{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T21:14:19Z","timestamp":1773090859833,"version":"3.50.1"},"reference-count":72,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T00:00:00Z","timestamp":1611187200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002383","name":"King Saud University","doi-asserted-by":"publisher","award":["RSP-2020\/167"],"award-info":[{"award-number":["RSP-2020\/167"]}],"id":[{"id":"10.13039\/501100002383","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Using the discrete fractional calculus, a novel discrete fractional-order food chain model for the case of strong pressure on preys map is proposed. Dynamical behaviors of the model involving stability analysis of its equilibrium points, bifurcation diagrams and phase portraits are investigated. It is demonstrated that the model can exhibit a variety of dynamical behaviors including stable steady states, periodic and quasiperiodic dynamics. Then, a hybrid encryption scheme based on chaotic behavior of the model along with elliptic curve key exchange scheme is proposed for colored plain images. The hybrid scheme combines the characteristics of noise-like chaotic dynamics of the map, including high sensitivity to values of parameters, with the advantages of reliable elliptic curves-based encryption systems. Security analysis assures the efficiency of the proposed algorithm and validates its robustness and efficiency against possible types of attacks.<\/jats:p>","DOI":"10.3390\/sym13020161","type":"journal-article","created":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T09:49:21Z","timestamp":1611222561000},"page":"161","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1167-2430","authenticated-orcid":false,"given":"Sameh","family":"Askar","sequence":"first","affiliation":[{"name":"Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5242-0280","authenticated-orcid":false,"given":"Abdulrahman","family":"Al-khedhairi","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7125-9675","authenticated-orcid":false,"given":"Amr","family":"Elsonbaty","sequence":"additional","affiliation":[{"name":"Department of Engineering Mathematics and Physics, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5010-601X","authenticated-orcid":false,"given":"Abdelalim","family":"Elsadany","sequence":"additional","affiliation":[{"name":"Basic Science Department, Faculty of Computers and Informatics, Suez Canal University, Ismailia 41522, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific.","DOI":"10.1142\/9789812817747"},{"key":"ref_2","unstructured":"Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Elsevier."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Baleanu, D., G\u00fcven\u00e7, Z.B., and Machado, J.T. (2010). New Trends in Nanotechnology and Fractional Calculus Applications, Springer.","DOI":"10.1007\/978-90-481-3293-5"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1109\/JETCAS.2013.2265797","article-title":"A survey of fractional-order circuit models for biology and biomedicine","volume":"3","author":"Freeborn","year":"2013","journal-title":"IEEE J. Emerg. Sel. Top. Circuits Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1305","DOI":"10.1016\/j.chaos.2006.07.051","article-title":"Nonlinear dynamics and chaos in a fractional-order financial system","volume":"36","author":"Chen","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"219","DOI":"10.7153\/fdc-06-14","article-title":"Elasticity for economic processes with memory: Fractional differential calculus approach","volume":"6","author":"Tarasova","year":"2016","journal-title":"Fractional Differential Calc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1586","DOI":"10.1016\/j.camwa.2009.08.039","article-title":"Fractional calculus models of complex dynamics in biological tissues","volume":"59","author":"Magin","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1109\/81.404062","article-title":"Chaos in a fractional order Chua\u2019s system","volume":"42","author":"Hartley","year":"1995","journal-title":"IEEE Trans. Circuits Syst. Fundam. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1016\/j.chaos.2009.03.016","article-title":"Dynamic analysis of a fractional-order Lorenz chaotic system","volume":"42","author":"Yu","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"8299","DOI":"10.3390\/e17127882","article-title":"Complexity analysis and DSP implementation of the fractional-order Lorenz hyperchaotic system","volume":"17","author":"He","year":"2015","journal-title":"Entropy"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1209","DOI":"10.1142\/S0218127410026411","article-title":"Bifurcations and chaos in fractional-order simplified Lorenz system","volume":"20","author":"Sun","year":"2010","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1016\/j.chaos.2004.02.035","article-title":"Chaos in the fractional order Chen system and its control","volume":"22","author":"Li","year":"2004","journal-title":"Chaos Solitons Fractals"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.physa.2006.03.021","article-title":"Extending synchronization scheme to chaotic fractional-order Chen systems","volume":"370","author":"Wang","year":"2006","journal-title":"Phys. Stat. Mech. