American Institute of Mathematical Sciences

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

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

Pan Zheng ,

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 & 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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[12]

S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.  doi: 10.1137/0134064.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

E. F. Keller and L. A. Segel, Traveling bans of chemotactic bacteria: a theoretical analysis, J. Theor. Biol., 30 (1971), 377-380.   Google Scholar

[17]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Izdat. Nauka, Moscow, 1967.  Google Scholar

[18]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[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.  Google Scholar

[20]

A. J. Lotka, Elements of Physical Biology, Williams and Wilkins Co., Baltimore, 1925. Google Scholar

[21]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.  Google Scholar

[22]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

Y. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

V. Volterra, Variazioni e Fluttuazioni del Numero d'individui in Specie Animali Conviventi, Mem. R. Accad. Naz. Dei Lincei. Ser. VI, 1926. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

J. Xing and P. Zheng, Global boundedness and long-time behavior for a two-dimensional quasilinear chemotaxis system with indirect signal consumption, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[12]

S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.  doi: 10.1137/0134064.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

E. F. Keller and L. A. Segel, Traveling bans of chemotactic bacteria: a theoretical analysis, J. Theor. Biol., 30 (1971), 377-380.   Google Scholar

[17]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Izdat. Nauka, Moscow, 1967.  Google Scholar

[18]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[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.  Google Scholar

[20]

A. J. Lotka, Elements of Physical Biology, Williams and Wilkins Co., Baltimore, 1925. Google Scholar

[21]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.  Google Scholar

[22]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

Y. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

V. Volterra, Variazioni e Fluttuazioni del Numero d'individui in Specie Animali Conviventi, Mem. R. Accad. Naz. Dei Lincei. Ser. VI, 1926. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

J. Xing and P. Zheng, Global boundedness and long-time behavior for a two-dimensional quasilinear chemotaxis system with indirect signal consumption, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097
[2]

Masaaki Mizukami. Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 269-278. doi: 10.3934/dcdss.2020015
[3]

Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061
[4]

Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2211-2236. doi: 10.3934/cpaa.2021064
[5]

Yu Ma, Chunlai Mu, Shuyan Qiu. Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021218
[6]

Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220
[7]

Xinyu Tu, Chunlai Mu, Pan Zheng, Ke Lin. Global dynamics in a two-species chemotaxis-competition system with two signals. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3617-3636. doi: 10.3934/dcds.2018156
[8]

Tahir Bachar Issa, Rachidi Bolaji Salako. Asymptotic dynamics in a two-species chemotaxis model with non-local terms. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3839-3874. doi: 10.3934/dcdsb.2017193
[9]

Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 203-209. doi: 10.3934/dcdss.2020011
[10]

Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021017
[11]

Guo-Bao Zhang, Fang-Di Dong, Wan-Tong Li. Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1511-1541. doi: 10.3934/dcdsb.2018218
[12]

Georg Hetzer, Wenxian Shen. Two species competition with an inhibitor involved. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 39-57. doi: 10.3934/dcds.2005.12.39
[13]

Yuan Lou, Daniel Munther. Dynamics of a three species competition model. Discrete & Continuous Dynamical Systems, 2012, 32 (9) : 3099-3131. doi: 10.3934/dcds.2012.32.3099
[14]

Nahla Abdellatif, Radhouane Fekih-Salem, Tewfik Sari. Competition for a single resource and coexistence of several species in the chemostat. Mathematical Biosciences & Engineering, 2016, 13 (4) : 631-652. doi: 10.3934/mbe.2016012
[15]

Hua Nie, Yuan Lou, Jianhua Wu. Competition between two similar species in the unstirred chemostat. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 621-639. doi: 10.3934/dcdsb.2016.21.621
[16]

Chiu-Ju Lin. Competition of two phytoplankton species for light with wavelength. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 523-536. doi: 10.3934/dcdsb.2016.21.523
[17]

Jifa Jiang, Fensidi Tang. The complete classification on a model of two species competition with an inhibitor. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 659-672. doi: 10.3934/dcds.2008.20.659
[18]

Alexander Kurganov, Mária Lukáčová-Medvidová. Numerical study of two-species chemotaxis models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 131-152. doi: 10.3934/dcdsb.2014.19.131
[19]

Hua Nie, Sze-Bi Hsu, Jianhua Wu. Coexistence solutions of a competition model with two species in a water column. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2691-2714. doi: 10.3934/dcdsb.2015.20.2691
[20]

Yuyue Zhang, Jicai Huang, Qihua Huang. The impact of toxins on competition dynamics of three species in a polluted aquatic environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3043-3068. doi: 10.3934/dcdsb.2020219

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (201)
  • HTML views (223)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences