Long time behavior of the field–road diffusion model: an entropy method and a finite volume scheme

References

[1] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice. Oxford Series in Ecology and Evolution, Oxford, Oxford University Press, 1997.10.1093/oso/9780198548522.001.0001Search in Google Scholar

[2] J. D. Murray, Mathematical Biology. Vol. 1: An Introduction, ser. Interdisciplinary Applied Mathematics, vol. 17, 3rd ed. New York, Springer, 2002.Search in Google Scholar

[3] J. D. Murray, Mathematical Biology. Vol. 2: Spatial Models and Biomedical Applications, ser. Interdisciplinary Applied Mathematics, vol. 18, 3rd ed. New York, Springer, 2003.10.1007/b98869Search in Google Scholar

[4] C. Robinet, et al.., “Human-mediated long-distance jumps of the pine processionary moth in Europe,” Biol. Invasions, vol. 14, no. 8, pp. 1557–1569, 2012, https://doi.org/10.1007/s10530-011-9979-9.Search in Google Scholar

[5] H. W. McKenzie, E. H. Merrill, R. J. Spiteri, and M. A. Lewis, “How linear features alter predator movement and the functional response,” Interface Focus, vol. 2, no. 2, pp. 205–216, 2012, https://doi.org/10.1098/rsfs.2011.0086.Search in Google Scholar PubMed PubMed Central

[6] B. V. Schmid, et al.., “Climate-driven introduction of the black death and successive plague reintroductions into Europe,” Proc. Natl. Acad. Sci. U. S. A., vol. 112, no. 10, pp. 3020–3025, 2015, https://doi.org/10.1073/pnas.1412887112.Search in Google Scholar PubMed PubMed Central

[7] M. Gatto, et al.., “Spread and dynamics of the COVID-19 epidemic in Italy: effects of emergency containment measures,” Proc. Natl. Acad. Sci. U. S. A., vol. 117, no. 19, pp. 10484–10491, 2020, https://doi.org/10.1073/pnas.2004978117.Search in Google Scholar PubMed PubMed Central

[8] H. Berestycki, J.-M. Roquejoffre, and L. Rossi, “The influence of a line with fast diffusion on Fisher-KPP propagation,” J. Math. Biol., vol. 66, no. 4, pp. 743–766, 2013, https://doi.org/10.1007/s00285-012-0604-z.Search in Google Scholar PubMed

[9] H. Berestycki, J.-M. Roquejoffre, and L. Rossi, “Fisher–KPP propagation in the presence of a line: further effects,” Nonlinearity, vol. 26, no. 9, pp. 2623–2640, 2013, https://doi.org/10.1088/0951-7715/26/9/2623.Search in Google Scholar

[10] H. Berestycki, J.-M. Roquejoffre, and L. Rossi, “The shape of expansion induced by a line with fast diffusion in Fisher-KPP equations,” Commun. Math. Phys., vol. 343, no. 1, pp. 207–232, 2016, https://doi.org/10.1007/s00220-015-2517-3.Search in Google Scholar

[11] H. Berestycki, J.-M. Roquejoffre, and L. Rossi, “Travelling waves, spreading and extinction for Fisher–KPP propagation driven by a line with fast diffusion,” Nonlinear Anal., vol. 137, no. 1, pp. 171–189, 2016. https://doi.org/10.1016/j.na.2016.01.023.Search in Google Scholar

[12] H. Berestycki, A.-C. Coulon, J.-M. Roquejoffre, and L. Rossi, “The effect of a line with nonlocal diffusion on Fisher-KPP propagation,” Math. Models Methods Appl. Sci., vol. 25, no. 13, pp. 2519–2562, 2015. https://doi.org/10.1142/S0218202515400175.Search in Google Scholar

[13] H. Berestycki, A.-C. Coulon, J.-M. Roquejoffre, and L. Rossi, “Speed-up of reaction–diffusion fronts by a line of fast diffusion,” in Séminaire Laurent Schwartz — EDP et applications, 2013, pp. 1–25.10.5802/slsedp.62Search in Google Scholar