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1532","DOI":"10.3934\/math.2020105","article-title":"Dynamical behaviour of fractional-order atmosphere-soil-land plant carbon cycle system","volume":"5","author":"Liu","year":"2020","journal-title":"AIMS Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1165","DOI":"10.1080\/00207179.2015.1124290","article-title":"A review and evaluation of numerical tools for fractional calculus and fractional order controls","volume":"90","author":"Li","year":"2017","journal-title":"Int. J. Control"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"012101","DOI":"10.1088\/1742-6596\/1074\/1\/012101","article-title":"September. Control and synchronization of fractional order complex valued chaotic Chen systems","volume":"1074","author":"Wang","year":"2018","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Song, C., Fei, S., Cao, J., and Huang, C. (2019). Robust synchronization of fractional-order uncertain chaotic systems based on output feedback sliding mode control. Mathematics, 7.","DOI":"10.3390\/math7070599"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Li, G., Zhang, X., and Yang, H. (2019). Numerical Analysis, Circuit Simulation, and Control Synchronization of Fractional-Order Unified Chaotic System. Mathematics, 7.","DOI":"10.3390\/math7111077"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1186\/s13662-020-2527-0","article-title":"Analyzing transient response of the parallel RCL circuit by using the Caputo\u2014Fabrizio fractional derivative","volume":"2020","author":"Alizadeh","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"083103","DOI":"10.1063\/1.5096645","article-title":"New variable-order fractional chaotic systems for fast image encryption","volume":"29","author":"Wu","year":"2019","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Tsirimokou, G., Psychalinos, C., and Elwakil, A. (2017). Design of CMOS Analog Integrated Fractional-Order Circuits: Applications in Medicine and Biology, Springer.","DOI":"10.1007\/978-3-319-55633-8"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"180403","DOI":"10.1103\/PhysRevLett.115.180403","article-title":"Propagation dynamics of a light beam in a fractional Schrodinger equation","volume":"115","author":"Zhang","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"043802","DOI":"10.1103\/PhysRevA.63.043802","article-title":"Higher-order correlation on polarization beats in Markovian stochastic fields","volume":"63","author":"Zhang","year":"2001","journal-title":"Phys. Rev. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"023802","DOI":"10.1103\/PhysRevA.71.023802","article-title":"Coherent laser control in attosecond sum-frequency polarization beats using twin noisy driving fields","volume":"71","author":"Zhang","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/s11071-013-1065-7","article-title":"Discrete fractional logistic map and its chaos","volume":"75","author":"Wu","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.physleta.2009.11.008","article-title":"Fractional standard map","volume":"374","author":"Edelman","year":"2009","journal-title":"Phys. Lett. A"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Khennaoui, A.A., Ouannas, A., Bendoukha, S., Wang, X., and Pham, V.T. (2018). On chaos in the fractional-order discrete-time unified system and its control synchronization. Entropy, 20.","DOI":"10.3390\/e20070530"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s12648-015-0742-4","article-title":"Chaotic synchronization between linearly coupled discrete fractional H\u00e9non maps","volume":"90","author":"Liu","year":"2016","journal-title":"Indian J. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/j.aeue.2017.05.009","article-title":"Investigation of chaos in fractional order generalized hyperchaotic Henon map","volume":"78","author":"Shukla","year":"2017","journal-title":"AEU Int. J. Electron. Commun."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1016\/j.cnsns.2017.10.009","article-title":"Bifurcation and chaos of a new discrete fractional-order logistic map","volume":"57","author":"Ji","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_31","unstructured":"Fridrich, J. (1997, January 12\u201315). Image encryption based on chaotic maps. Proceedings of the 1997 IEEE International Conference on Systems, Man, and Cybernetics. Computational Cybernetics and Simulation, Orlando, FL, USA."