Research article Special Issues

Traveling waves in delayed reaction-diffusion equations in biology

  • Received: 22 July 2020 Accepted: 27 August 2020 Published: 25 September 2020
  • This paper represents a literature review on traveling waves described by delayed reactiondiffusion (RD, for short) equations. It begins with the presentation of different types of equations arising in applications. The main results on wave existence and stability are presented for the equations satisfying the monotonicity condition that provides the applicability of the maximum and comparison principles. Other methods and results are described for the case where the monotonicity condition is not satisfied. The last two sections deal with delayed RD equations in mathematical immunology and in neuroscience. Existence, stability, and dynamics of wavefronts and of periodic waves are discussed.

    Citation: Sergei Trofimchuk, Vitaly Volpert. Traveling waves in delayed reaction-diffusion equations in biology[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6487-6514. doi: 10.3934/mbe.2020339

    Related Papers:

  • This paper represents a literature review on traveling waves described by delayed reactiondiffusion (RD, for short) equations. It begins with the presentation of different types of equations arising in applications. The main results on wave existence and stability are presented for the equations satisfying the monotonicity condition that provides the applicability of the maximum and comparison principles. Other methods and results are described for the case where the monotonicity condition is not satisfied. The last two sections deal with delayed RD equations in mathematical immunology and in neuroscience. Existence, stability, and dynamics of wavefronts and of periodic waves are discussed.


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    [1] A. Kolmogorov, I. Petrovskii, N. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter and its application to a biological problem, Byul. Mosk. Gos. Univ. Ser. A Mat. Mekh., 1 (1937) 1-26.
    [2] A. Friedman, Cancer as multifaceted disease, Math. Model. Nat. Phenom., 7 (2012), 3-28. doi: 10.1051/mmnp/20127102
    [3] A. V. Panfilov, H. Dierckx, V. Volpert, Reaction-diffusion waves in cardiovascular diseases, Phys. D Amsterdam, Neth. 399 (2019), 1-34.
    [4] J. Bebernes, D. Eberly, Mathematical Problems from Combustion Theory, Springer-Verlag, New York, (1989).
    [5] R. A. Fisher, The wave of advance of advantageous genes, Ann. Hum. Genet., 7 (1937), 353-369.
    [6] Y. B. Zeldovich, D. A. Frank-Kamenetskii, A theory of thermal propagation of flame, Acta Physicochim. USSR, 9 (1938), 341-350.
    [7] N. N. Semenov, To the theory of combustion processes, Fiz. Khim., 4 (1939), 4-7.
    [8] V. A. Mikhelson, On normal combustion velocity of explosive gaseous mixtures, Imp. Moscow Univ. Sci. Bull. Phys. Math. Ser., 10 (1893), 1-92.
    [9] R. Luther, Propagation of chemical reactions in space, J. Chem. Educ., 64 (9), 740.
    [10] J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296
    [11] A. I. Volpert, V. A. Volpert, V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, in American Mathematical Society, 1994.
    [12] A. I. Volpert, V. A. Volpert, Application of the theory of the rotation of vector fields to the investigation of wave solutions of parabolic equations, Proc. Moscow Math. Soc., 52 (1989), 58-109.
    [13] P. C. Fife, J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. doi: 10.1007/BF00250432
    [14] P. C. Fife, J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusion, Arch. Ration. Mech. Anal., 75 (1981), 281-314. doi: 10.1007/BF00256381
    [15] V. A. Volpert, Asymptotic behavior of solutions of a nonlinear diffusion equation with a source of general type, Sib. Math. J., 30 (1989), 25-36. doi: 10.1007/BF01054212
    [16] V. A. Volpert, Convergence to a wave of solutions of a nonlinear diffusion equation with a source of general type, Sib. Math. J., 30 (1989), 203-210. doi: 10.1007/BF00971374
    [17] M. Artola, Equations paraboliques à retardement, C. R. Acad. Sci. Paris, 264 (1967), 668-671.
    [18] M. Artola, Sur les perturbations des équations d'évolution: Application à des problèmes de retard, Annales scientifiques de l'école Normale Supérieure, Serie 4, 1969. Available from: http://www.numdam.org/article/ASENS_1969_4_2_2_137_0.
