{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T04:43:40Z","timestamp":1773117820171,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,10,22]],"date-time":"2022-10-22T00:00:00Z","timestamp":1666396800000},"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>In this article, a new competitive neural network (CNN) with reaction-diffusion terms and mixed delays is proposed. Because this network system contains reaction-diffusion terms, it belongs to a partial differential system, which is different from the existing classic CNNs. First, taking into account the spatial diffusion effect, we introduce spatial diffusion for CNNs. Furthermore, since the time delay has an essential influence on the properties of the system, we introduce mixed delays including time-varying discrete delays and distributed delays for CNNs. By constructing suitable Lyapunov\u2013Krasovskii functionals and virtue of the theories of delayed partial differential equations, we study the global asymptotic stability for the considered system. The effectiveness and correctness of the proposed CNN model with reaction-diffusion terms and mixed delays are verified by an example. Finally, some discussion and conclusions for recent developments of CNNs are given.<\/jats:p>","DOI":"10.3390\/sym14112224","type":"journal-article","created":{"date-parts":[[2022,10,24]],"date-time":"2022-10-24T11:53:55Z","timestamp":1666612435000},"page":"2224","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Global Asymptotic Stability of Competitive Neural Networks with Reaction-Diffusion Terms and Mixed Delays"],"prefix":"10.3390","volume":"14","author":[{"given":"Shuxiang","family":"Shao","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Suqian College, Suqian 223800, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4484-8789","authenticated-orcid":false,"given":"Bo","family":"Du","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Huaiyin Normal University, Huaian 223300, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1731","DOI":"10.1162\/neco.1996.8.8.1731","article-title":"Singular perturbation analysis of competitive neural networks with different time scales","volume":"8","author":"Oh","year":"1996","journal-title":"Neural Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/j.neucom.2013.03.030","article-title":"Multistability and instability of delayed competitive neural networks with nondecreasing piecewise linear activation functions","volume":"119","author":"Nie","year":"2013","journal-title":"Neurocomputing"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.neunet.2004.11.009","article-title":"Global exponential stability of delayed competitive neural networks with different time scales","volume":"18","author":"Lu","year":"2005","journal-title":"Neural Netw."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1152","DOI":"10.1109\/TNN.2006.875995","article-title":"Global exponential stability of multitime scale competitive neural networks with nonsmooth functions","volume":"17","author":"Lu","year":"2006","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"928","DOI":"10.1016\/j.nonrwa.2007.11.014","article-title":"Multi stability of competitive neural networks with time-varying and distributed delays","volume":"10","author":"Nie","year":"2009","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1016\/j.jfranklin.2009.03.005","article-title":"Existence and global exponential stability of equilibrium of competitive neural networks with different time scales and multiple delays","volume":"347","author":"Gu","year":"2010","journal-title":"J. Frankl. Inst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1016\/j.neucom.2009.10.003","article-title":"Local uniform stability of competitive neural networks with different time-scales under vanishing perturbations","volume":"73","author":"Roberts","year":"2010","journal-title":"Neurocomputing"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.neucom.2013.02.020","article-title":"Stochastic stability analysis of competitive neural networks with different time-scales","volume":"118","author":"Botella","year":"2013","journal-title":"Neurocomputing"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1016\/j.neucom.2007.08.014","article-title":"Robust stability analysis of competitive neural networks with different time-scales under perturbations","volume":"71","author":"Roberts","year":"2007","journal-title":"Neurocomputing"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1109\/TNN.2009.2034318","article-title":"Impulsive control and synchronization for delayed neural networks with reaction-diffusion terms","volume":"21","author":"Hu","year":"2010","journal-title":"IEEE Trans. Neural. Netw."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4658","DOI":"10.1109\/TCYB.2019.2949468","article-title":"Global stabilization of fuzzy memristor-based reaction-diffusion neural networks","volume":"50","author":"He","year":"2020","journal-title":"IEEE Trans. Cybern."