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1016\/j.physa.2005.01.001","article-title":"Security analysis of a chaos-based image encryption algorithm","volume":"351","author":"Lian","year":"2005","journal-title":"Phys. Stat. Mech. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1007\/s11071-013-0832-9","article-title":"Cryptanalysis of a parallel sub-image encryption method with high-dimensional chaos","volume":"73","author":"Wang","year":"2013","journal-title":"Nonlinear Dyn."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1016\/j.jss.2011.10.051","article-title":"The novel bilateral\u2013Diffusion image encryption algorithm with dynamical compound chaos","volume":"85","author":"Tong","year":"2012","journal-title":"J. Syst. Softw."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1202","DOI":"10.1016\/j.sigpro.2011.11.004","article-title":"A fast color image encryption algorithm based on coupled two-dimensional piecewise chaotic map","volume":"92","author":"Seyedzadeh","year":"2012","journal-title":"Signal Process."},{"key":"ref_36","first-page":"47","article-title":"Discrete-time fractional-order controllers","volume":"4","author":"Machado","year":"2001","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.sigpro.2014.02.022","article-title":"Chaos synchronization of the discrete fractional logistic map","volume":"102","author":"Wu","year":"2014","journal-title":"Signal Process."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1016\/j.physleta.2013.12.010","article-title":"Discrete chaos in fractional sine and standard maps","volume":"378","author":"Wu","year":"2014","journal-title":"Phys. Lett. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2243","DOI":"10.4236\/am.2014.515218","article-title":"Discrete chaos in fractional H\u00e9non map","volume":"5","author":"Hu","year":"2014","journal-title":"Appl. Math."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.chaos.2018.09.022","article-title":"Different type kernel h- fractional differences and their fractional h- sums","volume":"116","author":"Abdeljawad","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.chaos.2018.12.019","article-title":"On fractional\u2014Order discrete\u2014Time systems: Chaos, stabilization and synchronization","volume":"119","author":"Khennaoui","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1016\/j.optlaseng.2012.11.001","article-title":"Double optical image encryption using discrete Chirikov standard map and chaos-based fractional random transform","volume":"51","author":"Zhang","year":"2013","journal-title":"Opt. Lasers Eng."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"42227","DOI":"10.1109\/ACCESS.2018.2858839","article-title":"An efficient chaotic image cryptosystem based on simultaneous permutation and diffusion operations","volume":"6","author":"Diab","year":"2018","journal-title":"IEEE Access"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.optlaseng.2017.11.005","article-title":"Color image encryption using random transforms, phase retrieval, chaotic maps, and diffusion","volume":"103","author":"Annaby","year":"2018","journal-title":"Opt. Lasers Eng."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"33527","DOI":"10.1007\/s11042-019-08112-z","article-title":"Joint block and stream cipher based on a modified skew tent map","volume":"78","author":"Qumsieh","year":"2019","journal-title":"Multimed. Tools Appl."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1016\/j.sigpro.2017.11.005","article-title":"Cryptanalysis and enhancements of image encryption using combination of the 1D chaotic map","volume":"144","author":"Wang","year":"2018","journal-title":"Signal Process."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"163698","DOI":"10.1016\/j.ijleo.2019.163698","article-title":"Discrete tempered fractional calculus for new chaotic systems with short memory and image encryption","volume":"218","author":"Abdeljawad","year":"2019","journal-title":"Optik"},{"key":"ref_48","first-page":"96","article-title":"Image Scramble based on Discrete Cosine Transform and Henon Map","volume":"9","author":"Ridha","year":"2019","journal-title":"Iraqi J. Inf. Technol."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s12043-016-1231-9","article-title":"Chaos in discrete fractional difference equations","volume":"87","author":"Deshpande","year":"2016","journal-title":"Pramana"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Tarasova, V.V., and Tarasov, V.E. (2017). Exact discretization of an economic accelerator and multiplier with memory. Fractal Fract., 1.","DOI":"10.3390\/fractalfract1010006"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Girejko, E., Pawl uszewicz, E., and Wyrwas, M. (2016). The Z-transform method for sequential fractional difference operators. Theoretical Developments and Applications of Non-Integer Order Systems, Springer.","DOI":"10.1007\/978-3-319-23039-9_5"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Danca, M.F., and