Travelling fronts in time-delayed reaction-diffusion systems. (English) Zbl 1464.35150
Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci. 377, No. 2153, Article ID 20180127, 13 p. (2019).
Summary: We review a series of key travelling front problems in reaction-diffusion systems with a time-delayed feedback, appearing in ecology, nonlinear optics and neurobiology. For each problem, we determine asymptotic approximations for the wave shape and its speed. Particular attention is devoted to their validity and all analytical solutions are compared to solutions obtained numerically. We also extend the work by the last author et al. [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 368, No. 1911, 483–493 (2010; Zbl 1197.35301)] by considering the case of a slowly propagating front subject to a weak delayed feedback. The delay may either speed up the front in the same direction or reverse its direction.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35C07 | Traveling wave solutions |
| 35K58 | Semilinear parabolic equations |
Keywords:
delay differential equations; travelling fronts; reaction-diffusion equations; applied mathematicsCitations:
Zbl 1197.35301References:
| [1] | Erneux T. 2009 Applied delay differential equations. Berlin, Germany: Springer. · Zbl 1201.34002 |
| [2] | Stepan G (ed.) 2009 Delay effects in brain dynamics. Phil. Trans. R. Soc. A 367, 1059-1212. (doi:10.1098/rsta.2008.0279) · Zbl 1185.92025 · doi:10.1098/rsta.2008.0279 |
| [3] | Lakshmanan M, Senthilkumar DV. 2010 Dynamics of nonlinear time-delay systems. Springer Series in Synergetics, Springer Complexity. Berlin, Germany: Springer. · Zbl 1230.37001 |
| [4] | Smith H. 2011 An introduction to delay differential equations with applications to the life sciences. Berlin, Germany: Springer. · Zbl 1227.34001 |
| [5] | Sipahi R, Niculescu S-I, Abdallah CT, Michiels W, Gu K. 2011 Stability and stabilization of systems with time delay. IEEE Control Syst. Mag. 31, 38-65. (doi:10.1109/MCS.2010.939135) · Zbl 1395.93271 · doi:10.1109/MCS.2010.939135 |
| [6] | Milton J, Ohira T. 2014 Mathematics as a laboratory tool. Berlin, Germany: Springer. · Zbl 1319.92001 |
| [7] | Koch G, Krzyzanski W, Pérez-Ruixo JJ, Schropp J. 2014 Modeling of delays in PKPD: classical approaches and a tutorial for delay differential equations. J. Pharmacokinet. Pharmacodyn. 41, 291-318. (doi:10.1007/s10928-014-9368-y) · doi:10.1007/s10928-014-9368-y |
| [8] | Bélair J, Rimbu A. 2015 Time delays in drug administration: effect, transit, tricks and oscillations. IFAC-PapersOnLine 48, 111-116. (doi:10.1016/j.ifacol.2015.09.362) · doi:10.1016/j.ifacol.2015.09.362 |
| [9] | Mackey MC, Santillán M, Tyran-Kamińska M, Zeron ES. 2016 Simple mathematical models of gene regulatory dynamics. Lecture Notes on Mathematical Modelling in the Life Sciences. Berlin, Germany: Springer. · Zbl 1360.92002 |
| [10] | Al-Omari J, Gourley SA. 2002 Monotone traveling fronts in an age-structured reaction-diffusion model of a single species. J. Math. Biol. 45, 294-312. (doi:10.1007/s002850200159) · Zbl 1013.92032 · doi:10.1007/s002850200159 |