    [19] C. C. Travis, G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3
    [20] R. H. Martin, H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
    [21] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, 1996.
    [22] K. Schumacher, Travelling-front solutions for integro-differential equations I, J. Reine Angew. Math., 316 (1980), 54-70.
    [23] K. Schaaf, Asymptotic behavior and travelling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.
    [24] R. Benguria, A. Solar, An iterative estimation for disturbances of semi-wavefronts to the delayed Fisher-KPP equation, Proc. Amer. Math. Soc., 147 (2019), 2495-2501. doi: 10.1090/proc/14381
    [25] J. Billingham, Slow travelling wave solutions of the nonlocal Fisher-KPP equation, Nonlinearity, 33 (2020), 2106-2142. doi: 10.1088/1361-6544/ab6f4f
    [26] A. Ducrot, G. Nadin, Asymptotic behaviour of traveling waves for the delayed Fisher-KPP equation, J. Differ. Equations, 256 (2014), 3115-3140. doi: 10.1016/j.jde.2014.01.033
    [27] K. Hasík, S. Trofimchuk, An extension of Wright's 3/2-theorem for the KPP-Fisher delayed equation, Proc. Amer. Math. Soc., 143 (2015), 3019-3027. doi: 10.1090/S0002-9939-2015-12496-3
    [28] E. Hernández, S. Trofimchuk, Nonstandard quasi-monotonicity: an application to the wave existence in a neutral KPP-Fisher equation, J. Dynam. Differ. Equations, 32 (2020), 921-939. doi: 10.1007/s10884-019-09748-z
    [29] Y. Liu, P. Weng, Asymptotic pattern for a partial neutral functional differential equation, J. Differ. Equations, 258 (2015), 3688-3741. doi: 10.1016/j.jde.2015.01.016
    [30] S. Ma, X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reactiondiffusion monostable equation with delay, J. Differ. Equations, 217 (2005), 54-87.
    [31] S. A. Gourley, J. W. H. So, J.H. Wu, Non-locality of reaction-diffusion equations induced by delay: Biological modeling and nonlinear dynamics, J. Math. Sci., 124 (2004), 5119-5153. doi: 10.1023/B:JOTH.0000047249.39572.6d
    [32] J. D. Murray, Mathematical Biology, Vol. 1, 3rd edition, Springer-Verlag, (2002).
    [33] M. Aguerrea, C. Gomez, S. Trofimchuk, On uniqueness of semi-wavefronts Diekmann-Kaper theory of a nonlinear convolution equation re-visited, Math. Ann., 354 (2012), 73-109. doi: 10.1007/s00208-011-0722-8
    [34] J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dyn. Differ. Equations, 11 (1999), 1-48.
    [35] E. Trofimchuk, M. Pinto, S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction, Discrete Contin. Dyn. Syst. A, 33 (2013), 2169-2187. doi: 10.3934/dcds.2013.33.2169
    [36] T. Xu, S. Ji, R. Huang, M. Mei, J. Yin, Theoretical and numerical studies on global stability of traveling waves with oscillations for time-delayed nonlocal dispersion equations, Int. J. Numer. Anal. Mod., 17 (2020), 68-86.
    [37] T. Yi, X. Zou, Asymptotic behavior, spreading speeds, and traveling waves of nonmonotone dynamical systems, SIAM J. Math. Anal., 47 (2015), 3005-3034.
