Asymptotic stability in a chemotaxis-competition system with indirect signal production

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March 2021, 41(3): 1207-1223. doi: 10.3934/dcds.2020315

Asymptotic stability in a chemotaxis-competition system with indirect signal production

College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

^* Corresponding author: Pan Zheng

Received May 2019 Revised October 2019 Published March 2021 Early access August 2020

Fund Project: This work is partially supported by National Natural Science Foundation of China (Grant Nos: 11601053, 11526042), Natural Science Foundation of Chongqing (Grant No: cstc2019jcyj-msxmX0082) and China-South Africa Young Scientist Exchange Programme

This paper deals with a fully parabolic inter-species chemotaxis-competition system with indirect signal production

$ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} &u_{t} = \text{div}(d_{u}\nabla u+\chi u\nabla w)+\mu_{1}u(1-u-a_{1}v), &(x,t)\in \Omega\times (0,\infty), \\ &v_{t} = d_{v}\Delta v+\mu_{2}v(1-v-a_{2}u), &(x,t)\in \Omega\times (0,\infty), \\ & w_{t} = d_{w}\Delta w-\lambda w+\alpha v, &(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} $

under zero Neumann boundary conditions in a smooth bounded domain

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

(

$ N\geq 1 $

), where

$ d_{u}>0, d_{v}>0 $

and

$ d_{w}>0 $

are the diffusion coefficients,

$ \chi\in \mathbb{R} $

is the chemotactic coefficient,

$ \mu_{1}>0 $

and

$ \mu_{2}>0 $

are the population growth rates,

$ a_{1}>0, a_{2}>0 $

denote the strength coefficients of competition, and

$ \lambda $

and

$ \alpha $

describe the rates of signal degradation and production, respectively. Global boundedness of solutions to the above system with

$ \chi>0 $

was established by Tello and Wrzosek in [J. Math. Anal. Appl. 459 (2018) 1233-1250]. The main purpose of the paper is further to give the long-time asymptotic behavior of global bounded solutions, which could not be derived in the previous work.

Keywords: Asymptotic stability, chemotaxis, inter-species competition.

Mathematics Subject Classification: Primary:35B35, 35B40, 35B45;Secondary:35K55, 92C17.

Citation: Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315

References:

[1]	X. Bai and M. Winkler, Equilibration in a fully parabolic two-spescies chemotaxis system with competitive kinetics, Indiana University Math. J., 65 (2016), 553-583. doi: 10.1512/iumj.2016.65.5776.
[2]	T. Black, Global existence and asymptotic behavior in a competition two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1253-1272. doi: 10.3934/dcdsb.2017061.
[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.
[4]	X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904. doi: 10.3934/dcds.2015.35.1891.
[5]	X. Cao, S. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, Math. Meth. Appl. Sci., 41 (2018), 3138-3154. doi: 10.1002/mma.4807.
[6]	X. Cao, S. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller-Segel-Stokes system with competitive kinetics, preprint, arXiv: 1706.07910v1.
[7]	M. Ding and W. Wang, Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4665-4684. doi: 10.3934/dcdsb.2018328.
[8]	M. Fuest, Analysis of a chemotaxis model with indirect signal absorbtion, J. Differential Equations, 267 (2019), 4778-4806. doi: 10.1016/j.jde.2019.05.015.
[9]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
[10]	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.
[11]	M. Hirata, S. Kurima, M. Mizukami and T. Yokota, Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, preprint, arXiv: 1710.00957v1.
[12]	S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763. doi: 10.1137/0134064.
[13]	B. Hu and Y. Tao, To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128. doi: 10.1142/S0218202516400091.
[14]	H.-Y. Jin and T. Xiang, Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes with competition kinetics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1919-1942. doi: 10.3934/dcdsb.2018249.
[15]	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.
[16]	E. F. Keller and L. A. Segel, Traveling bans of chemotactic bacteria: a theoretical analysis, J. Theor. Biol., 30 (1971), 377-380.
[17]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Izdat. Nauka, Moscow, 1967.
[18]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.
[19]	K. Lin, C. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Meth. Appl. Sci., 38 (2015), 5085-5096. doi: 10.1002/mma.3429.
[20]	A. J. Lotka, Elements of Physical Biology, Williams and Wilkins Co., Baltimore, 1925.
[21]	J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.
[22]	C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.
[23]	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.
[24]	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.
[25]	Y. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, preprint.
[26]	Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc. (JEMS), 19 (2017), 3641-3678. doi: 10.4171/JEMS/749.
[27]	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.
[28]	J. I. Tello and D. Wrzosek, Inter-species competition and chemorepulsion, J. Math. Anal. Appl., 459 (2018), 1233-1250. doi: 10.1016/j.jmaa.2017.11.021.
[29]	V. Volterra, Variazioni e Fluttuazioni del Numero d'individui in Specie Animali Conviventi, Mem. R. Accad. Naz. Dei Lincei. Ser. VI, 1926.
[30]	W. Wang, A quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source, J. Math. Anal. Appl., 477 (2019), 488-522. doi: 10.1016/j.jmaa.2019.04.043.
[31]	Q. Wang, L. Zhang, J. Yang and J. Hu, Global existence and steady states of a two competing species Keller-Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807. doi: 10.3934/krm.2015.8.777.
[32]	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.
[33]	J. Xing and P. Zheng, Global boundedness and long-time behavior for a two-dimensional quasilinear chemotaxis system with indirect signal consumption, preprint.
[34]	P. Zheng and C. Mu, Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177. doi: 10.1007/s10440-016-0083-0.
[35]	P. Zheng, C. Mu and X. Hu, Global dynamics for an attraction-repulsion chemotaxis-(Navier)-Stokes system with logistic source, Nonl. Anal. Real World Appl., 45 (2019), 557-580. doi: 10.1016/j.nonrwa.2018.07.028.
[36]	P. Zheng, C. Mu and Y. Mi, Global stability in a two-competing-species chemotaxis system with two chemicals, Diff. Inte. Equa., 31 (2018), 547-558.
[37]	P. Zheng, C. Mu, R. Willie and X. Hu, Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity, Comput. Math. Appl., 75 (2018), 1667-1675. doi: 10.1016/j.camwa.2017.11.032.
[38]	P. Zheng, R. Willie and C. Mu, Global boundedness and stabilization in a two-competing-species chemotaxis-fluid system with two chemicals, J. Dyn. Differential Equations, 32 (2020), 1371-1399. doi: 10.1007/s10884-019-09797-4.

show all references

References:

[1]	X. Bai and M. Winkler, Equilibration in a fully parabolic two-spescies chemotaxis system with competitive kinetics, Indiana University Math. J., 65 (2016), 553-583. doi: 10.1512/iumj.2016.65.5776.
[2]	T. Black, Global existence and asymptotic behavior in a competition two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1253-1272. doi: 10.3934/dcdsb.2017061.
[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.
[4]	X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904. doi: 10.3934/dcds.2015.35.1891.
[5]	X. Cao, S. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, Math. Meth. Appl. Sci., 41 (2018), 3138-3154. doi: 10.1002/mma.4807.
[6]	X. Cao, S. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller-Segel-Stokes system with competitive kinetics, preprint, arXiv: 1706.07910v1.
[7]	M. Ding and W. Wang, Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4665-4684. doi: 10.3934/dcdsb.2018328.
[8]	M. Fuest, Analysis of a chemotaxis model with indirect signal absorbtion, J. Differential Equations, 267 (2019), 4778-4806. doi: 10.1016/j.jde.2019.05.015.
[9]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
[10]	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.
[11]	M. Hirata, S. Kurima, M. Mizukami and T. Yokota, Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, preprint, arXiv: 1710.00957v1.
[12]	S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763. doi: 10.1137/0134064.
[13]	B. Hu and Y. Tao, To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128. doi: 10.1142/S0218202516400091.
[14]	H.-Y. Jin and T. Xiang, Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes with competition kinetics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1919-1942. doi: 10.3934/dcdsb.2018249.
[15]	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.
[16]	E. F. Keller and L. A. Segel, Traveling bans of chemotactic bacteria: a theoretical analysis, J. Theor. Biol., 30 (1971), 377-380.
[17]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Izdat. Nauka, Moscow, 1967.
[18]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.
[19]	K. Lin, C. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Meth. Appl. Sci., 38 (2015), 5085-5096. doi: 10.1002/mma.3429.
[20]	A. J. Lotka, Elements of Physical Biology, Williams and Wilkins Co., Baltimore, 1925.
[21]	J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.
[22]	C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.
[23]	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.
[24]	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.
[25]	Y. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, preprint.
[26]	Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc. (JEMS), 19 (2017), 3641-3678. doi: 10.4171/JEMS/749.
[27]	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.
[28]	J. I. Tello and D. Wrzosek, Inter-species competition and chemorepulsion, J. Math. Anal. Appl., 459 (2018), 1233-1250. doi: 10.1016/j.jmaa.2017.11.021.
[29]	V. Volterra, Variazioni e Fluttuazioni del Numero d'individui in Specie Animali Conviventi, Mem. R. Accad. Naz. Dei Lincei. Ser. VI, 1926.
[30]	W. Wang, A quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source, J. Math. Anal. Appl., 477 (2019), 488-522. doi: 10.1016/j.jmaa.2019.04.043.
[31]	Q. Wang, L. Zhang, J. Yang and J. Hu, Global existence and steady states of a two competing species Keller-Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807. doi: 10.3934/krm.2015.8.777.
[32]	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.
[33]	J. Xing and P. Zheng, Global boundedness and long-time behavior for a two-dimensional quasilinear chemotaxis system with indirect signal consumption, preprint.
[34]	P. Zheng and C. Mu, Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177. doi: 10.1007/s10440-016-0083-0.
[35]	P. Zheng, C. Mu and X. Hu, Global dynamics for an attraction-repulsion chemotaxis-(Navier)-Stokes system with logistic source, Nonl. Anal. Real World Appl., 45 (2019), 557-580. doi: 10.1016/j.nonrwa.2018.07.028.
[36]	P. Zheng, C. Mu and Y. Mi, Global stability in a two-competing-species chemotaxis system with two chemicals, Diff. Inte. Equa., 31 (2018), 547-558.
[37]	P. Zheng, C. Mu, R. Willie and X. Hu, Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity, Comput. Math. Appl., 75 (2018), 1667-1675. doi: 10.1016/j.camwa.2017.11.032.
[38]	P. Zheng, R. Willie and C. Mu, Global boundedness and stabilization in a two-competing-species chemotaxis-fluid system with two chemicals, J. Dyn. Differential Equations, 32 (2020), 1371-1399. doi: 10.1007/s10884-019-09797-4.

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