[14] T. Giletti, L. Monsaingeon, and M. Zhou, “A KPP road–field system with spatially periodic exchange terms,” Nonlinear Anal., vol. 128, pp. 273–302, 2015, https://doi.org/10.1016/j.na.2015.07.021.Search in Google Scholar

[15] A. Pauthier, “The influence of nonlocal exchange terms on Fisher-KPP propagation driven by a line of fast diffusion,” Commun. Math. Sci., vol. 14, no. 2, pp. 535–570, 2016. https://doi.org/10.4310/CMS.2016.v14.n2.a10.Search in Google Scholar

[16] A. Pauthier, “Uniform dynamics for Fisher-KPP propagation driven by a line of fast diffusion under a singular limit,” Nonlinearity, vol. 28, no. 11, pp. 3891–3920, 2015, https://doi.org/10.1088/0951-7715/28/11/3891.Search in Google Scholar

[17] A. Pauthier, “Road–field reaction–diffusion system: a new threshold for long range exchanges,” arXiv:1504.05437, 2015.Search in Google Scholar

[18] A. Tellini, “Propagation speed in a strip bounded by a line with different diffusion,” J. Differ. Equ., vol. 260, no. 7, pp. 5956–5986, 2016, https://doi.org/10.1016/j.jde.2015.12.028.Search in Google Scholar

[19] L. Rossi, A. Tellini, and E. Valdinoci, “The effect on Fisher-KPP propagation in a cylinder with fast diffusion on the boundary,” SIAM J. Math. Anal., vol. 49, no. 6, pp. 4595–4624, 2017, https://doi.org/10.1137/17m1125388.Search in Google Scholar

[20] R. Ducasse, “Influence of the geometry on a field–road model: the case of a conical field,” J. Lond. Math. Soc., vol. 97, no. 3, pp. 441–469, 2018, https://doi.org/10.1112/jlms.12114.Search in Google Scholar

[21] H. Berestycki, R. Ducasse, and L. Rossi, “Generalized principal eigenvalues for heterogeneous road–field systems,” Commun. Contemp. Math, vol. 22, no. 1, p. 1950013, 2020. https://doi.org/10.1142/S0219199719500135.Search in Google Scholar

[22] H. Berestycki, R. Ducasse, and L. Rossi, “Influence of a road on a population in an ecological niche facing climate change,” J. Math. Biol., vol. 81, nos. 4–5, pp. 1059–1097, 2020. https://doi.org/10.1007/s00285-020-01537-3.Search in Google Scholar PubMed

[23] M. Zhang, “Spreading speeds and pulsating fronts for a field–road model in a spatially periodic habitat,” J. Differ. Equ., vol. 304, pp. 191–228, 2021, https://doi.org/10.1016/j.jde.2021.09.038.Search in Google Scholar

[24] B. Bogosel, T. Giletti, and A. Tellini, “Propagation for KPP bulk-surface systems in a general cylindrical domain,” Nonlinear Anal., vol. 213, p. 112528, 2021, https://doi.org/10.1016/j.na.2021.112528.Search in Google Scholar

[25] E. Affili, “A Fisher-KPP model with a fast diffusion line in periodic media,” arXiv:2009.14760, 2020.Search in Google Scholar

[26] M. Alfaro, R. Ducasse, and S. Tréton, “The field–road diffusion model: fundamental solution and asymptotic behavior,” J. Differ. Equ., vol. 367, pp. 332–365, 2023, https://doi.org/10.1016/j.jde.2023.05.002.Search in Google Scholar

[27] K. Fellner, E. Latos, and B. Q. Tang, “Well-posedness and exponential equilibration of a volume-surface reaction–diffusion system with nonlinear boundary coupling,” Ann. Inst. H. Poincaré C Anal. Non Linéaire, vol. 35, no. 3, pp. 643–673, 2018. https://doi.org/10.1016/j.anihpc.2017.07.002.Search in Google Scholar

