{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,15]],"date-time":"2025-03-15T04:07:51Z","timestamp":1742011671816,"version":"3.38.0"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,1,29]],"date-time":"2025-01-29T00:00:00Z","timestamp":1738108800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,1,29]],"date-time":"2025-01-29T00:00:00Z","timestamp":1738108800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00f3n","doi-asserted-by":"publisher","award":["PDC2022- 400 133115-I00"],"award-info":[{"award-number":["PDC2022- 400 133115-I00"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006365","name":"Universidad de Cantabria","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006365","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Free-boundary diffusive logistic model finds applications in diverse fields associated with population dynamics. These processes often possess stochastic characteristics and involve parameters with uncertainties. This study focuses on enhancing a two-dimensional diffusive logistic partial differential model with free boundary by incorporating randomness in the mean square sense, considering the conditions for well-posedness in the random case, which is crucial for the further analysis. Both unknown stochastic processes the solution and its moving front, and the parameters involved in the random problem as random variables, are constrained by a finite degree of randomness. To tackle this challenge, we propose a random level set method. Given the complexity of the problem, we employ alternating  direction explicit methods for the interior solvers, to effectively address computational challenges. Since computing the mean and the standard deviation of both unknown stochastic processes are required, we combine the sample approach of the difference schemes together with Monte Carlo technique avoiding the storage accumulation of symbolic expressions of all the previous levels of the iteration process. Parallel computing is employed to enhance performance. A careful numerical analysis is performed in the mean square context to ensure stability, positivity, and boundedness. The set of presented examples illustrates these qualitative properties, assess numerical convergence and enables us to gain a deeper understanding of the system\u2019s behavior attending to the geometry of the initial habitat. This approach provides valuable tools for analyzing and predicting spreading-vanishing dichotomy.<\/jats:p>","DOI":"10.1007\/s40314-025-03107-z","type":"journal-article","created":{"date-parts":[[2025,1,29]],"date-time":"2025-01-29T18:23:47Z","timestamp":1738175027000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Tracking interfaces in a random logistic free-boundary diffusion problems: a random level set method"],"prefix":"10.1007","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3024-3033","authenticated-orcid":false,"given":"Vera","family":"Egorova","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5708-5709","authenticated-orcid":false,"given":"M. C.","family":"Casab\u00e1n","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5217-1889","authenticated-orcid":false,"given":"Rafael","family":"Company","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9672-6249","authenticated-orcid":false,"given":"Lucas","family":"J\u00f3dar","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,1,29]]},"reference":[{"key":"3107_CR1","doi-asserted-by":"publisher","DOI":"10.3390\/axioms12050460","author":"MS Arif","year":"2023","unstructured":"Arif MS, Abodayeh K, Nawaz Y (2023) A reliable computational scheme for stochastic reaction-diffusion nonlinear chemical model. Axioms. https:\/\/doi.org\/10.3390\/axioms12050460","journal-title":"Axioms"},{"key":"3107_CR2","doi-asserted-by":"publisher","first-page":"167","DOI":"10.4171\/IFB\/299","volume":"15","author":"T Aiki","year":"2013","unstructured":"Aiki T, Muntean A (2013) A free-boundary problem for concrete carbonation: front nucleation and rigorous justification of the $$\\sqrt{t}$$ -law of propagation. Interfaces Free Bound 15:167\u2013180. https:\/\/doi.org\/10.4171\/IFB\/299","journal-title":"Interfaces Free Bound"},{"key":"3107_CR3","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/0001-8708(78)90130-5","volume":"30","author":"DG Aronson","year":"1978","unstructured":"Aronson DG, Weinberger HF (1978) Multidimensional nonlinear diffusion arising in population genetics. Adv Math 30:33\u201376. https:\/\/doi.org\/10.1016\/0001-8708(78)90130-5","journal-title":"Adv