| [11] | Gourley SA, So JW-H, Wu JH. 2004 Nonlocality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics. J. Math. Sci. 124, 5119-5153. (doi:10.1023/B:JOTH.0000047249.39572.6d) · Zbl 1128.35360 · doi:10.1023/B:JOTH.0000047249.39572.6d |
| [12] | Gourley SA, Kuang Y. 2005 A delay reaction-diffusion model of the spread of bacteriophage infection. SIAM J. Appl. Math. 65, 550-566. (doi:10.1137/S0036139903436613) · Zbl 1068.92042 · doi:10.1137/S0036139903436613 |
| [13] | Fort J, Méndez V. 2002 Time-delayed spread of viruses in growing plaques. Phys. Rev. Lett. 89, 178101. (doi:10.1103/PhysRevLett.89.178101) · doi:10.1103/PhysRevLett.89.178101 |
| [14] | Fort J, Méndez V. 2002 Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment. Rep. Prog. Phys. 65, 895-954. (doi:10.1088/0034-4885/65/6/201) · doi:10.1088/0034-4885/65/6/201 |
| [15] | Amor DR, Fort J. 2010 Virus infection speeds: theory versus experiment. Phys. Rev. E 82, 061905. (doi:10.1103/PhysRevE.82.061905) · doi:10.1103/PhysRevE.82.061905 |
| [16] | Jones DA, Smith HL, Thieme HR, Röst G. 2012 On spread of phage infection of bacteria in a petri dish. SIAM J. Appl. Math. 72, 670-688. (doi:10.1137/110848360) · Zbl 1243.92041 · doi:10.1137/110848360 |
| [17] | Cattaneo C. 1948 Sulla conduzione del calore. Atti Semin. Mat. Fis. Univ. Modena 3, 83-101. (doi:10.1007/978-3-642-11051-1_5) · Zbl 0035.26203 · doi:10.1007/978-3-642-11051-1_5 |
| [18] | Keller JB. 2004 Diffusion at finite speed and random walks. Proc. Natl Acad. Sci. USA 101, 1120-1122. (doi:10.1073/pnas.0307052101) · Zbl 1069.60066 · doi:10.1073/pnas.0307052101 |
| [19] | Gourley SA, So JW-H. 2002 Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain. J. Math. Biol. 44, 49-78. (doi:10.1007/s002850100109) · Zbl 0993.92027 · doi:10.1007/s002850100109 |
| [20] | Bestehorn M, Grigorieva EV, Kaschenko SA. 2004 Spatiotemporal structures in a model with delay and diffusion. Phys. Rev. E 70, 026202. (doi:10.1103/PhysRevE.70.026202) · doi:10.1103/PhysRevE.70.026202 |
| [21] | Li QS, Hu HX. 2007 Pattern transitions induced by delay feedback. J. Chem. Phys. 127, 154510. (doi:10.1063/1.2792877) · doi:10.1063/1.2792877 |
| [22] | Tian C, Zhang L. 2013 Delay-driven irregular spatiotemporal patterns in a plankton system. Phys. Rev. E 88, 012713. (doi:10.1103/PhysRevE.88.012713) · doi:10.1103/PhysRevE.88.012713 |
| [23] | Gurevich SV. 2013 Dynamics of localized structures in reaction-diffusion systems induced by delayed feedback. Phys. Rev. E 87, 052922. (doi:10.1103/PhysRevE.87.052922) · Zbl 1327.35158 · doi:10.1103/PhysRevE.87.052922 |
| [24] | Iri R, Tonosaki Y, Shitara K, Ohta T. 2014 Dynamics of Turing pattern under time-delayed feedback. J. Phys. Soc. Jpn. 83, 024001. (doi:10.7566/JPSJ.83.024001) · doi:10.7566/JPSJ.83.024001 |
| [25] | Murray JD. 2002 Mathematical biology I: an introduction, 3rd edn. Interdisciplinary Applied Mathematics 17. Berlin, Germany: Springer. · Zbl 1006.92001 |
| [26] | Lewis MA, Kareiva P. 1993 Allee dynamics and the spread of invading organisms. Theor. Popul. Biol. 43, 141-158. (doi:10.1006/tpbi.1993.1007) · Zbl 0769.92025 · doi:10.1006/tpbi.1993.1007 |
| [27] | McKean HP. 1975 Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Commun. Pure Appl. Math. 28, 323-331. (doi:10.1002/(ISSN)1097-0312) · Zbl 0316.35053 · doi:10.1002/(ISSN)1097-0312 |
