# American Institute of Mathematical Sciences

December  2021, 26(12): 6155-6171. doi: 10.3934/dcdsb.2021011

## Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity

 1 School of Mathematics, South China University of Technology, Guangzhou 510640, China 2 College of Mathematics and Statistics, Shenzhen University, Shenzhen 518061, China

* Corresponding author: yrchen@szu.edu.cn

Received  August 2020 Revised  November 2020 Published  December 2021 Early access  December 2020

Fund Project: Z. Liu was partially supported by the National Natural Science Foundation of China (No. 11971176 and No. 12026608). Y.Chen was partially supported by the National Natural Science Foundation of China (No. 12001377 and No. 11971176) and the Outstanding Innovative Young Talents of Guangdong Province, China (No. 2019KQNCX122)

In this paper, we shall study the initial-boundary value problem of a chemotaxis model with signal-dependent diffusion and sensitivity as follows
 $\begin{cases} u_t = \nabla\cdot(\gamma(v)\nabla u-\chi(v)u\nabla v)+\alpha u F(w) +\theta u-\beta u^2, &x\in \Omega, \; \; t>0,\\ v_t = D\Delta v+u-v,& x\in \Omega, \; \; t>0,\\ w_t = \Delta w-uF(w),& x\in \Omega, \; \; t>0,\\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0,&x\in \partial\Omega, \; \; t>0,\\ u(x,0) = u_0(x), v(x,0) = v_0(x),w(x,0) = w_0(x), & x\in\Omega, \end{cases} \;\;(*)$
in a bounded domain
 $\Omega\subset \mathbb{R}^2$
with smooth boundary, where
 $\alpha,\beta, D$
are positive constants,
 $\theta\in \mathbb{R}$
and
 $\nu$
denotes the outward normal vector of
 $\partial \Omega$
. The functions
 $\chi(v),\gamma(v)$
and
 $F(v)$
satisfy
 $(\gamma(v),\chi(v))\in [C^2[0,\infty)]^2$
with
 $\gamma(v)>0,\gamma'(v)<0$
and
 $\frac{|\chi(v)|+|\gamma'(v)|}{\gamma(v)}$
is bounded;
 $F(w)\in C^1([0,\infty)), F(0) = 0,F(w)>0 \ \mathrm{in}\; (0,\infty)\; \mathrm{and}\; F'(w)>0 \ \mathrm{on}\ \ [0,\infty).$
We first prove that the existence of globally bounded solution of system (*) based on the method of weighted energy estimates. Moreover, by constructing Lyapunov functional, we show that the solution
 $(u,v,w)$
will converge to
 $(0,0,w_*)$
in
 $L^\infty$
with some
 $w_*\geq0$
as time tends to infinity in the case of
 $\theta\leq 0$
, while if
 $\theta>0$
, the solution
 $(u,v,w)$
will asymptotically converge to
 $(\frac{\theta}{\beta},\frac{\theta}{\beta},0)$
in
 $L^\infty$
-norm provided
 $D>\max\limits_{0\leq v\leq \infty}\frac{\theta|\chi(v)|^2}{16\beta^2\gamma(v)}$
.
Citation: Hui Zhao, Zhengrong Liu, Yiren Chen. Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6155-6171. doi: 10.3934/dcdsb.2021011
##### References:
 [1] J. Ahn and C. Yoon, Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing, Nonlinearity, 32 (2019), 1327-1351.  doi: 10.1088/1361-6544/aaf513. [2] N. D. Alikakos, $L^{p}$ bounds of solutions of reaction-diffusion equations, Commun. Partial Differential Equations, 4 (1979), 827–868. doi: 10.1080/03605307908820113. [3] H. Amann, Dynamic theory of quasilinear parabolic equations, Ⅱ. Reaction-diffusion systems, Differ. Integral Equ., 3 (1990), 13-75. [4] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte zur Math., Stuttgart-Leipzig, 133 (1993), 9-126.  doi: 10.1007/978-3-663-11336-2_1. [5] X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776. [6] N. Bellomo, A. Bellouquid, Y. S. Tao and M. Winkler, Towards 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. [7] X. Fu, L.-H. Tang, C. Liu, J.-D. Huang, T. Hwa and P. Lenz, Stripe formation in bacterial system with density-suppressed motility, Phys. Rev. Lett., 108 (2012), 198102. doi: 10.1103/PhysRevLett.108.198102. [8] K. Fujie and J. Jiang, Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities, arXiv: 2001.01288. [9] K. Fujie and J. Jiang, Global existence for a kinetic model of pattern formation with density-suppressed motilities, J. Differential Equations, 269 (2020), 5338-5378.  