Math"},{"key":"3107_CR4","doi-asserted-by":"publisher","DOI":"10.3390\/math11061296","author":"MC Casab\u00e1n","year":"2023","unstructured":"Casab\u00e1n MC, Company R, Egorova VN, J\u00f3dar L (2023) Qualitative numerical analysis of a free-boundary diffusive logistic model. Mathematics. https:\/\/doi.org\/10.3390\/math11061296","journal-title":"Mathematics"},{"key":"3107_CR5","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2024.02.016","author":"M-C Casab\u00e1n","year":"2024","unstructured":"Casab\u00e1n M-C, Company R, Egorova VN, J\u00f3dar L (2024) A random free-boundary diffusive logistic differential model: numerical analysis, computing and simulation. Math Comput Simul. https:\/\/doi.org\/10.1016\/j.matcom.2024.02.016","journal-title":"Math Comput Simul"},{"key":"3107_CR6","doi-asserted-by":"publisher","unstructured":"Casab\u00e1n MC, Company R, J\u00f3dar L (2020) Numerical solutions of random mean square Fisher-KPP models with advection. Math Methods Appl Sci 43(14):8015\u20138031. https:\/\/doi.org\/10.1002\/mma.5942. https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/mma.5942","DOI":"10.1002\/mma.5942"},{"key":"3107_CR7","doi-asserted-by":"publisher","DOI":"10.3390\/math9030206","author":"MC Casab\u00e1n","year":"2021","unstructured":"Casab\u00e1n MC, Company R, J\u00f3dar L (2021) Reliable efficient difference methods for random heterogeneous diffusion reaction models with a finite degree of randomness. Mathematics. https:\/\/doi.org\/10.3390\/math9030206","journal-title":"Mathematics"},{"key":"3107_CR8","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1016\/j.physd.2019.03.004","volume":"397","author":"M Cartwright","year":"2019","unstructured":"Cartwright M, Gottwald GA (2019) A collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations. Physica D: Nonlinear Phenom 397:54\u201364. https:\/\/doi.org\/10.1016\/j.physd.2019.03.004","journal-title":"Physica D: Nonlinear Phenom"},{"key":"3107_CR9","doi-asserted-by":"publisher","first-page":"4336","DOI":"10.1016\/j.jde.2011.02.011","volume":"250","author":"Y Du","year":"2011","unstructured":"Du Y, Guo Z (2011) Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II. J Differ Equ 250:4336\u20134366. https:\/\/doi.org\/10.1016\/j.jde.2011.02.011","journal-title":"J Differ Equ"},{"key":"3107_CR10","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1137\/090771089","volume":"42","author":"Y Du","year":"2010","unstructured":"Du Y, Lin Z (2010) Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J Math Anal 42:377\u2013405. https:\/\/doi.org\/10.1137\/090771089","journal-title":"SIAM J Math Anal"},{"key":"3107_CR11","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1080\/00207160.2015.1108409","volume":"94","author":"VN Egorova","year":"2017","unstructured":"Egorova VN, Tan SH, Lai CH, Company R, J\u00f3dar L (2017) Moving boundary transformation for American call options with transaction cost: finite difference methods and computing. Int J Comput Math 94:345\u2013362. https:\/\/doi.org\/10.1080\/00207160.2015.1108409","journal-title":"Int J Comput Math"},{"key":"3107_CR12","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1111\/j.1469-1809.1937.tb02153.x","volume":"7","author":"RA Fisher","year":"1937","unstructured":"Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugenics 7:355\u2013369","journal-title":"Ann Eugenics"},{"key":"3107_CR13","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1016\/j.jcp.2017.10.006","volume":"353","author":"F Gibou","year":"2018","unstructured":"Gibou F, Fedkiw R, Osher S (2018) A review of level-set methods and some recent applications. J Comput Phys 353:82\u2013109. https:\/\/doi.org\/10.1016\/j.jcp.2017.10.006","journal-title":"J Comput Phys"},{"key":"3107_CR14","volume-title":"Random ordinary differential equations and their numerical solution","author":"P Kloeden","year":"2017","unstructured":"Kloeden P, Han X (2017) Random ordinary differential equations and their numerical solution. Springer, New York"},{"key":"3107_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/s00285-021-01641-y","author":"K Khan","year":"2021","unstructured":"Khan K, Liu S, Schaerf TM, Du Y (2021) Invasive behaviour under competition via a free boundary model: a numerical approach. J Math Biol. https:\/\/doi.org\/10.1007\/s00285-021-01641-y","journal-title":"J Math Biol"},{"key":"3107_CR16","first-page":"1","volume":"1","author":"A Kolmogorov","year":"1937","unstructured":"Kolmogorov A, Petrovsky