| [28] | Gourley SA. 2005 Linear stability of traveling fronts in an age-structured reaction-diffusion population model. Q. J. Mech. Appl. Math. 58, 257-268. (doi:10.1093/qjmamj/hbi012) · Zbl 1069.92018 · doi:10.1093/qjmamj/hbi012 |
| [29] | Lewis MA, Petrovskii SV, Potts JR. 2016 The mathematics behind biological invasions. Interdisciplinary Applied Mathematics 44. Berlin, Germany: Springer. · Zbl 1338.92001 |
| [30] | Tanguy Y, Ackemann T, Jäger R. 2006 Characteristics of bistable localized emission states in broad-area vertical-cavity surface-emitting lasers with frequency-selective feedback. Phys. Rev. A 74, 053824. (doi:10.1103/PhysRevA.74.053824) · doi:10.1103/PhysRevA.74.053824 |
| [31] | Tanguy Y, Radwell N, Ackemann T, Jäger R. 2008 Characteristics of cavity solitons and drifting excitations in broad-area vertical-cavity surface-emitting lasers with frequency-selective feedback. Phys. Rev. A 78, 023810. (doi:10.1103/PhysRevA.78.023810) · doi:10.1103/PhysRevA.78.023810 |
| [32] | Paulau PV, Gomila D, Ackemann T, Loiko NA, Firth WJ. 2008 Self-localized structures in vertical-cavity surface-emitting lasers with external feedback. Phys. Rev. E 78, 016212. (doi:10.1103/PhysRevE.78.016212) · doi:10.1103/PhysRevE.78.016212 |
| [33] | Tlidi M, Vladimirov AG, Pieroux D, Turaev D. 2009 Spontaneous motion of cavity solitons induced by a delayed feedback. Phys. Rev. Lett. 103, 103904. (doi:10.1103/PhysRevLett.103.103904) · doi:10.1103/PhysRevLett.103.103904 |
| [34] | Gurevich SV, Friedrich R. 2013 Instabilities of localized structures in dissipative systems with delayed feedback. Phys. Rev. Lett. 110, 014101. (doi:10.1103/PhysRevLett.110.014101) · doi:10.1103/PhysRevLett.110.014101 |
| [35] | Panajotov K, Tlidi M. 2010 Spontaneous motion of cavity solitons in vertical-cavity lasers subject to optical injection and to delayed feedback. Eur. Phys. J. D 59, 67-72. (doi:10.1140/epjd/e2010-00111-y) · doi:10.1140/epjd/e2010-00111-y |
| [36] | Tlidi M, Averlant E, Vladimirov AG, Panajotov K. 2012 Delay feedback induces a spontaneous motion of two-dimensional cavity solitons in driven semiconductor microcavities. Phys. Rev. A 86, 033822. (doi:10.1103/PhysRevA.86.033822) · doi:10.1103/PhysRevA.86.033822 |
| [37] | Pimenov A, Vladimirov AG, Gurevich SV, Panajotov K, Huyet G, Tlidi M. 2013 Delayed feedback control of self-mobile cavity solitons. Phys. Rev. A 88, 053830. (doi:10.1103/PhysRevA.88.053830) · doi:10.1103/PhysRevA.88.053830 |
| [38] | Pyragas K. 1992 Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421-428. (doi:10.1016/0375-9601(92)90745-8) · doi:10.1016/0375-9601(92)90745-8 |
| [39] | Schöll E, Hiller G, Hövel P, Dahlem MA. 2009 Time-delayed feedback in neurosystems. Phil. Trans. R. Soc. A 367, 1079-1096. (doi:10.1098/rsta.2008.0258) · Zbl 1185.34108 · doi:10.1098/rsta.2008.0258 |
| [40] | Michiels W, Niculescu S-I. 2007 Stability and stabilization of time-delay systems: an eigenvalue-based approach. Advances in Design and Control, no. 12. Philadelphia, PA: SIAM. · Zbl 1140.93026 |
| [41] | Boubendir Y, Méndez V, Rotstein HG. 2010 Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: propagation failure and control mechanisms. Phys. Rev. E 82, 036601. (doi:10.1103/PhysRevE.82.036601) · doi:10.1103/PhysRevE.82.036601 |