    [38] N. F. Britton, Aggregation and the competitive exclusion principle, J. Theor. Biol., 136 (1989), 57-66. doi: 10.1016/S0022-5193(89)80189-4
    [39] N. F. Britton, Spatial structures and periodic travelling waves in an integro- differential reactiondiffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099
    [40] H. L. Smith, A structured population model and a related functional-differential equation: Global attractors and uniform persistence, J. Dyn. Differ. Equations, 6 (1994), 71-99. doi: 10.1007/BF02219189
    [41] H. L. Smith, H. Thieme, Strongly order preserving semiflows generated by functional differential equations, J. Differ. Equations, 93 (1991), 332-363. doi: 10.1016/0022-0396(91)90016-3
    [42] J. W. H. So, J. H. Wu, X. F. Zou, A reaction-diffusion model for a single species with age structure I: Traveling wavefronts on unbounded domains, Proc. R. Soc. London, Ser. A, 457 (2001), 1841-1853. doi: 10.1098/rspa.2001.0789
    [43] M. Bani-Yaghoub, G. M. Yao, M. Fujiwara, D.E. Amundsen, Understanding the interplay between density dependent birth function and maturation time delay using a reaction-diffusion population model, Ecol. Complexity, 21 (2015), 14-26. doi: 10.1016/j.ecocom.2014.10.007
    [44] V. Volpert, S. Petrovskii, Reaction-diffusion waves in biology, Phys. Life Rev., 6 (2009), 267-310. doi: 10.1016/j.plrev.2009.10.002
    [45] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differ. Equations, 2 (1997), 125-160.
    [46] T. Ogiwara, H. Matano, Monotonicity and convergence in order-preserving systems, Discrete Contin. Dyn. Syst. A, 5 (1999), 1-34. doi: 10.3934/dcds.1999.5.1
    [47] J. Fang, X. Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. Eur. Math. Soc., 17 (2015), 2243-2288. doi: 10.4171/JEMS/556
    [48] J. Fang, X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704. doi: 10.1137/140953939
    [49] T. Faria, W. Huang, J. Wu, Traveling waves for delayed reaction-diffusion equations with nonlocal response, Proc. R. Soc. Ser. A, 462 (2006), 229-261. doi: 10.1098/rspa.2005.1554
    [50] X. Liang, X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903. doi: 10.1016/j.jfa.2010.04.018
    [51] H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953. doi: 10.2977/prims/1260476648
    [52] H. Yagisita, Existence of traveling waves solutions for a nonlocal bistable equation:an abstract approach, Publ. RIMS Kyoto Univ., 45 (2009), 955-979. doi: 10.2977/prims/1260476649
    [53] W. van Saarloos, Front propagation into unstable states, Phys. Rep., 386 (2003), 29-222.
    [54] B. Sandstede, Stability of travelling waves, in Handbook of dynamical systems Ⅱ (eds. B. Fiedler), Elsevier, (2002), 983-1055.
    [55] I. L. Chern, M. Mei, X. F. Yang, Q. F. Zhang, Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay, J. Differ. Equations, 259 (2015), 1503-1541. doi: 10.1016/j.jde.2015.03.003
    [56] C. K. Lin, C. T. Lin, Y. Lin, M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084. doi: 10.1137/120904391
    [57] M. Mei, C. K. Lin, C. T. Lin, J. W. H. So, Traveling wavefronts for time-delayed reactiondiffusion equation: (I) Local nonlinearity, J. Differ. Equations, 247 (2009), 495-510. doi: 10.1016/j.jde.2008.12.026
    [58] M. Mei, C. K. Lin, C. T. Lin, J. W.-H. So, Traveling wavefronts for time-delayed reactiondiffusion equation: (Ⅱ) Nonlocal nonlinearity, J. Differ. Equations, 247 (2009), 511-529. doi: 10.1016/j.jde.2008.12.020
    [59] M. Mei, J. W. H. So, M. Y. Li, S. S. P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. R. Soc. Edinburgh, Sect. A, 134 (2004), 579-594. doi: 10.1017/S0308210500003358
    [60] M. Mei, Y. Wang, Remark on stability of traveling waves for nonlocal Fisher-KPP equations, Int. J. Numer. Anal. Model. Ser. B, 2 (2011), 379-401.