Feckan, M. (2019). Chaos suppression in a Gompertz-like discrete system of fractional order. arXiv.","DOI":"10.1142\/S0218127420500492"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Brandibur, O., Kaslik, E., Mozyrska, D., and Wyrwas, M. (2020). Stability of Systems of Fractional-Order Difference Equations and Applications to a Rulkov-Type Neuronal Model. New Trends in Nonlinear Dynamics, Springer.","DOI":"10.1007\/978-3-030-34724-6_31"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1186\/s13662-019-2064-x","article-title":"The fractional form of a new three-dimensional generalized H\u00e9non map","volume":"2019","author":"Jouini","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Xin, B., Peng, W., and Kwon, Y. (2019). A fractional-order difference Cournot duopoly game with long memory. arXiv.","DOI":"10.1016\/j.physa.2020.124993"},{"key":"ref_56","first-page":"31","article-title":"Variazioni e fluttuazioni del numero di individui in specie animali conviventi","volume":"2","author":"Volterra","year":"1926","journal-title":"Mem. Acad. Lincei."},{"key":"ref_57","unstructured":"Lotka, A.J. (1925). Elements of Physical Biology, Williams and Wilkins."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Kot, M. (2001). Elements of Mathematical Ecology, Cambridge University Press.","DOI":"10.1017\/CBO9780511608520"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"105187","DOI":"10.1016\/j.cnsns.2020.105187","article-title":"Dynamics in a time-discrete food-chain model with strong pressure on preys","volume":"84","author":"Vidiella","year":"2020","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1109\/TIT.1985.1057074","article-title":"A public key cryptosystem and a signature scheme based on discrete logarithms","volume":"31","author":"ElGamal","year":"1985","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_61","first-page":"553","article-title":"A novel pseudo-random bit generator based on a new couple of chaotic systems","volume":"11","author":"Boriga","year":"2011","journal-title":"Ann. Ovidius Univ. Econ. Sci. Ser."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1007\/s11071-014-1492-0","article-title":"A fast color image encryption algorithm based on hyper-chaotic systems","volume":"78","author":"Norouzi","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s11071-016-3024-6","article-title":"An image encryption scheme combining chaos with cycle operation for DNA sequences","volume":"87","author":"Hu","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/j.cnsns.2014.05.022","article-title":"A symmetrical image encryption scheme in wavelet and time domain","volume":"20","author":"Luo","year":"2015","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_65","first-page":"31","article-title":"NPCR and UACI randomness tests for image encryption","volume":"2","author":"Wu","year":"2011","journal-title":"Cyber J. Multidisciplinary J. Sci. Technol."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/j.ins.2014.02.156","article-title":"A symmetric image encryption algorithm based on mixed linear-nonlinear coupled map lattice","volume":"273","author":"Zhang","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1016\/j.cnsns.2004.03.006","article-title":"Cryptography using multiple one- dimensional chaotic maps","volume":"10","author":"Pareek","year":"2005","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1007\/s11071-018-4391-y","article-title":"An efficient pixel-level chaotic image encryption algorithm","volume":"94","author":"Ye","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"1101","DOI":"10.1016\/j.sigpro.2011.10.023","article-title":"Anovel colour image encryption algorithm based on chaos","volume":"92","author":"Wang","year":"2012","journal-title":"Signal Process"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1016\/j.ijleo.2018.04.068","article-title":"Cryptanalysis of multimedia encryption using elliptic curve Cryptography","volume":"168","author":"Singh","year":"2018","journal-title":"Optik"},{"key":"ref_71","first-page":"415","article-title":"Class number, a theory of factorization, and genera","volume":"Volume XX","author":"Shanks","year":"1969","journal-title":"Proceedings of the Symposia in Pure Mathematics"},{"key":"ref_72","first-page":"918","article-title":"Monte Carlo methods for index computation (mod p)","volume":"32","author":"Pollard","year":"1978","journal-title":"Math. Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/161\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:13:26Z","timestamp":1760159606000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/161"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,21]]},"references-count":72,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["sym13020161"],"URL":"https:\/\/doi.org\/10.3390\/sym13020161","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,21]]}}}