Traveling wave solutions of a singular Keller-Segel system with logistic source

Tong Li; Zhi-An Wang; Tong Li; Zhi-An Wang

doi:10.3934/mbe.2022379

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 8: 8107-8131. doi: 10.3934/mbe.2022379

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Research article

Traveling wave solutions of a singular Keller-Segel system with logistic source

Tong Li ¹,
Zhi-An Wang ^{2
,
,}

1.
Department of Mathematics, The University of Iowa, Iowa City IA 52242, USA
2.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong S.A.R., China

Received: 07 April 2022 Revised: 17 May 2022 Accepted: 23 May 2022 Published: 06 June 2022

This paper is concerned with the traveling wave solutions of a singular Keller-Segel system modeling chemotactic movement of biological species with logistic growth. We first show the existence of traveling wave solutions with zero chemical diffusion in $ \mathbb{R} $. We then show the existence of traveling wave solutions with small chemical diffusion by the geometric singular perturbation theory and establish the zero diffusion limit of traveling wave solutions. Furthermore, we show that the traveling wave solutions are linearly unstable in the Sobolev space $ H^1(\mathbb{R}) \times H^2(\mathbb{R}) $ by the spectral analysis. Finally we use numerical simulations to illustrate the stabilization of traveling wave profiles with fast decay initial data and numerically demonstrate the effect of system parameters on the wave propagation dynamics.
- Keller-Segel model,
- traveling waves,
- minimal wave speed,
- singular perturbation method,
- linear instability
Citation: Tong Li, Zhi-An Wang. Traveling wave solutions of a singular Keller-Segel system with logistic source[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 8107-8131. doi: 10.3934/mbe.2022379

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Abstract

This paper is concerned with the traveling wave solutions of a singular Keller-Segel system modeling chemotactic movement of biological species with logistic growth. We first show the existence of traveling wave solutions with zero chemical diffusion in $ \mathbb{R} $. We then show the existence of traveling wave solutions with small chemical diffusion by the geometric singular perturbation theory and establish the zero diffusion limit of traveling wave solutions. Furthermore, we show that the traveling wave solutions are linearly unstable in the Sobolev space $ H^1(\mathbb{R}) \times H^2(\mathbb{R}) $ by the spectral analysis. Finally we use numerical simulations to illustrate the stabilization of traveling wave profiles with fast decay initial data and numerically demonstrate the effect of system parameters on the wave propagation dynamics.