    [61] G. Lv, M. Wang, Nonlinear stability of travelling wave fronts for delayed reaction diffusion equations, Nonlinearity, 23 (2010), 845-873. doi: 10.1088/0951-7715/23/4/005
    [62] S. L. Wu, W. T. Li, S. Y. Liu, Exponential stability of traveling fronts in monostable reactionadvection-diffusion equations with non-local delay, Discrete Contin. Dyn. Syst. B, 17 (2012), 347-366. doi: 10.3934/dcdsb.2012.17.347
    [63] A. Solar, S. Trofimchuk, Speed selection and stability of wavefronts for delayed monostable reaction-diffusion equations, J. Dyn. Differ. Equations, 28 (2016), 1265-1292. doi: 10.1007/s10884-015-9482-6
    [64] Z. C. Wang, W. T. Li, S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dyn. Differ. Equations, 20 (2008), 573-607. doi: 10.1007/s10884-008-9103-8
    [65] A. Solar, S. Trofimchuk, Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction, Nonlinearity, 28 (2015), 2027-2052. doi: 10.1088/0951-7715/28/7/2027
    [66] H. Berestycki, L. Nirenberg, Traveling waves in cylinders, Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497-572.
    [67] H. L. Smith, X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reactiondiffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534. doi: 10.1137/S0036141098346785
    [68] S. L. Wu, T. C. Niu, C. H. Hsu, Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations, Discrete Contin. Dyn. Syst. A, 37 (2017), 3467-3486. doi: 10.3934/dcds.2017147
    [69] O. Diekmann, H. G. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9
    [70] E. Trofimchuk, M. Pinto, S. Trofimchuk, Monotone waves for non-monotone and non-local monostable reaction-diffusion equations, J. Differ. Equations, 261 (2016), 203-1236.
    [71] O. Bonnefon, J. Garnier, F. Hamel, L. Roques, Inside dynamics of delayed traveling waves, Math. Model. Nat. Phenom., 8 (2013), 42-59. doi: 10.1051/mmnp/20138305
    [72] M. Jankovic, S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect, Theor. Ecol., 7 (2014), 335-349. doi: 10.1007/s12080-014-0222-z
    [73] S. Ma, J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a nonlocal delayed diffusion equation, J. Dyn. Differ. Equations, 19 (2007), 391-436.
    [74] Z. C. Wang, W. T. Li, S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differ. Equations, 238 (2007), 153-200. doi: 10.1016/j.jde.2007.03.025
    [75] A. Ducrot, M. Marion, V. Volpert, Spectrum of some integro-differential operators and stability of travelling waves, Nonlinear Anal. 74 (2011), 4455-4473.
    [76] J. Wu, X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Differ. Equations, 13 (2001), 651-687. doi: 10.1023/A:1016690424892
    [77] T. Faria, S. Trofimchuk, Non-monotone travelling waves in a single species reaction-diffusion equation with delay, J. Differ. Equations, 228 (2006), 357-376. doi: 10.1016/j.jde.2006.05.006
    [78] K. Hasík, J. Kopfová, P. Nábělková, S. Trofimchuk, Traveling waves in the nonlocal KPP-Fisher equation: different roles of the right and the left interactions, J. Differ. Equations, 260 (2016), 6130-6175. doi: 10.1016/j.jde.2015.12.035
    [79] K. Hasík, S. Trofimchuk, Slowly oscillating wavefronts of the KPP-Fisher delayed equation, Discrete Contin. Dyn. Syst. A, 34 (2014), 3511-3533. doi: 10.3934/dcds.2014.34.3511
    [80] E. Trofimchuk, V. Tkachenko and S. Trofimchuk, Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay, J. Differ. Equations, 245 (2008) 2307-2332.
    [81] Z. Chladná, K. Hasík, J. Kopfova, P. Nábělková, S. Trofimchuk, Nonlinearly determined wavefronts of the Nicholson's diffusive equation: when small delays are not harmless, J. Differ. Equations, 268 (2020), 5156-5178. doi: 10.1016/j.jde.2019.11.007
    [82] K. Hasík, J. Kopfova, P. Nábělková, S. Trofimchuk, On the geometric diversity of wavefronts for the scalar Kolmogorov ecological equation, J. Nonlinear Sci., 2020 (2020), 1-38.
    [83] E. Trofimchuk, M. Pinto, S. Trofimchuk, Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model, Proc. R. Soc. Edinburgh, Sect. A, 2020 (2020) 1-22.