doi: 10.1016/j.jde.2020.04.001. [10] T Hillen, K. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Method Appl. Sci., 23 (2013), 165-198.  doi: 10.1142/S0218202512500480. [11] H. Y. Jin, Y. J. Kim and Z. A. Wang, Boundedness, stabilization, and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 78 (2018), 1632-1657.  doi: 10.1137/17M1144647. [12] H. Y. Jin and Z. A. Wang, Global stability of prey-taxis systems, J. Differential Equations, 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010. [13] H. Y. Jin and Z. A. Wang, Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion, Euro. J. Appl. Math., doi: 10.1017/S0956792520000248,2020. doi: 10.1017/S0956792520000248. [14] H. Y. Jin, S. Shi and Z. A. Wang, Boundedness and asymptotics of a reaction-diffusion system with density-dependent motility, J. Differential Equations, 269 (2020), 6758-6793.  doi: 10.1016/j.jde.2020.05.018. [15] H. Y. Jin and Z. A. Wang, Critical mass on the Keller-Segel system with signal-dependent motility, Proc. Amer. Math. Soc., 148 (2020), 4855-4873.  doi: 10.1090/proc/15124. [16] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [17] K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Phys. D, 241 (2012), 1629-1639.  doi: 10.1016/j.physd.2012.06.009. [18] Y. Lou and M. Winkler, Global existence and uniform boundedness of smooth solutions to a cross-diffusion system with equal diffusion rates, Comm. Partial Differential Equations, 40 (2015), 1905-1941.  doi: 10.1080/03605302.2015.1052882. [19] K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.  doi: 10.3934/dcds.2016018. [20] C. Liu, Sequtential establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238-241. [21] M. Ma, R. Peng and Z. Wang, Stationary and non-stationary patterns of the density-suppressed motility model, Physica D, 402 (2020), 132259, 13pp. doi: 10.1016/j.physd.2019.132259. [22] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. TMA, 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X. [23] K.J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Phys. D, 240 (2011), 363-375.  doi: 10.1016/j.physd.2010.09.011. [24] M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045. [25] J. Smith-Roberge, D. Iron and T. Kolokolnikov, Pattern formation in bacterial colonies with density-dependent diffusion, Eur. J. Appl. Math., 30 (2019), 196-218.  doi: 10.1017/S0956792518000013. [26] C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.  doi: 10.1137/13094058X. [27] Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.  doi: 10.1016/j.jmaa.2011.02.041. [28] 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. [29] Y. S. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115. [30] Y. Tao and M. Winkler, Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Meth. Appl. Sci., 27 (2017), 1645-1683.  doi: 10.1142/S0218202517500282. [31] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [32] J. Wang and M. Wang, Boundedness in the higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth, J. Math. Phys., 60 (2019), 011507, 14pp. doi: 10.1063/1.5061738. [33] Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13pp. doi: 10.1063/1.2766864. [34] S. Wang, J. Wang and J. Shi, Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math. Models Methods Appl. Sci., 28 (2018), 2275-2312.  doi: 10.1142/S0218202518400158. [35] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [36] 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. [37] M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9. [38] S. Wu, J. Shi and B. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differential Equations, 260 (2016), 5847-5874.  doi: 10.1016/j.jde.2015.12.024. [39] T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323.  doi: 10.1016/j.jde.2015.01.032. [40] C. Yoon and Y.-J. Kim, Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Application Mathematics, 149 (2017), 101-123.  doi: 10.1007/s10440-016-0089-7.

