A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals

Liangchen Wang; Chunlai Mu

doi:10.3934/dcdsb.2020114

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December 2020, 25(12): 4585-4601. doi: 10.3934/dcdsb.2020114

A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals

Liangchen Wang ^1,, and Chunlai Mu ^2,

1.	Key Lab of Intelligent Analysis and Decision on Complex Systems, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2.	College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

^* Corresponding author: Liangchen Wang

Received August 2019 Revised December 2019 Published March 2020

This paper deals with the following competitive two-species chemotaxis system with two chemicals

$ \left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - {\chi _1}\nabla \cdot (u\nabla v) + {\mu _1}u\left( {1 - u - {a_1}w} \right),}&{x \in \Omega ,t > 0,}\\{0 = \Delta v - v + w,}&{x \in \Omega ,t > 0,}\\{{w_t} = \Delta w - {\chi _2}\nabla \cdot (w\nabla z) + {\mu _2}w\left( {1 - w - {a_2}u} \right),}&{x \in \Omega ,t > 0,}\\{0 = \Delta z - z + u,}&{x \in \Omega ,t > 0}\end{array}} \right. $

under homogeneous Neumann boundary conditions in a bounded domain

$ \Omega\subset \mathbb{R}^n $

(

$ n\geq1 $

), where the parameters

$ \chi_i>0 $

$ \mu_i>0 $

and

$ a_i>0 $

(

$ i = 1, 2 $

). It is proved that the corresponding initial-boundary value problem possesses a unique global bounded classical solution if one of the following cases holds:

(ⅰ)

$ q_1\leq a_1; $

(ⅱ)

$ q_2\leq a_2 $

;

(ⅲ)

$ q_1>a_1 $

and

$ q_2> a_2 $

as well as

$ (q_1-a_1)(q_2-a_2)<1 $

where

$ q_1: = \frac{\chi_1}{\mu_1} $

and

$ q_2: = \frac{\chi_2}{\mu_2} $

, which partially improves the results of Zhang et al. [53] and Tu et al. [34].

Moreover, it is proved that when

$ a_1, a_2\in(0, 1) $

and

$ \mu_1 $

and

$ \mu_2 $

are sufficiently large, then any global bounded solution exponentially converges to

$ \left(\frac{1-a_1}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_1}{1-a_1a_2}\right) $

$ t\rightarrow\infty $

; When

$ a_1>1>a_2>0 $

and

$ \mu_2 $

is sufficiently large, then any global bounded solution exponentially converges to

$ (0, 1, 1, 0) $

$ t\rightarrow\infty $

; When

$ a_1 = 1>a_2>0 $

and

$ \mu_2 $

is sufficiently large, then any global bounded solution algebraically converges to

$ (0, 1, 1, 0) $

$ t\rightarrow\infty $

. This result improves the conditions assumed in [34] for asymptotic behavior.

Keywords: Two-species chemotaxis, boundedness, asymptotic stability, Lotka-Volterra competition.

Mathematics Subject Classification: Primary: 35K35, 35A01, 35B44; Secondary: 35B35, 92C17.

Citation: Liangchen Wang, Chunlai Mu. A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4585-4601. doi: 10.3934/dcdsb.2020114

References:

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show all references

References:

[1]	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. Google Scholar
[2]	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
[3]	T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876. doi: 10.1093/imamat/hxw036. Google Scholar
[4]	T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1253-1272. doi: 10.3934/dcdsb.2017061. Google Scholar
[5]	M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004. Google Scholar
[6]	T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar
[7]	M. Hirata, S. Kurima, M. Mizukami and T. Yokota, Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490. doi: 10.1016/j.jde.2017.02.045. Google Scholar
[8]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. Google Scholar
[9]	D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x. Google Scholar
[10]	B. Hu and Y. Tao, Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Appl. Math. Lett., 64 (2017), 1-7. doi: 10.1016/j.aml.2016.08.003. Google Scholar
[11]	K. Kang and A. Stevens, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72. doi: 10.1016/j.na.2016.01.017. Google Scholar
[12]	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
[13]	J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191. doi: 10.1016/j.jde.2014.10.016. Google Scholar
[14]	J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527. doi: 10.3934/dcdsb.2015.20.1499. Google Scholar
[15]	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. Google Scholar
[16]	K. Lin and C. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260. doi: 10.3934/dcdsb.2017094. Google Scholar
[17]	K. Lin, C. Mu and and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096. doi: 10.1002/mma.3429. Google Scholar
[18]	M. Mizukami, Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249. doi: 10.1002/mma.4607. Google Scholar
[19]	M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669. doi: 10.1016/j.jde.2016.05.008. Google Scholar
[20]	M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319. doi: 10.3934/dcdsb.2017097. Google Scholar
[21]	M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 269-278. doi: 10.3934/dcdss.2020015. Google Scholar
[22]	M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781. doi: 10.1137/140971853. Google Scholar
[23]	M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617. doi: 10.1016/j.jde.2014.11.009. Google Scholar
[24]	K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. Google Scholar
[25]	K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X. Google Scholar
[26]	K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147. doi: 10.1007/s11538-009-9396-8. Google Scholar
[27]	H. Qiu and S. Guo, Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1569-1587. doi: 10.3934/dcdsb.2018220. Google Scholar
[28]	P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2007. Google Scholar
[29]	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. Google Scholar
[30]	C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626. doi: 10.1007/s00285-013-0681-7. Google Scholar
[31]	Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183. doi: 10.3934/dcdsb.2015.20.3165. Google Scholar
[32]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. Google Scholar
[33]	J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413. Google Scholar
[34]	X. Tu, C. Mu, P. Zheng and K. Lin, Global dynamics in a two-species chemotaxis-competition system with two signals, Discrete Contin. Dyn. Syst., 38 (2018), 3617-3636. doi: 10.3934/dcds.2018156. Google Scholar
[35]	Z. Wang and T. Xiang, A class of chemotaxis systems with growth source and nonlinear secretion, preprint, arXiv: 1510.07204. Google Scholar
[36]	L. Wang, Improvement of conditions for boundedness in a two-species chemotaxis competition system of parabolic-parabolic-elliptic type, J. Math. Anal. Appl., 484 (2020), 123705, 10 pp. doi: 10.1016/j.jmaa.2019.123705. Google Scholar
[37]	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
[38]	L. Wang, C. Mu, X. Hu and P. Zheng, Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differential Equations, 264 (2018), 3369-3401. doi: 10.1016/j.jde.2017.11.019. Google Scholar
[39]	L. Wang, J. Zhang, C. Mu and X. Hu, Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 191-221. doi: 10.3934/dcdsb.2019178. Google Scholar
[40]	M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057. Google Scholar
[41]	M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Art. 69, 40 pp. doi: 10.1007/s00033-018-0935-8. Google Scholar
[42]	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. Google Scholar
[43]	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
[44]	M. Winkler, Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793. doi: 10.3934/dcdsb.2017135. Google Scholar
[45]	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
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A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals

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A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals

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