| [42] | Rubinstein BY, Nepomnyashchy AA, Golovin AA. 2007 Stability of localized solutions in a subcritically unstable pattern-forming system under a global delayed control. Phys. Rev. E 75, 046213. (doi:10.1103/PhysRevE.75.046213) · doi:10.1103/PhysRevE.75.046213 |
| [43] | Schneider FM, Schöll E, Dahlem MA. 2009 Controlling the onset of traveling pulses in excitable media by nonlocal spatial coupling and time-delayed feedback. Chaos 19, 015110. (doi:10.1063/1.3096411) · doi:10.1063/1.3096411 |
| [44] | Dahlem MA, Graf R, Strong AJ, Dreier JP, Dahlem YA, Sieber M, Podoll HWK, Scholl E. 2010 Two-dimensional wave patterns of spreading depolarization: retracting, re-entrant, and stationary waves. Physica D 239, 889-903. (doi:10.1016/j.physd.2009.08.009) · Zbl 1187.92051 · doi:10.1016/j.physd.2009.08.009 |
| [45] | Erneux T, Kozyreff G, Tlidi M. 2010 Bifurcation to fronts due to delay. Phil. Trans. R. Soc. A 368, 483-493. (doi:10.1098/rsta.2009.0228) · Zbl 1197.35301 · doi:10.1098/rsta.2009.0228 |
| [46] | Coombes S, Laing CR. 2009 Instabilities in threshold-diffusion equations with delay. Physica D 238, 264-272. (doi:10.1016/j.physd.2008.10.014) · Zbl 1156.37312 · doi:10.1016/j.physd.2008.10.014 |
| [47] | Gelens L, Gomila D, Van der Sande G, Matías MA, Colet P. 2010 Nonlocality-induced front-interaction enhancement. Phys. Rev. Lett. 104, 154101. (doi:10.1103/PhysRevLett.104.154101) · doi:10.1103/PhysRevLett.104.154101 |
| [48] | Fernandez-Oto C, Clerc MG, Escaff D, Tlidi M. 2013 Strong nonlocal coupling stabilizes localized structures: an analysis based on front dynamics. Phys. Rev. Lett. 110, 174101. (doi:10.1103/PhysRevLett.110.174101) · doi:10.1103/PhysRevLett.110.174101 |
| [49] | Colet P, Matías MA, Gelens L, Gomila D. 2014 Formation of localized structures in bistable systems through nonlocal spatial coupling. I. General framework. Phys. Rev. E 89, 012914. (doi:10.1103/PhysRevE.89.012914) · doi:10.1103/PhysRevE.89.012914 |
| [50] | Gelens L, Matías MA, Gomila D, Dorissen T, Colet P. 2014 Formation of localized structures in bistable systems through nonlocal spatial coupling. II. The nonlocal Ginzburg-Landau equation. Phys. Rev. E 89, 012915. (doi:10.1103/PhysRevE.89.012915) · doi:10.1103/PhysRevE.89.012915 |
| [51] | Clerc MG, Escaff D, Kenkre VM. 2010 Analytical studies of fronts, colonies, and patterns: combination of the Allee effect and nonlocal competition interactions. Phys. Rev. E 82, 036210. (doi:10.1103/PhysRevE.82.036210) · doi:10.1103/PhysRevE.82.036210 |
| [52] | Chang J, Ferrell JE Jr. 2013 Mitotic trigger waves and the spatial coordination of the Xenopus cell cycle. Nature 500, 603-607. (doi:10.1038/nature12321) · doi:10.1038/nature12321 |
| [53] | Yang Q, Ferrell JE Jr. 2013 The Cdk1-APC/C cell cycle oscillator circuit functions as a time-delayed, ultrasensitive switch. Nat. Cell Biol. 15, 519-525. (doi:10.1038/ncb2737) · doi:10.1038/ncb2737 |
| [54] | Rombouts J, Vandervelde A, Gelens L. 2018 Delay models for the early embryonic cell cycle oscillator. PLoS ONE 13, e0194769. (doi:10.1371/journal.pone.0194769) · doi:10.1371/journal.pone.0194769 |
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