N, Piscounov S (1937) \u00c9tude de l\u2019\u00e9quations de la diffusion avec croissance de la quantit\u00e9 de mati\u00e8re et son application a un probl\u00e8me biologique. Bull Univ Moskou 1:1\u201325","journal-title":"Bull Univ Moskou"},{"key":"3107_CR17","volume-title":"Numerical schemes for conservation laws","author":"D Kr\u00f6ner","year":"1997","unstructured":"Kr\u00f6ner D (1997) Numerical schemes for conservation laws. Wiley and Teubner, Hoboken"},{"issue":"86","key":"3107_CR18","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1090\/S0025-5718-1964-0164450-X","volume":"18","author":"BK Larkin","year":"1964","unstructured":"Larkin BK (1964) Some stable explicit difference approximations to the diffusion equation. Math Comput 18(86):196\u2013202","journal-title":"Math Comput"},{"key":"3107_CR19","doi-asserted-by":"publisher","first-page":"959","DOI":"10.1080\/00207160.2019.1599868","volume":"97","author":"S Liu","year":"2020","unstructured":"Liu S, Du Y, Liu X (2020) Numerical studies of a class of reaction\u2013diffusion equations with Stefan conditions. Int J Comput Math 97:959\u2013979. https:\/\/doi.org\/10.1080\/00207160.2019.1599868","journal-title":"Int J Comput Math"},{"key":"3107_CR20","doi-asserted-by":"publisher","first-page":"354","DOI":"10.1007\/s42967-023-00261-1","volume":"6","author":"S Liu","year":"2024","unstructured":"Liu S, Liu X (2024) Exponential time differencing method for a reaction\u2013diffusion system with free boundary. Commun Appl Math Comput 6:354\u2013371. https:\/\/doi.org\/10.1007\/s42967-023-00261-1","journal-title":"Commun Appl Math Comput"},{"key":"3107_CR21","doi-asserted-by":"publisher","DOI":"10.1201\/9781482286137","volume-title":"Spatiotemporal patterns in ecology and epidemiology","author":"H Malchow","year":"2008","unstructured":"Malchow H (2008) Spatiotemporal patterns in ecology and epidemiology. Chapman and Hall\/CRC, New York. https:\/\/doi.org\/10.1201\/9781482286137"},{"key":"3107_CR22","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/j.jtbi.2011.04.025","volume":"281","author":"V M\u00e9ndez","year":"2011","unstructured":"M\u00e9ndez V, Llopis I, Campos D, Horsthemke W (2011) Effect of environmental fluctuations on invasion fronts. J Theor Biol 281:31\u201338. https:\/\/doi.org\/10.1016\/j.jtbi.2011.04.025","journal-title":"J Theor Biol"},{"key":"3107_CR23","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/s11587-016-0272-1","volume":"65","author":"A Mentrelli","year":"2016","unstructured":"Mentrelli A, Pagnini G (2016) Modelling and simulation of wildland fire in the framework of the level set method. Ricerche di Matematica 65:523\u2013533. https:\/\/doi.org\/10.1007\/s11587-016-0272-1","journal-title":"Ricerche di Matematica"},{"key":"3107_CR24","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1016\/j.cam.2015.12.021","volume":"300","author":"SL Mitchell","year":"2016","unstructured":"Mitchell SL, Vynnycky M (2016) On the accurate numerical solution of a two-phase Stefan problem with phase formation and depletion. J Comput Appl Math 300:259\u2013274. https:\/\/doi.org\/10.1016\/j.cam.2015.12.021","journal-title":"J Comput Appl Math"},{"key":"3107_CR25","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2021.126719","author":"S Nandi","year":"2022","unstructured":"Nandi S, Sanyasiraju YVSS (2022) A second order accurate fixed-grid method for multi-dimensional Stefan problem with moving phase change materials. Appl Math Comput. https:\/\/doi.org\/10.1016\/j.amc.2021.126719","journal-title":"Appl Math Comput"},{"key":"3107_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.probengmech.2023.103546","author":"S Nepal","year":"2023","unstructured":"Nepal S, \u00d6gren M, Wondmagegne Y, Muntean A (2023) Random walks and moving boundaries: estimating the penetration of diffusants into dense rubbers. Probab Eng Mech. https:\/\/doi.org\/10.1016\/j.probengmech.2023.103546","journal-title":"Probab Eng Mech"},{"key":"3107_CR27","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1006\/JCPH.2000.6636","volume":"169","author":"S Osher","year":"2001","unstructured":"Osher S, Fedkiw RP (2001) Level set methods: an overview and some recent results. J Comput Phys 169:463\u2013502. https:\/\/doi.org\/10.1006\/JCPH.2000.6636","journal-title":"J Comput Phys"},{"key":"3107_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/b98879","volume-title":"Level set methods and dynamic implicit surfaces","author":"S Osher","year":"2003","unstructured":"Osher S, Fedkiw R (2003) Level set methods and dynamic implicit surfaces. Springer, New York"},{"key":"3107_CR29","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1016\/0021-9991(88)90002-2","volume":"79","author":"S Osher","year":"1988","unstructured":"Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton\u2013Jacobi formulations. J Comput Phys 79:12\u201349. https:\/\/doi.org\/10.1016\/0021-9991(88)90002-2","journal-title":"J Comput Phys"},{"key":"3107_CR30","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1016\/j.cam.2016.02.029","volume":"309","author":"MA Piqueras","year":"2017","unstructured":"Piqueras MA, Company R, J\u00f3dar L (2017) A front-fixing numerical method for a free boundary nonlinear diffusion logistic population model. J Comput Appl Math 309:473\u2013481. https:\/\/doi.org\/10.1016\/j.cam.2016.02.029","journal-title":"J Comput Appl Math"},{"key":"3107_CR31","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/0024-3795(77)90073-8","volume":"18","author":"RJ Plemmons","year":"1977","unstructured":"Plemmons RJ (1977) M-matrix characterizations. I-Nonsingular M-matrices. Linear Algebra Appl 18:175\u2013188. https:\/\/doi.org\/10.1016\/0024-3795(77)90073-8","journal-title":"Linear Algebra Appl"},{"key":"3107_CR32","doi-asserted-by":"crossref","unstructured":"Sethian JA (1996) A fast marching level set method for monotonically advancing fronts. Proc Natl Acad Sci USA 93:1591\u20131595","DOI":"10.1073\/pnas.93.4.1591"},{"key":"3107_CR33","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198548522.001.0001","volume-title":"Biological invasions: theory and practice","author":"N Shigesada","year":"1997","unstructured":"Shigesada N, Kawasaki K (1997) Biological invasions: theory and practice. Oxford University Press, Oxford"},{"key":"3107_CR34","doi-asserted-by":"publisher","first-page":"1355","DOI":"10.1007\/s12648-023-02886-7","volume":"98","author":"T Singla","year":"2024","unstructured":"Singla T, Kumar B, Sharma S (2024) Numerical study of the melting process of spherical phase change material with variable thermal conductivity. Indian J Phys 98:1355\u20131363. https:\/\/doi.org\/10.1007\/s12648-023-02886-7","journal-title":"Indian J Phys"},{"key":"3107_CR35","volume-title":"Numerical solution of partial differential equations: finite difference methods","author":"GD Smith","year":"1985","unstructured":"Smith GD (1985) Numerical solution of partial differential equations: finite difference methods, 3rd edn. Clarendon Press, Oxford","edition":"3"},{"key":"3107_CR36","volume-title":"Random differential equations in science and engineering","author":"TT Soong","year":"1973","unstructured":"Soong TT (1973) Random differential equations in science and engineering. Academic Press, New York"},{"key":"3107_CR37","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1016\/j.camwa.2009.08.061","volume":"59","author":"L Villafuerte","year":"2010","unstructured":"Villafuerte L, Braumann CA, Cort\u00e9s JC, J\u00f3dar L (2010) Random differential operational calculus: theory and applications. Comput Math Appl 59:115\u2013125. https:\/\/doi.org\/10.1016\/j.camwa.2009.08.061","journal-title":"Comput Math Appl"},{"key":"3107_CR38","doi-asserted-by":"publisher","first-page":"390","DOI":"10.1007\/s11465-020-0588-0","volume":"15","author":"P Wei","year":"2020","unstructured":"Wei P, Wang W, Yang Y, Wang MY (2020) Level set band method: a combination of density-based and level set methods for the topology optimization of continuums. Front Mech Eng 15:390\u2013405. https:\/\/doi.org\/10.1007\/s11465-020-0588-0","journal-title":"Front Mech Eng"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03107-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03107-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03107-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,14]],"date-time":"2025-03-14T06:08:34Z","timestamp":1741932514000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03107-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,29]]},"references-count":38,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,4]]}},"alternative-id":["3107"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03107-z","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2025,1,29]]},"assertion":[{"value":"19 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 October 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 January 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 January 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"148"}}