show all references

##### References:
 [1] J. Ahn and C. Yoon, Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing, Nonlinearity, 32 (2019), 1327-1351.  doi: 10.1088/1361-6544/aaf513. [2] N. D. Alikakos, $L^{p}$ bounds of solutions of reaction-diffusion equations, Commun. Partial Differential Equations, 4 (1979), 827–868. doi: 10.1080/03605307908820113. [3] H. Amann, Dynamic theory of quasilinear parabolic equations, Ⅱ. Reaction-diffusion systems, Differ. Integral Equ., 3 (1990), 13-75. [4] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte zur Math., Stuttgart-Leipzig, 133 (1993), 9-126.  doi: 10.1007/978-3-663-11336-2_1. [5] X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776. [6] N. Bellomo, A. Bellouquid, Y. S. Tao and M. Winkler, Towards 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. [7] X. Fu, L.-H. Tang, C. Liu, J.-D. Huang, T. Hwa and P. Lenz, Stripe formation in bacterial system with density-suppressed motility, Phys. Rev. Lett., 108 (2012), 198102. doi: 10.1103/PhysRevLett.108.198102. [8] K. Fujie and J. Jiang, Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities, arXiv: 2001.01288. [9] K. Fujie and J. Jiang, Global existence for a kinetic model of pattern formation with density-suppressed motilities, J. Differential Equations, 269 (2020), 5338-5378.  doi: 10.1016/j.jde.2020.04.001. [10] T Hillen, K. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Method Appl. Sci., 23 (2013), 165-198.  doi: 10.1142/S0218202512500480. [11] H. Y. Jin, Y. J. Kim and Z. A. Wang, Boundedness, stabilization, and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 78 (2018), 1632-1657.  doi: 10.1137/17M1144647. [12] H. Y. Jin and Z. A. Wang, Global stability of prey-taxis systems, J. Differential Equations, 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010. [13] H. Y. Jin and Z. A. Wang, Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion, Euro. J. Appl. Math., doi: 10.1017/S0956792520000248,2020. doi: 10.1017/S0956792520000248. [14] H. Y. Jin, S. Shi and Z. A. Wang, Boundedness and asymptotics of a reaction-diffusion system with density-dependent motility, J. Differential Equations, 269 (2020), 6758-6793.  doi: 10.1016/j.jde.2020.05.018. [15] H. Y. Jin and Z. A. Wang, Critical mass on the Keller-Segel system with signal-dependent motility, Proc. Amer. Math. Soc., 148 (2020), 4855-4873.  doi: 10.1090/proc/15124. [16] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [17] K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Phys. D, 241 (2012), 1629-1639.  doi: 10.1016/j.physd.2012.06.009. [18] Y. Lou and M. Winkler, Global existence and uniform boundedness of smooth solutions to a cross-diffusion system with equal diffusion rates, Comm. Partial Differential Equations, 40 (2015), 1905-1941.  doi: 10.1080/03605302.2015.1052882. [19] K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.  doi: 10.3934/dcds.2016018. [20] C. Liu, Sequtential establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238-241. [21] M. Ma, R. Peng and Z. Wang, Stationary and non-stationary patterns of the density-suppressed motility model, Physica D, 402 (2020), 132259, 13pp. doi: 10.1016/j.physd.2019.132259. [22] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. TMA, 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X. [23] K.J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Phys. D, 240 (2011), 363-375.  doi: 10.1016/j.physd.2010.09.011. [24] M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045. [25] J. Smith-Roberge, D. Iron and T. Kolokolnikov, Pattern formation in bacterial colonies with density-dependent diffusion, Eur. J. Appl. Math., 30 (2019), 196-218.  doi: 10.1017/S0956792518000013. [26] C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.  doi: 10.1137/13094058X. [27] Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.  doi: 10.1016/j.jmaa.2011.02.041. [28] 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. [29] Y. S. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115. [30] Y. Tao and M. Winkler, Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Meth. Appl. Sci., 27 (2017), 1645-1683.  doi: 10.1142/S0218202517500282. [31] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [32] J. Wang and M. Wang, Boundedness in the higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth, J. Math. Phys., 60 (2019), 011507, 14pp. doi: 10.1063/1.5061738. [33] Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13pp. doi: 10.1063/1.2766864. [34] S. Wang, J. Wang and J. Shi, Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math. Models Methods Appl. Sci., 28 (2018), 2275-2312.  doi: 10.1142/S0218202518400158. [35] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [36] 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. [37] M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9. [38] S. Wu, J. Shi and B. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differential Equations, 260 (2016), 5847-5874.  doi: 10.1016/j.jde.2015.12.024. [39] T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323.  doi: 10.1016/j.jde.2015.01.032. [40] C. Yoon and Y.-J. Kim, Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Application Mathematics, 149 (2017), 101-123.  doi: 10.1007/s10440-016-0089-7.
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