    [84] S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differ. Equations, 232 (2007), 104-133. doi: 10.1016/j.jde.2006.08.015
    [85] K. Hasík, J. Kopfova, P. Nábělková, S. Trofimchuk, On pushed wavefronts in monostable equation with unimodal delayed reaction, work in progress.
    [86] T. Yi, Y. Chen, J. Wu, Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves, J. Differ. Equations, 254 (2013), 3538-3572.
    [87] C. Gomez, H. Prado, S. Trofimchuk, Separation dichotomy and wavefronts for a nonlinear convolution equation, J. Math. Anal. Appl., 420 (2014), 1-19. doi: 10.1016/j.jmaa.2014.05.064
    [88] S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differ. Equations, 237 (2007), 259-277. doi: 10.1016/j.jde.2007.03.014
    [89] E. Trofimchuk, S. Trofimchuk, Admissible wavefront speeds for a single species reactiondiffusion equation with delay, Discrete Contin. Dyn. Syst. A, 20 (2008), 407-423. doi: 10.3934/dcds.2008.20.407
    [90] E. Trofimchuk, P. Alvarado, S. Trofimchuk, On the geometry of wave solutions of a delayed reaction-diffusion equation, J. Differ. Equations, 246 (2009), 1422-1444. doi: 10.1016/j.jde.2008.10.023
    [91] J. Fang, X. Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054. doi: 10.1088/0951-7715/24/11/002
    [92] A. Gomez, S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differ. Equations, 250 (2011), 1767-1787. doi: 10.1016/j.jde.2010.11.011
    [93] J. K. Hale, X. B. Lin, Heteroclinic orbits for retarded functional differential equations, J. Differ. Equations, 65 (1985), 175-202.
    [94] T. Faria, W. Huang, J. Wu, Traveling waves for delayed reaction-diffusion equations with nonlocal response, Proc. R. Soc. A, 462 (2006), 229-261. doi: 10.1098/rspa.2005.1554
    [95] W. Huang, Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response, J. Differ. Equations, 244 (2008), 1230-1254. doi: 10.1016/j.jde.2007.10.001
    [96] W. Huang, D. Duehring, Periodic travelling wave solutions for a reaction-diffusion equation with time delay and non-local response, J. Dyn. Differ. Equations, 19 (2007), 457-477. doi: 10.1007/s10884-006-9048-8
    [97] C. Ou, J. Wu, Persistence of wavefronts in delayed nonlocal reaction-diffusion equations, J. Differ. Equations, 235 (2007), 219-261. doi: 10.1016/j.jde.2006.12.010
    [98] C. Ou, J. Wu, Traveling wavefronts in a delayed food-limited population model, SIAM J. Math. Anal., 39 (2007), 103-125. doi: 10.1137/050638011
    [99] T. Faria, S. Trofimchuk, Positive traveling fronts for reaction-diffusion systems with distributed delay, Nonlinearity, 23 (2010), 2457-2481. doi: 10.1088/0951-7715/23/10/006
    [100] M. Aguerrea, S. Trofimchuk, G. Valenzuela, Uniqueness of fast traveling fronts in a single species reaction-diffusion equation with delay, Proc. R. Soc. Ser. A, 464 (2008), 2591-2608. doi: 10.1098/rspa.2008.0011
    [101] A. Gomez, S. Trofimchuk, Global continuation of monotone wavefronts, J. London Math. Soc., 89 (2014), 47-68. doi: 10.1112/jlms/jdt050
    [102] V. Volpert, S. Trofimchuk, Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities, Nonlinearity, 32 2019, 2593-2632.
    [103] V. Volpert, Elliptic Partial Differential Equations. Volume 1. Fredholm Theory of Elliptic Problems in Unbounded Domains, Birkhäuser, (2011).
    [104] V. Volpert, Elliptic Partial Differential Equations. Volume 2. Reaction-Diffusion Equations, Birkhäuser, (2014).