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1007\/s11063-019-10131-8","article-title":"Stability of impulsive stochastic reaction diffusion recurrent neural network","volume":"51","author":"Vidhya","year":"2020","journal-title":"Neural Process Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.neunet.2019.11.008","article-title":"Global exponential synchronization of delayed memristive neural networks with reaction-diffusion terms","volume":"123","author":"Cao","year":"2020","journal-title":"Neural Netw."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s00521-021-06358-z","article-title":"Inverse optimal synchronization control of competitive neural networks with constant time delays","volume":"34","author":"Liu","year":"2022","journal-title":"Neural Comput. Appl."},{"key":"ref_15","first-page":"12548","article-title":"Global asymptotic stability of fractional-order competitive neural networks with multiple time-varying-delay links","volume":"389","author":"Xu","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1016\/j.neunet.2022.06.002","article-title":"Fixed-time synchronization of discontinuous competitive neural networks with time-varying delays","volume":"153","author":"Zheng","year":"2022","journal-title":"Neural Netw."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1887","DOI":"10.1016\/j.camwa.2020.08.020","article-title":"Dynamical behavior of reaction-diffusion neural networks and their synchronization arising in modeling epileptic seizure: A numerical simulation study","volume":"80","author":"Moayeri","year":"2020","journal-title":"Comput. Math. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1109\/TNNLS.2016.2618899","article-title":"State estimation for delayed genetic regulatory networks with reaction-diffusion terms","volume":"29","author":"Zhang","year":"2018","journal-title":"IEEE Trans. Neural Netw. Learn Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3161","DOI":"10.1007\/s00034-015-0006-8","article-title":"Asymptotic stability criteria for genetic regulatory networks with time-varying delays and reaction-diffusion terms","volume":"34","author":"Han","year":"2015","journal-title":"Circuits Syst. Signal Process"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1007\/s11063-017-9682-7","article-title":"Passivity of reaction-diffusion genetic regulatory networks with time-varying delays","volume":"47","author":"Zou","year":"2018","journal-title":"Neural Process Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1109\/TNB.2017.2675446","article-title":"Oscillatory behaviors in genetic regulatory networks mediated by microRNA with time delays and reaction-diffusion terms","volume":"16","author":"Zhang","year":"2017","journal-title":"IEEE Trans. Nanobiosci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1109\/TNB.2020.2964900","article-title":"Stability and oscillation analysis of a gene regulatory network with multiple time delays and diffusion rate","volume":"19","author":"Dong","year":"2020","journal-title":"IEEE Trans. Nanobiosci."},{"key":"ref_23","first-page":"1465","article-title":"Finite-time exponential synchronization of reaction-diffusion delayed complex-dynamical networks","volume":"14","author":"Ali","year":"2021","journal-title":"Discret. Contin. Dyn. Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1016\/j.neucom.2014.06.048","article-title":"Design of controller on synchronization of memristor-based neural networks with time-varying delays","volume":"147","author":"Wang","year":"2015","journal-title":"Neurocomputing"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1016\/j.ins.2012.11.023","article-title":"Global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays","volume":"232","author":"Zhang","year":"2013","journal-title":"Inf. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Salau, A.O., and Jain, S. (2019, January 7\u20139). A Survey of the Types, Techniques, Applications. Proceedings of the 5th IEEE International Conference on Signal Processing and Communication (ICSC), Noida, India.","DOI":"10.1109\/ICSC45622.2019.8938371"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/s12190-020-01346-3","article-title":"Fixed-time synchronization for competitive neural networks with Gaussian-wavelet-type activation functions and discrete delays","volume":"64","author":"Zhou","year":"2020","journal-title":"J. Appl. Math. Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2224\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:00:49Z","timestamp":1760144449000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2224"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,22]]},"references-count":27,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["sym14112224"],"URL":"https:\/\/doi.org\/10.3390\/sym14112224","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,22]]}}}