"Nature"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1038\/nature06111","article-title":"Spatial Vegetation Patterns and Imminent Desertification in Mediterranean Arid Ecosystems","volume":"449","author":"Kefi","year":"2007","journal-title":"Nature"},{"key":"ref_10","first-page":"279","article-title":"An Overview on Progess of Interdisciplinary Studies of Dynamics and Life Sciences","volume":"15","author":"Jia","year":"2017","journal-title":"J. Dyn. Control."},{"key":"ref_11","first-page":"278","article-title":"Competition Analysis of Travel Agencies Based on the Population Ecology Model","volume":"2","author":"Zhang","year":"2014","journal-title":"J. Henan Sci. Tech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1362","DOI":"10.2307\/1940066","article-title":"Complex Interactions Between Dispersal and Dynamics: Lessons from Coupled Logistic Equations","volume":"74","author":"Hastings","year":"1993","journal-title":"Ecology"},{"key":"ref_13","first-page":"137","article-title":"Dynamics of Coupled Nonlinear Maps and Its Application to Ecological Modeling","volume":"82","author":"Udwadia","year":"1997","journal-title":"Appl. Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1006\/jtbi.1995.0058","article-title":"The Coupled Logistic Map: A Simple Model for the Effects of Spatial Heterogeneity on Population Dynamics","volume":"173","author":"Alun","year":"1995","journal-title":"J. Theor. Biol."},{"key":"ref_15","first-page":"99","article-title":"Chaos Control of Two Species Discrete Coupling Logistic Model with Symbiotic Interaction","volume":"26","author":"Fu","year":"2011","journal-title":"J. Biomath."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"307","DOI":"10.3934\/mbe.2004.1.307","article-title":"Complex Behavior in a Discrete Coupled Logistic Model for the Symbiotic Interaction of Two Species","volume":"1","author":"Ricardo","year":"2004","journal-title":"Math. Biosci. Eng."},{"key":"ref_17","first-page":"84","article-title":"The Stability and Bifurcation of a Class of Two Dimensional Discrete Dynamical Systems","volume":"28","author":"Hou","year":"2010","journal-title":"Math. Theory Appl."},{"key":"ref_18","first-page":"22","article-title":"The Bifurcation Analysis of one Discrete-Time system","volume":"28","author":"Huang","year":"2010","journal-title":"J. Jiaying Univ. Nat. Sci."},{"key":"ref_19","first-page":"343","article-title":"Bifurcation Control of the Coupled Logistic Mapping","volume":"40","author":"Zong","year":"2011","journal-title":"Inf. Control"},{"key":"ref_20","first-page":"162","article-title":"Effects of Noises and Habitat Complexity in the Prey-Predator Ecosystem","volume":"34","author":"Xu","year":"2013","journal-title":"Appl. Math. Mech."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"170509","DOI":"10.7498\/aps.61.170509","article-title":"Information Entropies and Dynamics in the Stochastic Ecosystem of Two Competing Species","volume":"61","author":"Xie","year":"2012","journal-title":"Acta Phys. Sin."},{"key":"ref_22","first-page":"581","article-title":"Chaotic Characteristics of Two-Dimensional Random Coupled Logistic Map","volume":"56","author":"Yu","year":"2019","journal-title":"J. Sichuan Univ. Nat. Sci. Ed."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Jia, W.T., Xu, Y., and Li, D.X. (2018). Stochastic Dynamics of a Time-Delayed Ecosystem Driven by Poisson White Noise Excitation. Entropy, 20.","DOI":"10.3390\/e20020143"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1017\/S0027763000017232","article-title":"Small Random Perturbations of Dynamical Systems with Reflecting Boundary","volume":"60","author":"Anderson","year":"1976","journal-title":"Nagoya Math. J."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1512\/iumj.1975.24.24039","article-title":"Small Random Pertubations of Dynamical Systems and Applications to Parabolic Equations","volume":"24","author":"Freidman","year":"1974","journal-title":"Indiana Univ. Math. J."},{"key":"ref_26","first-page":"580","article-title":"The Fundamental Solution of a Linear Parabolic, Equation Containing a Small Parameter","volume":"3","author":"Anderson","year":"1959","journal-title":"Ill. J. Math."},{"key":"ref_27","first-page":"991","article-title":"The Asymptotic Behavior of the First Real Eigenvalue of the Second Order Elliptic Operator with a Small Parameter in the Higher Derivatives","volume":"74","author":"Devinatz","year":"1973","journal-title":"Indiana Univ. Math. J."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Wang, Y., Zhao, M., Yu, H., Dai, C., Mei, D., Wang, Q., and Ma, Z. (2015). Analysis of Spatiotemporal Dynamic and Bifurcation in a Wetland Ecosystem. Discret. Dyn. Nat. Soc., 2015.","DOI":"10.1155\/2015\/185432"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1016\/j.physa.2004.01.026","article-title":"A Logistic Map Approach to Economic Cycles","volume":"336","author":"Ausloos","year":"2004","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_30","first-page":"99","article-title":"Stochastic Dynamics of a Predator-prey System with Disease in Predator","volume":"36","author":"Feng","year":"2017","journal-title":"J. Shandong Univ. Sci. Technol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"0130041","DOI":"10.1063\/2.1401304","article-title":"The Asymptotic Stability Analysis in Stochastic Logistic Model with Poisson Growth Coefficient","volume":"4","author":"Ma","year":"2014","journal-title":"Theor. Appl. Mech. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jtbi.2013.03.005","article-title":"Dynamical Behaviors of a Stochastic Delay Logistic System with Impulsive Toxicant Input in a Polluted Environment","volume":"329","author":"Liu","year":"2013","journal-title":"J. Theor. Biol."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Ma, S.J., Dong, D., and Zheng, J. (2013). Generalized Synchronization of Stochastic Discrete Chaotic System with Poisson Distribution Coefficient. Discret. Dyn. Nat. Soc., 2013.","DOI":"10.1155\/2013\/981503"},{"key":"ref_34","unstructured":"Xu, W. (2013). Numerical Analysis Methods for Stochastic Dynamics System, Science Press."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1002\/cpa.3160280406","article-title":"Asymptotic for the Wiener Saussage","volume":"28","author":"Donsker","year":"1975","journal-title":"Comm. Pure Appl. Math."},{"key":"ref_36","unstructured":"Elaydi, S. (2005). An. Introduction to Difference Equations, Springer. [3rd ed.]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1547\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:49:47Z","timestamp":1760165387000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1547"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,23]]},"references-count":36,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["sym13081547"],"URL":"https:\/\/doi.org\/10.3390\/sym13081547","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,8,23]]}}}