    [105] H. Berestycki, G. Nadin, B. Perthame, L. Ryzhik, The non-local Fisher-KPP equation: travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002
    [106] S. Trofimchuk, V. Volpert, Traveling waves for a bistable reaction-diffusion equation with delay, SIAM J. Math. Anal., 50 (2018), 1175-1199. doi: 10.1137/17M1115587
    [107] S. Trofimchuk, V. Volpert, Existence of bistable waves for a nonlocal and nonmonotone reactiondiffusion equation, Proc. R. Soc. Edinburgh, Sect. A, 150 (2020), 721-739. doi: 10.1017/prm.2018.164
    [108] G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk, V. Volpert, Spatiotemporal dynamics of virus infection spreading in tissues, PLoS ONE, 11 (2016), e0168576.
    [109] M. Alfaro, J. Coville, G. Raoul, Bistable travelling waves for nonlocal reaction diffusion equations, Discrete Contin. Dyn. Syst. A, 34 (2014), 1775-1791. doi: 10.3934/dcds.2014.34.1775
    [110] M. Alfaro, A. Ducrot, T. Giletti, Travelling waves for a non- monotone bistable equation with delay: existence and oscillations, Proc. London Math. Soc., 116 (2018), 729-759. doi: 10.1112/plms.12092
    [111] V. Volpert, Existence of waves for a bistable reaction-diffusion system with delay, J. Dyn. Differ. Equations, 32 (2020), 615-629. doi: 10.1007/s10884-019-09751-4
    [112] A. Solar, S. Trofimchuk, A simple approach to the wave uniqueness problem, J. Differ. Equations, 266 (2019), 6647-6660. doi: 10.1016/j.jde.2018.11.012
    [113] J. Coville, J.Dávila, S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differ. Equations, 244 (2008), 3080-3118. doi: 10.1016/j.jde.2007.11.002
    [114] G. Nadin, L. Rossi, L. Ryzhik, B. Perthame, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model, Math. Model. Nat. Phenom., 8 (2013), 33-41. doi: 10.1051/mmnp/20138304
    [115] M. Mei, Ch. Ou, X. Q. Zhao, Global stability of monostable traveling waves for nonlocal timedelayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762-2790. doi: 10.1137/090776342
    [116] M. Mei, K. Zhang, Q. Zhang, Global stability of critical traveling waves with oscillations for time-delayed reaction-diffusion equation, Int. J. Numer. Anal. Model., 16 (2019), 375-397.
    [117] A. Solar, Stability of semi-wavefronts for delayed reaction-diffusion equations, Nonlinear Differ. Equations Appl. NoDEA, 26 (2019), 33. doi: 10.1007/s00030-019-0580-8
    [118] A. Solar, Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations, Discrete Contin. Dyn. Syst. A, 39 (2019), 5799-5823. doi: 10.3934/dcds.2019255
    [119] P. Ashwin, M. V. Bartuccelli, T. J. Bridges, S. A. Gourley, Travelling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys., 53 (2002), 103-122. doi: 10.1007/s00033-002-8145-8
    [120] M. K. Kwong, C. Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation, J. Differ. Equations, 249 (2010), 728-745. doi: 10.1016/j.jde.2010.04.017
    [121] G. Bocharov, V. Volpert, B. Ludewig, A. Meyerhans, Mathematical Immunology of Virus Infections, Springer-Verlag, (2018).
    [122] G. A. Bocharov, V. A. Volpert, A. L. Tasevich, Reaction-diffusion equations in Immunology, Comput. Math. Math. Phys., 58 (2018), 1967-1976. doi: 10.1134/S0965542518120059
    [123] N. Bessonov, G. Bocharov, T. M. Touaoula, S. Trofimchuk, V. Volpert, Delay reaction-diffusion equation for infection dynamics, Discrete Contin. Dyn. Syst. B, 24 (2019) 2073-2091.
    [124] T. M. Touaoula, M. N. Frioui, N. Bessonov, V. Volpert, Dynamics of solutions of a reactiondiffusion equation with delayed inhibition, Discrete Contin. Dyn. Syst. S, 13 2020, 2425-2442.
    [125] N. Bessonov, G. Bocharov, V. Volpert, Space and genotype-dependent virus distribution during infection progression, work in progress.
    [126] N. Bessonov, G. Bocharov, A. Meyerhans, V. Popov, V. Volpert, Nonlocal reaction-diffusion model of viral evolution: emergence of virus strains, Mathematics, 8 2020,117.
