Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary

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doi: 10.3934/dcdsb.2020154

Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary

Lei Yang ^1,2, and Lianzhang Bao ^2,3,,

1.	School of Mathematics, Jilin University, Changchun, Jilin 130012, China
2.	School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P. R. China
3.	Department of Mathematics and Statistics, Auburn University, AL 36849, USA

* Corresponding author: Lianzhang Bao

Received July 2019 Revised December 2019 Published May 2020

The current paper is to investigate the numerical approximation of logistic type chemotaxis models in one space dimension with a free boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front (see Bao and Shen [1], [2]). The main challenges in the numerical studies lie in tracking the moving free boundary and the nonlinear terms from the chemical. To overcome them, a front-fixing framework coupled with the finite difference method is introduced. The accuracy of the proposed method, the positivity of the solution, and the stability of the scheme are discussed. The numerical simulations agree well with theoretical results such as the vanishing spreading dichotomy, local persistence, and stability. These simulations also validate some conjectures in our future theoretical studies such as the dependence of the vanishing-spreading dichotomy on the initial value $ u_0 $, initial habitat $ h_0 $, the moving speed $ \nu $ and the chemotactic sensitivity coefficients $ \chi_1, \chi_2 $.

Keywords: Chemoattraction-repulsion system, nonlinear parabolic equations, free boundary problem, spreading-vanishing dichotomy, front-fixing, finite difference, invasive population.

Mathematics Subject Classification: Primary: 35R35, 35J65, 35K20; Secondary: 78M20, 92B05.

Citation: Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020154

References:

[1]	L. Bao and W. Shen, Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 1107-1130. doi: 10.3934/dcds.2020072. Google Scholar
[2]	L. Bao and W. Shen, Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. II, Spreading-vanishing dichotomy in a domain with a free boundary, preprint. Google Scholar
[3]	N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. Google Scholar
[4]	C. C. Chiu and J. L. Yu, An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems, Math. Biosci. Eng., 4 (2007), 187-203. doi: 10.3934/mbe.2007.4.187. Google Scholar
[5]	J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Science and Applications, 5 (1995), 659-680. Google Scholar
[6]	J. I. Diaz, T. Nagai and J.-M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^N$, J. Differential Equations, 145 (1998), 156-183. doi: 10.1006/jdeq.1997.3389. Google Scholar
[7]	Y.-H. Du and Z.-G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089. Google Scholar
[8]	Y.-H. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 32 (2015), 279–305. doi: 10.1016/j.anihpc.2013.11.004. Google Scholar
[9]	E. Galakhov, O. Salieva and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647. doi: 10.1016/j.jde.2016.07.008. Google Scholar
[10]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. Google Scholar
[11]	T. B. Issa and W. Shen, Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst., 16 (2017), 926-973. doi: 10.1137/16M1092428. Google Scholar
[12]	H. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196. doi: 10.1016/j.jde.2015.08.040. Google Scholar
[13]	K. Kanga and A. Steven, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72. doi: 10.1016/j.na.2016.01.017. Google Scholar
[14]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar
[15]	E. F. Keller and L. A. Segel, A Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar
[16]	H. G. Landau, Heat conduction in a melting solid, Quaterly of Applied Mathematics, 8 (1950), 81-94. doi: 10.1090/qam/33441. Google Scholar
[17]	F. Li, X. Liang and W. Shen, Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy, Discrete Contin. Dyn. Syst, 36 (2016), 3317-3338. doi: 10.3934/dcds.2016.36.3317. Google Scholar
[18]	F. Li, X. Liang and W. Shen, Diffusive KPP equations with free boundaries in time almost periodic environments: II. Spreading speeds and semi-wave solutions, J. Differential Equations, 261 (2016), 2403-2445. doi: 10.1016/j.jde.2016.04.035. Google Scholar
[19]	R. H. Li, Z. Y. Chen and W. Wu, Generalized Difference Methods for Differential Equations- Numerical Analysis of Finite Volume Methods, Marcel Dekker, Inc, 2000. Google Scholar
[20]	X. J. Li, C. W. Shu and Y. Yang, Local discontinuous Galerkin method for the Keller-Segel chemotaxis model, J. Sci. Comput., 73 (2017), 943-967. doi: 10.1007/s10915-016-0354-y. Google Scholar
[21]	J. G. Liu, L. Wang and Z. N. Zhou, Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations, Math. Comp., 87 (2018), 1165-1189. doi: 10.1090/mcom/3250. Google Scholar
[22]	S. Liu and X. F. Liu, Numerical methods for a wwo-species competition-diffusion model with free boundaries, Mathematics, 6 (2018), 72-96. Google Scholar
[23]	S. Liu, Y. H. Du and X. F. Liu, Numerical studies of a class of reaction-diffusion equations with stefan conditions, International Journal of Computer Mathematics, 97 (2020), 959-979. doi: 10.1080/00207160.2019.1599868. Google Scholar
[24]	J. L. Lockwood, M. F. Hoopes and M. P. Marchetti, Invasion Ecology, Blackwell Publishing, 2007. Google Scholar
[25]	M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730. doi: 10.1016/S0092-8240(03)00030-2. Google Scholar
[26]	M.-A. Piqueras, R. Company and L. L$\acute{o}$dar, A front-fixing numerical method for a free boundary nonlinear diffusion logistic population model, J. Comput. Appl. Math., 309 (2017), 473-481. doi: 10.1016/j.cam.2016.02.029. Google Scholar
[27]	T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433. Google Scholar
[28]	N. Saito and T. Suzuki, Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis, Appl. Math. Comput., 171 (2005), 72-90. doi: 10.1016/j.amc.2005.01.037. Google Scholar
[29]	R. B. Salako and 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. doi: 10.3934/dcds.2017268. Google Scholar
[30]	R. B. Salako, W. Shen and S. W. Xue, Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic chemotaxis systems with logistic source?, J. Math. Biol., 79 (2019), 1455-1490. doi: 10.1007/s00285-019-01400-0. Google Scholar
[31]	N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press., Oxford, 1997. Google Scholar
[32]	Y. Sugiyama, Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876. Google Scholar
[33]	Y. Sugiyama and H. Kunii, Global Existence and decay properties for a degenerate keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003. Google Scholar
[34]	Y.-S. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443. Google Scholar
[35]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. Google Scholar
[36]	L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007. Google Scholar
[37]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. Google Scholar
[38]	M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057. Google Scholar
[39]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020. Google Scholar
[40]	M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023. Google Scholar
[41]	M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855. doi: 10.1007/s00332-014-9205-x. Google Scholar
[42]	T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst. Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 2015 (2015), 1125-1133. doi: 10.3934/proc.2015.1125. Google Scholar
[43]	P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, Math. Anal. Appl., 424 (2015), 509-522. doi: 10.1016/j.jmaa.2014.11.031. Google Scholar

