Traveling waves for reaction-diffusion PDE coupled to difference equation with nonlocal dispersal term and time delay

Mostafa Adimy¹ ; Abdennasser Chekroun² ; Bogdan Kazmierczak³

¹ Inria, CNRS UMR 5208, Institut Camille Jordan, Université Lyon 1, 69200 Villeurbanne, France
² Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, University of Tlemcen, Tlemcen 13000, Algeria
³ Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawińskiego 5B, 02-106 Warsaw, Poland

Mathematical modelling of natural phenomena, Tome 17 (2022), article no. 17.

Résumé

We consider a class of biological models represented by a system composed of reactiondiffusion PDE coupled with difference equations (renewal equations) in n-dimensional space, with nonlocal dispersal terms and implicit time delays. The difference equation generally arises, by means of the method of characteristics, from an age-structured partial differential system. Using upper and lower solutions, we study the existence of monotonic planar traveling wave fronts connecting the extinction state to the uniform positive state. The corresponding minimum wave speed is also obtained. In addition, we investigate the effect of the parameters on this minimum wave speed and we give a detailed analysis of its asymptotic behavior.

Détail

DOI : 10.1051/mmnp/2022021

Bibliographie

[1] M. Adimy, A. Chekroun, C.P. Ferreira Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase Math. Biosci. Eng. 2020 1329 1354

[2] M. Adimy, A. Chekroun, B. Kazmierczak Traveling waves in a coupled reaction-diffusion and difference model of hematopoiesis J. Differ. Equ. 2017 4085 4128

[3] M. Adimy, A. Chekroun, T. Kuniya Traveling waves of a differential-difference diffusive Kermack-McKendrick epidemic model with age-structured protection phase J. Math. Anal. Appl. 2022 125464

[4] M. Adimy, A. Chekroun, T.-M. Touaoula Age-structured and delay differential-difference model of hematopoietic stem cell dynamics Discr. Continu. Dyn. Syst. Ser. B 2015 2765 2791

[5] D.G. Aronson, H.F. Weinberger Multidimensional nonlinear diffusion arising in population genetics Adv. Math. 1978 33 76

[6] H. Berestycki, G. Nadin, B. Perthame, L. Ryzhik The non-local Fisher-KPP equation: travelling waves and steady states Nonlinearity 2009 2813 2844

[7] G. Bocharov, K. Hadeler Structured population models, conservation laws, and delay equations J. Differ. Equ. 2000 212 237

[8] O. Bonnefon, J. Garnier, F. Hamel, L. Roques Inside dynamics of delayed traveling waves MMNP 2013 42 59

[9] E. Bouin, V. Calvez, G. Nadin Propagation in a kinetic reaction-transport equation: travelling waves and accelerating fronts Arch. Ratl. Mech. Anal. 2015 571 617

[10] K.J. Brown, J. Carr Deterministic epidemic waves of critical velocity Math. Proc. Cambridge Philos. Soc. 1977 431 433

[11] A. Chekroun Existence and asymptotics of traveling wave fronts for a coupled nonlocal diffusion and difference system with delay Electr. J. Qual. Theory Differ. Equ. 2019 1 23

[12] J. Fang, X.-Q. Zhao Existence and uniqueness of traveling waves for non-monotone integral equations with applications J. Differ. Equ. 2010 2199 2226

[13] R.A. Fisher The wave of advance of advantageous genes Ann. Eugenics 1937 353 369

[14] C. Gomez, H. Prado, S. Trofimchuk Separation dichotomy and wavefronts for a nonlinear convolution equation J. Math. Anal. Appl. 2014 1 19

[15] S.-A. Gourley, J. Wu Delayed non-local diffusive systems in biological invasion and disease spread Nonlinear Dyn. Evol. Equ. Am. Math. Soc. 2006 137 200

[16] K.P. Hadeler, T. Hillen and M.A. Lewis, Biological Modeling with Quiescent Phases, in Spatial Ecology, edited by R.S. Cantrell, C. Cosner and S. Ruan. Mathematical and Computational Biology series, Chapman Hall/CRC Press (2009) 101–127.

[17] K.P. Hadeler, M.A. Lewis Spatial dynamics of the diffusive logistic equation ith a sedentary compartment Can. Appl. Math. Quart. 2002 473 499

[18] F. Hamel, N. Nadirashvili Travelling fronts and entire solutions of the Fisher-KPP equation in Rn Arch. Ratl. Mech. Anal. 2001 91 163

[19] R. Huang Stability of travelling fronts of the Fisher-KPP equation in Rn Nonlinear Differ. Equ. Appl. 2008 599 622

[20] T. Kapitula Multidimensional stability of planar travelling waves Trans. Am. Math. Soc. 1997 257 269

[21] 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. 1937 1 26

[22] W. Li, S. Ruan, Z. Wang On the diffusive Nicholson’s blowflies equation with nonlocal delay J. Nonlinear Sci. 2007 505 525

[23] X. Liang, X.-Q. Zhao Asymptotic speeds of spread and traveling waves for monotone semiflows with applications Commun. Pure Appl. Math. 2007 1 40

[24] Z. Ling, Z. Lin Traveling wavefront in a hematopoiesis model with time delay Appl. Math. Lett. 2010 426 431

[25] M.C. Mackey Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis Blood 1978 941 956

[26] J.D. Murray, Mathematical Biology: I. An Introduction. Springer, New York, 3rd edition (2002).

[27] D.H. Sattinger On the stability of waves of nonlinear parabolic systems Adv. Math. 1976 312 355

[28] K.W. Schaaf Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations Trans. Am. Math. Soc. 1987 587 615

[29] K. Schumacher Travelling-front solutions for integro-differential equations. I J. Reine Angew. Math. 1980 54 70

[30] J.W.-H. So, J. Wu, X. Zou A reaction-diffusion model for a single species with age structure. I Travelling wavefronts on unbounded domains Proc. R. Soc. Lond. A 2001 1841 1853

[31] H.R. Thieme, X.-Q. Zhao Asymptotic speeds of spread and traveling waves for integral equations and delayed reactiondiffusion models J. Differ. Equ. 2003 430 470

[32] A.I. Volpert, V.A. Volpert and V.A. Volpert, Traveling Wave Solutions of Parabolic System. American Mathematical Soc. (1994).

[33] Z. Wang, W. Li, S. Ruan Traveling fronts in monostable equations with nonlocal delayed effects J. Dyn. Differ. Equ. 2008 573 607

[34] Z. Wang, W. Li, S. Ruan Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay J. Differ. Equ. 2007 153 200

[35] G.F. Webb, Population Models Structured by Age, Size, and Spatial Position, Lecture Notes in Mathematics, 1936. Springer (2008) 1–49.

[36] D.V. Widder, Laplace Transform. Princeton University Press (1946).

[37] J. Wu, Theory and Applications of Partial Functional Differential Equations. Springer (1996).

[38] J. Wu, X. Zou Traveling wave fronts of reaction-diffusion systems with delay J. Dyn. Differ. Equ. 2001 651 687

[39] J. Wu, X. Zou Erratum to "Traveling wave fronts of reaction-diffusion systems with delay" J. Dyn. Differ. Equ. 2008 531 533

[40] J. Xin Front propagation in heterogeneous media SIAM Rev. 2000 161 230

[41] X.-Q. Zhao, W. Wang Fisher waves in an epidemic model Discr. Continu. Dyn. Syst. Ser. B 2004 1117 1128

[42] X.-Q. Zhao, D. Xiao The asymptotic speed of spread and traveling waves for a vector disease model J. Dyn. Diff. Equ. 2006 1001 1019