    [127] N. Bessonov, G. A. Bocharov, C. Leon, V. Popov, V. Volpert, Genotype-dependent virus distribution and competition of virus strains, Math. Mech. Compl. Syst., 8 (2020), 101-126. doi: 10.2140/memocs.2020.8.101
    [128] V. Volpert, Existence of reaction-diffusion waves in a model of immune response, Mediterr. J. Math., 17 (2020), 1-20. doi: 10.1007/s00009-019-1430-y
    [129] G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk, V. Volpert, Modelling the dynamics of virus infection and immune response in space and time, Int. J. Parallel, Emergent Distrib. Syst., 34 (2019), 341-355. doi: 10.1080/17445760.2017.1363203
    [130] V. Botella-Soler, M. Valderrama, B. Crépon, V. Navarro, M. Le Van Quyen, Large-scale cortical dynamics of sleep slow waves, PLoS ONE, 7 (2012), e30757.
    [131] L. Muller, F. Chavane, J. Reynolds, T. J. Sejnowski, Cortical travelling waves: mechanisms and computational principles, Nat. Rev. Neurosci., 19 (2018), 255-268. doi: 10.1038/nrn.2018.20
    [132] J. Y. Wu, X. Huang, C. Zhang, Propagating waves of activity in the neocortex: what they are, what they do, Neuroscientist, 14 (2008), 487-502.
    [133] H. R. Wilson, J. D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, 80 (1973), 55-80.
    [134] H. Zhang, A. J. Watrous, A. Patel, J. Jacobs, Theta and alpha oscillations are travelling waves in the human neocortex, Neuron, 98 (2018), 1-13. doi: 10.1016/j.neuron.2018.03.027
    [135] J. Senk, K. Korvasova, J. Schuecker, E. Hagen, T. Tetzlaff, M. Diesmann, et al., Conditions for travelling waves in spiking neural networks, Phys. Rev. Res., 2 (2020), 023174.
    [136] D. J. Pinto, G. B. Ermentrout, Spatially structured activity in synapticalaly coupled neuronal networks: Ⅱ. Lateral inhibition and standing pulses, SIAM J. Appl. Math., 62 (2001), 226-243. doi: 10.1137/S0036139900346465
    [137] A. Beuter, A. Balossier, S. Trofimchuk, V. Volpert, Modeling of post-stroke stimulation of cortical tissue, Math. Biosci., 305 (2018), 146-159. doi: 10.1016/j.mbs.2018.08.014
    [138] A. Moussaoui, V. Volpert, Speed of wave propagation for a nonlocal reaction-diffusion equation, Appl. Anal., 2018 (2018), 1-15.
    [139] N. Bessonov, A. Beuter, S. Trofimchuk, V. Volpert, Dynamics of periodic waves in a neural field model, Mathematics, 8 (2020), 1076. doi: 10.3390/math8071076
    [140] F. M. Atay, A. Hutt, Neural Fields with Distributed Transmission Speeds and Long-Range Feedback Delays, SIAM J. Appl. Dyn. Syst., 5 (2006), 670-698. doi: 10.1137/050629367
    [141] N. A. Venkov, S. Coombes, P. C. Matthews, Dynamic instabilities in scalar neural field equations with space-dependent delays, Phys. D, 232 (2007), 1-15. doi: 10.1016/j.physd.2007.04.011
    [142] H. G. E. Meijer, S. Coombes, Travelling waves in a neural field model with refractoriness, J. Math. Biol., 68 (2014), 1249-1268. doi: 10.1007/s00285-013-0670-x
    [143] M. Adimy, A. Chekroun, B. Kazmierczak, Traveling waves in a coupled reaction-diffusion and difference model of hematopoiesis, J. Differ. Equations, 262 (2017), 4085-4128. doi: 10.1016/j.jde.2016.12.009
    [144] M. Jankovic, S. Petrovskii, M. Banerjee, Delay driven spatio-temporal chaos in single species population dynamics models, Theor. Popul. Biol., 110 (2016), 1-62. doi: 10.1016/j.tpb.2016.03.003
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