References

[1]	R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 355–369. https://doi.org/10.1111/j.1469-1809.1937.tb02153.x doi: 10.1111/j.1469-1809.1937.tb02153.x
[2]	A. N. Kolmogorov, I. G. Petrovskii, N. S. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ. Math. Mech., 1 (1937), 1–26.
[3]	M. El-Hachem, S. W. McCue, W. Jin, Y. Du, M. J. Simpson, Revisiting the Fisher–Kolmogorov–Petrovsky–Piskunov equation to interpret the spreading–extinction dichotomy, Proc. R. Soc. A, 475 (2019), 20190378. https://doi.org/10.1098/rspa.2019.0378 doi: 10.1098/rspa.2019.0378
[4]	D. G. Aronson, H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33–76. https://doi.org/10.1016/0001-8708(78)90130-5 doi: 10.1016/0001-8708(78)90130-5
[5]	E. F. Keller, L. A. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis, J. Theor. Biol., 30 (1971), 235–248. https://doi.org/10.1016/0022-5193(71)90051-8 doi: 10.1016/0022-5193(71)90051-8
[6]	G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen, Bull. Math. Biol., 40 (1978), 671–674. https://doi.org/10.1016/S0092-8240(78)80025-1 doi: 10.1016/S0092-8240(78)80025-1
[7]	G. Rosen, Theoretical significance of the condition $\delta = 2$ in bacterial chemotaxis, Bull. Math. Biol., 45 (1983), 151–153. https://doi.org/10.1016/s0092-8240(83)80048-2 doi: 10.1016/s0092-8240(83)80048-2
[8]	L. Corrias, B. Perthame, H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris. Ser. I., 336 (2003), 141–146. https://doi.org/10.1016/s1631-073x(02)00008-0 doi: 10.1016/s1631-073x(02)00008-0
[9]	H. A. Levine, B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683–730. https://doi.org/10.1137/S0036139995291106 doi: 10.1137/S0036139995291106
[10]	H. A. Levine, B. D. Sleeman, M. Nilsen-Hamilton, A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. I. the role of protease inhibitors in preventing angiogenesis, Math. Biosci., 168 (2000), 71–115. https://doi.org/10.1016/S0025-5564(00)00034-1 doi: 10.1016/S0025-5564(00)00034-1
[11]	H. A. Levine, S. Pamuk, B. D. Sleeman, M. Nilsen-Hamilton, Mathematical modeling of capillary formation and development in tumor angiogenesis: Penetration into the stroma, Bull. Math. Biol., 63 (2001), 801–863. https://doi.org/10.1006/bulm.2001.0240 doi: 10.1006/bulm.2001.0240
[12]	T. Li, Z. A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differ. Equations, 250 (2011), 1310–1333. https://doi.org/10.1016/j.jde.2010.09.020 doi: 10.1016/j.jde.2010.09.020
[13]	C. Deng, T. Li, Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the Sobolev space framework, J. Differ. Equations, 257 (2014), 1311–1332. https://doi.org/10.1016/j.jde.2014.05.014 doi: 10.1016/j.jde.2014.05.014
[14]	H. Y. Jin, J. Y. Li, Z. A. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differ. Equations, 255 (2013), 193–219. https://doi.org/10.1016/j.jde.2013.04.002 doi: 10.1016/j.jde.2013.04.002
[15]	D. Li, T. Li, K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis, Math. Models Methods Appl. Sci., 21 (2011), 1631–1650. https://doi.org/10.1142/S0218202511005519 doi: 10.1142/S0218202511005519
[16]	D. Li, R. H. Pan, K. Zhao, Quantitative decay of a hybrid type chemotaxis model with large data, Nonlinearity, 28 (2015), 2181–2210. https://doi.org/10.1088/0951-7715/28/7/2181 doi: 10.1088/0951-7715/28/7/2181
[17]	H. C. Li, K. Zhao, Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differ. Equations, 258 (2015), 302–338. https://doi.org/10.1016/j.jde.2014.09.014 doi: 10.1016/j.jde.2014.09.014
[18]	J. Y. Li, L. N. Wang, K. J. Zhang, Asymptotic stability of a composite wave of two traveling waves to a hyperbolic-parabolic system modeling chemotaxis, Math. Methods Appl. Sci., 36 (2013), 1862–1877. https://doi.org/10.1002/mma.2731 doi: 10.1002/mma.2731
[19]	T. Li, R. H. Pan, K. Zhao, Global dynamics of a chemotaxis model on bounded domains with large data, SIAM J. Appl. Math., 72 (2012), 417–443. https://doi.org/10.1137/110829453 doi: 10.1137/110829453
[20]	T. Li, J. Park, Traveling waves in a chemotaxis model with logistic growth, Discrete Contin. Dyn. Syst. B, 24 (2019), 6465–6480. https://doi.org/10.3934/dcdsb.2019147 doi: 10.3934/dcdsb.2019147
[21]	T. Li, Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522–1541. https://doi.org/10.1137/09075161x doi: 10.1137/09075161x
[22]	M. Zhang, C. J. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2007), 1017–1027. https://doi.org/10.1090/S0002-9939-06-08773-9 doi: 10.1090/S0002-9939-06-08773-9
[23]	J. Y. Li, T. Li, Z. A. Wang, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819–2849. https://doi.org/10.1142/S0218202514500389 doi: 10.1142/S0218202514500389
[24]	J. Y. Li, Z. A. Wang, Convergence to traveling waves of a singular PDE-ODE hybrid chemotaxis system in the half space, J. Differ. Equations, 268 (2020), 6940–6970. https://doi.org/10.1016/j.jde.2019.11.076 doi: 10.1016/j.jde.2019.11.076
[25]	V. Martinez, Z. A. Wang, K. Zhao, Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology, Indiana Univ. Math. J., 67 (2018), 1383–1424. https://doi.org/10.1512/iumj.2018.67.7394 doi: 10.1512/iumj.2018.67.7394
[26]	L. G. Rebholz, D. Wang, Z. Wang, K. Zhao, C. Zerfas, Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions, Discrete Cont. Dyn. Syst., 39 (2019), 3789–3838. https://doi.org/10.3934/dcds.2019154 doi: 10.3934/dcds.2019154
[27]	D. Wang, Z. A. Wang, K. Zhao, Cauchy problem of a system of parabolic conservation laws arising from a Keller-Segel type chemotaxis model in multi-dimensions, Indiana Univ. Math. J., 70 (2021), 1–47. https://doi.org/10.1512/iumj.2021.70.8075 doi: 10.1512/iumj.2021.70.8075
[28]	Z. A. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst. Series B, 18 (2013), 601–641. https://doi.org/10.3934/dcdsb.2013.18.601 doi: 10.3934/dcdsb.2013.18.601
[29]	J. A. Carrillo, J. Li, Z. A. Wang, Boundary spike-layer solutions of the singular Keller-Segel system: existence and stability, Proc. London Math. Soc., 122 (2021), 42–68. https://doi.org/10.1112/plms.12319
[30]	Q. Q. Hou, Z. A. Wang, K. Zhao, Boundary layer problem on a hyperbolic system arising from chemotaxis, J. Differ. Equations, 261 (2016), 5035–5070. https://doi.org/10.1016/j.jde.2016.07.018 doi: 10.1016/j.jde.2016.07.018
[31]	Q. Q. Hou, C. J. Liu, Y. G. Wang, Z. A. Wang, Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: one dimensional case, SIAM J. Math. Anal., 50 (2018), 3058–3091. https://doi.org/10.1137/17M112748X
[32]	Q. Hou, Z. A. Wang, Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half–plane, J. Math. Pures Appl., 130 (2019), 251–287. https://doi.org/10.1016/j.matpur.2019.01.008 doi: 10.1016/j.matpur.2019.01.008
[33]	T. Li, Z. A. Wang, Steadily propagating waves of a chemotaxis model, Math. Biosci., 240 (2012), 161–168. https://doi.org/10.1016/j.mbs.2012.07.003 doi: 10.1016/j.mbs.2012.07.003
[34]	Z. A. Wang, Z. Y. Xiang, P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differ. Equations, 260 (2016), 2225–2258. https://doi.org/10.1016/j.jde.2015.09.063 doi: 10.1016/j.jde.2015.09.063
[35]	S. Ai, W. Huang, Z. A. Wang, Reaction, diffusion and chemotaxis in wave propagation, Discrete Contin. Dyn. Syst. Series B, 20 (2015), 1–21. https://doi.org/10.3934/dcdsb.2015.20.1 doi: 10.3934/dcdsb.2015.20.1
[36]	Y. Zeng, K. Zhao, On the logarithmic Keller-Segel-Fisher/KPP system, Discrete Contin. Dyn. Syst., 39 (2019), 5365–5402. https://doi.org/10.3934/dcds.2019220 doi: 10.3934/dcds.2019220
[37]	Y. Zeng, K. Zhao, Optimal decay rates for a chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate, J. Differ. Equations, 268 (2020), 1379–1411. https://doi.org/10.1016/j.jde.2019.08.050 doi: 10.1016/j.jde.2019.08.050
[38]	Y. Zeng, Nonlinear stability of diffusive contact wave for a chemotaxis model, J. Differ. Equations, 308 (2022), 286–326. https://doi.org/10.1016/j.jde.2021.11.008 doi: 10.1016/j.jde.2021.11.008
[39]	J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer, New York, 1994. https://doi.org/10.1007/978-1-4612-0873-0
[40]	R. Salako, W. Shen, Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $ \mathbb{R}^N$, Discrete Contin. Dyn. Syst., 37 (2017), 6189–6225. https://doi.org/10.3934/dcds.2017268 doi: 10.3934/dcds.2017268
[41]	R. B. Salako, W. Shen, S. Xue, Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?, J. Math. Biol., 79 (2019), 1455–1490. https://doi.org/10.1007/s00285-019-01400-0 doi: 10.1007/s00285-019-01400-0
[42]	N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equations, 31 (1979), 53–98. https://doi.org/10.1016/0022-0396(79)90152-9 doi: 10.1016/0022-0396(79)90152-9
[43]	C. K. R. T. Jones, Geometric Singular Perturbation Theories, Lecture Notes in Mathematics, Vol. 1609, Springer-Verlag, Berlin, (1995), 44–118. https://doi.org/10.1007/bfb0095239
[44]	T. Kapitula, K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Springer, New York, 2013. https://doi.org/10.1007/978-1-4614-6995-7
[45]	A. I. Volpert, V. A. Volpert, V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, 1994.
[46]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol. 180, Springer-Verlag, New York/Berlin, 1981. https://doi.org/10.1007/bfb0089647
[47]	J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Dev., 17 (1973), 307–313. https://doi.org/10.1147/rd.174.0307 doi: 10.1147/rd.174.0307
[48]	D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312–355. https://doi.org/10.1016/0001-8708(76)90098-0 doi: 10.1016/0001-8708(76)90098-0

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