show all references

References:

[1]	L. Bao and W. Shen, Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 1107-1130. doi: 10.3934/dcds.2020072. Google Scholar
[2]	L. Bao and W. Shen, Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. II, Spreading-vanishing dichotomy in a domain with a free boundary, preprint. Google Scholar
[3]	N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. Google Scholar
[4]	C. C. Chiu and J. L. Yu, An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems, Math. Biosci. Eng., 4 (2007), 187-203. doi: 10.3934/mbe.2007.4.187. Google Scholar
[5]	J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Science and Applications, 5 (1995), 659-680. Google Scholar
[6]	J. I. Diaz, T. Nagai and J.-M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^N$, J. Differential Equations, 145 (1998), 156-183. doi: 10.1006/jdeq.1997.3389. Google Scholar
[7]	Y.-H. Du and Z.-G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089. Google Scholar
[8]	Y.-H. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 32 (2015), 279–305. doi: 10.1016/j.anihpc.2013.11.004. Google Scholar
[9]	E. Galakhov, O. Salieva and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647. doi: 10.1016/j.jde.2016.07.008. Google Scholar
[10]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. Google Scholar
[11]	T. B. Issa and W. Shen, Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst., 16 (2017), 926-973. doi: 10.1137/16M1092428. Google Scholar
[12]	H. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196. doi: 10.1016/j.jde.2015.08.040. Google Scholar
[13]	K. Kanga and A. Steven, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72. doi: 10.1016/j.na.2016.01.017. Google Scholar
[14]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar
[15]	E. F. Keller and L. A. Segel, A Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar
[16]	H. G. Landau, Heat conduction in a melting solid, Quaterly of Applied Mathematics, 8 (1950), 81-94. doi: 10.1090/qam/33441. Google Scholar
[17]	F. Li, X. Liang and W. Shen, Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy, Discrete Contin. Dyn. Syst, 36 (2016), 3317-3338. doi: 10.3934/dcds.2016.36.3317. Google Scholar
[18]	F. Li, X. Liang and W. Shen, Diffusive KPP equations with free boundaries in time almost periodic environments: II. Spreading speeds and semi-wave solutions, J. Differential Equations, 261 (2016), 2403-2445. doi: 10.1016/j.jde.2016.04.035. Google Scholar
[19]	R. H. Li, Z. Y. Chen and W. Wu, Generalized Difference Methods for Differential Equations- Numerical Analysis of Finite Volume Methods, Marcel Dekker, Inc, 2000. Google Scholar
[20]	X. J. Li, C. W. Shu and Y. Yang, Local discontinuous Galerkin method for the Keller-Segel chemotaxis model, J. Sci. Comput., 73 (2017), 943-967. doi: 10.1007/s10915-016-0354-y. Google Scholar
[21]	J. G. Liu, L. Wang and Z. N. Zhou, Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations, Math. Comp., 87 (2018), 1165-1189. doi: 10.1090/mcom/3250. Google Scholar
[22]	S. Liu and X. F. Liu, Numerical methods for a wwo-species competition-diffusion model with free boundaries, Mathematics, 6 (2018), 72-96. Google Scholar
[23]	S. Liu, Y. H. Du and X. F. Liu, Numerical studies of a class of reaction-diffusion equations with stefan conditions, International Journal of Computer Mathematics, 97 (2020), 959-979. doi: 10.1080/00207160.2019.1599868. Google Scholar
[24]	J. L. Lockwood, M. F. Hoopes and M. P. Marchetti, Invasion Ecology, Blackwell Publishing, 2007. Google Scholar
[25]	M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730. doi: 10.1016/S0092-8240(03)00030-2. Google Scholar
[26]	M.-A. Piqueras, R. Company and L. L$\acute{o}$dar, A front-fixing numerical method for a free boundary nonlinear diffusion logistic population model, J. Comput. Appl. Math., 309 (2017), 473-481. doi: 10.1016/j.cam.2016.02.029. Google Scholar
[27]	T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433. Google Scholar
[28]	N. Saito and T. Suzuki, Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis, Appl. Math. Comput., 171 (2005), 72-90. doi: 10.1016/j.amc.2005.01.037. Google Scholar