[28] H. Egger, K. Fellner, J.-F. Pietschmann, and B. Q. Tang, “Analysis and numerical solution of coupled volume-surface reaction–diffusion systems with application to cell biology,” Appl. Math. Comput., vol. 336, pp. 351–367, 2018, https://doi.org/10.1016/j.amc.2018.04.031.Search in Google Scholar

[29] R. Eymard, T. Gallouët, and R. Herbin, “Finite volume methods,” in Handbook of Numerical Analysis, ser. Handb. Numer. Anal., vol. VII, North-Holland/Elsevier, Amsterdam, 2000, pp. 713–1020.10.1016/S1570-8659(00)07005-8Search in Google Scholar

[30] C. Chainais-Hillairet, A. Jüngel, and S. Schuchnigg, “Entropy-dissipative discretization of nonlinear diffusion equations and discrete beckner inequalities,” ESAIM Math. Model. Numer. Anal., vol. 50, no. 1, pp. 135–162, 2016, https://doi.org/10.1051/m2an/2015031.Search in Google Scholar

[31] C. Chainais-Hillairet and M. Herda, “Large-time behaviour of a family of finite volume schemes for boundary-driven convection–diffusion equations,” IMA J. Numer. Anal., vol. 40, no. 4, pp. 2473–2504, 2020. https://doi.org/10.1093/imanum/drz037.Search in Google Scholar

[32] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5: Evolution problems I. With the Collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon. Transl. from the French by A. Craig., 2nd ed. Berlin, Springer, 2000.Search in Google Scholar

[33] M. Burger, I. Humpert, and J.-F. Pietschmann, “On Fokker–Planck equations with in- and outflow of mass,” Kinet. Relat. Models, vol. 13, no. 2, pp. 249–277, 2020. https://doi.org/10.3934/krm.2020009.Search in Google Scholar

[34] A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations, Bilbao, BCAM Springer Briefs. Springer, 2016.10.1007/978-3-319-34219-1Search in Google Scholar

[35] K. Rektorys, Variational Methods in Mathematics, Science and Engineering, 2nd ed. Dordrecht–Boston, Mass., D. Reidel Publishing Co., 1980, translated from the Czech by Michael Basch.Search in Google Scholar

[36] R. Duduchava, “On Poincaré, Friedrichs and Korns inequalities on domains and hypersurfaces,” arXiv preprint arXiv:1504.01677, 2015.Search in Google Scholar

[37] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, ser. Lecture Notes of the Unione Matematica Italiana, vol. 3, Berlin; UMI, Bologna, Springer, 2007.Search in Google Scholar

[38] K. Drury, J. Drake, D. Lodge, and G. Dwyer, “Immigration events dispersed in space and time: factors affecting invasion success,” Ecol. Model., vol. 206, nos. 1–2, pp. 63–78, 2007, https://doi.org/10.1016/j.ecolmodel.2007.03.017.Search in Google Scholar

[39] J. Garnier, L. Roques, and F. Hamel, “Success rate of a biological invasion in terms of the spatial distribution of the founding population,” Bull. Math. Biol., vol. 74, no. 2, pp. 453–473, 2012. https://doi.org/10.1007/s11538-011-9694-9.Search in Google Scholar PubMed

[40] I. Mazari, G. Nadin, and A. I. Toledo Marrero, “Optimisation of the total population size with respect to the initial condition for semilinear parabolic equations: two-scale expansions and symmetrisations,” Nonlinearity, vol. 34, no. 11, pp. 7510–7539, 2021. https://doi.org/10.1088/1361-6544/ac23b9.Search in Google Scholar

[41] M. Alfaro, F. Hamel, and L. Roques, “Propagation or extinction in bistable equations: the non-monotone role of initial fragmentation,” Discrete Contin. Dyn. Syst. Ser. S., vol. 17, no. 4, pp. 1460–1484, 2024.10.3934/dcdss.2023165Search in Google Scholar