[29]	R. B. Salako and 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. doi: 10.3934/dcds.2017268. Google Scholar
[30]	R. B. Salako, W. Shen and S. W. Xue, Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic chemotaxis systems with logistic source?, J. Math. Biol., 79 (2019), 1455-1490. doi: 10.1007/s00285-019-01400-0. Google Scholar
[31]	N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press., Oxford, 1997. Google Scholar
[32]	Y. Sugiyama, Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876. Google Scholar
[33]	Y. Sugiyama and H. Kunii, Global Existence and decay properties for a degenerate keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003. Google Scholar
[34]	Y.-S. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443. Google Scholar
[35]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. Google Scholar
[36]	L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007. Google Scholar
[37]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. Google Scholar
[38]	M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057. Google Scholar
[39]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020. Google Scholar
[40]	M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023. Google Scholar
[41]	M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855. doi: 10.1007/s00332-014-9205-x. Google Scholar
[42]	T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst. Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 2015 (2015), 1125-1133. doi: 10.3934/proc.2015.1125. Google Scholar
[43]	P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, Math. Anal. Appl., 424 (2015), 509-522. doi: 10.1016/j.jmaa.2014.11.031. Google Scholar

Figure 1. Evolution of the density $ u(t,x) $

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Figure 2. Evolution of the speed $ \frac{h(t)}{t} $

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Figure 3. Evolution of the density $ u(t,x) $

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Figure 4. Evolution of the speed $ \frac{h(t)}{t} $

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Figure 5. Evolution of the density $ u(t,x) $

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Figure 6. Evolution of the speed $ \frac{h(t)}{t} $

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Figure 7. Evolution of the density $ u(t,x) $

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Figure 8. Evolution of the speed $ \frac{h(t)}{t} $

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Figure 9. Evolution of the density $ u(t,x) $

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Figure 10. Evolution of the speed $ \frac{h(t)}{t} $

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Figure 11. Evolution of the habitat length $ h(t) $

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Figure 12. Evolution of the habitat length $ h(t) $

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Figure 13. Evolution of the density $ u(t,x) $

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Figure 14. Evolution of the speed $ \frac{h(t)}{t} $

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Figure 15. Evolution of the density $ u(t,x) $

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Figure 16. Evolution of the speed $ \frac{h(t)}{t} $

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Figure 17. Evolution of the density $ u(t,x) $

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Figure 18. Evolution of the speed $ \frac{h(t)}{t} $

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Figure 19. Evolution of the density $ u(t,x) $

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Figure 20. Evolution of the speed $ \frac{h(t)}{t} $

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Figure 21. Evolution of the density $ u(t,x) $

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Figure 22. Evolution of the density $ u(t,x) $

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Figure 23. Evolution of the speed $ \frac{h(t)}{t} $

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Figure 24. Evolution of the density $ u(t,x) $

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