Global dynamics in a two-species chemotaxis-competition system with two signals

Xinyu Tu; Chunlai Mu; Pan Zheng; Ke Lin

doi:10.3934/dcds.2018156

Previous Article
Homoclinic and stable periodic solutions for differential delay equations from physiology
DCDS Home
This Issue
Next Article
Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity

July 2018, 38(7): 3617-3636. doi: 10.3934/dcds.2018156

Global dynamics in a two-species chemotaxis-competition system with two signals

Xinyu Tu ^1,, , Chunlai Mu ^1, , Pan Zheng ^2, and Ke Lin ^3,

1.	College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
2.	Depart. of Appl. Math., Chongqing Univ. of Posts and Telecommun., Chongqing 400065, China
3.	College of Economic Math., Southwestern Univ. of Finance and Economics, Chengdu 611130, China

* Corresponding author: Xinyu Tu

Received November 2017 Revised February 2018 Published April 2018

Fund Project: The first author is partially supported by the China Scholarship Council (201706050065). The second author is partially supported by National Natural Science Foundation of China (Grant Nos: 11771062, 11371384, 11571062), the Basic and Advanced Research Project of CQCSTC (Grant No: cstc2015jcyjBX0007). Fundamental Research Funds for the Central Universities (Grant Nos. 10611CDJXZ238826). The third author is partially supported by National Natural Science Foundation of China (Grant Nos: 11601053, 11526042). The fourth author is partially supported by Chongqing Scientific & Technological Talents Program (Grant No. KJXX2017006).

In this paper, we consider a chemotaxis-competition system of parabolic-elliptic-parabolic-elliptic type

$\begin{eqnarray*}\label{1}\left\{\begin{array}{llll}u_t = Δ u-χ_{1}\nabla·(u\nabla v)+μ_{1}u(1-u-a_{1}w), &x∈ Ω, ~~~t>0, \\0 = Δ v-v+w, &x∈Ω, ~~~t>0, \\w_t = Δ w-χ_{2}\nabla·(w\nabla z)+μ_{2}w(1-a_{2}u-w), &x∈ Ω, ~~~ t>0, \\0 = Δ z-z+u, &x∈Ω, ~~~t>0, \\\end{array}\right.\end{eqnarray*}$

with homogeneous Neumann boundary conditions in an arbitrary smooth bounded domain

$Ω\subset R^n$

$n≥2$

, where

$χ_{i}$

$μ_{i}$

and

$a_{i}$

$(i = 1, 2)$

are positive constants. It is shown that for any positive parameters

$χ_{i}$

$μ_{i}$

$a_{i}$

$(i = 1, 2)$

and any suitably regular initial data

$(u_{0}, w_{0})$

, this system possesses a global bounded classical solution provided that

$\frac{χ_{i}}{μ_{i}}$

are small. Moreover, when

$a_{1}, a_{2}∈ (0, 1)$

and the parameters

$μ_{1}$

and

$μ_{2}$

are sufficiently large, it is proved that the global solution

$(u, v, w, z)$

of this system exponentially approaches to the steady state

$\left(\frac{1-a_{1}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{1}}{1-a_{1}a_{2}}\right)$

in the norm of

$L^{∞}(Ω)$

$t\to ∞$

; If

$a_{1}≥1>a_{2}>0$

and

$μ_{2}$

is sufficiently large, the solution of the system converges to the constant stationary solution

$\left(0, 1, 1, 0\right)$

as time tends to infinity, and the convergence rates can be calculated accurately.

Keywords: Chemotaxis, Lotka-Volterra-type competition, boundedness, Logistic source, asymptotic stability.

Mathematics Subject Classification: Primary: 35B35, 35B34; Secondary: 35K55, 92C17.

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

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.
[2]	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.
[3]	J. Cao, W. Wang and H. Yu, Asymptotic behavior of solutions to two-dimensional chemotaxis system with logistic source and singular sensitivity, J. Math. Anal. Appl., 436 (2016), 382-392. doi: 10.1016/j.jmaa.2015.11.058.
[4]	E. E. Espejo and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34. doi: 10.1016/j.aml.2014.04.007.
[5]	A. Friedman, Partoal Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que-London, 1969.
[6]	K. Fujie, A. Ito, M. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169. doi: 10.3934/dcds.2016.36.151.
[7]	C. Gai, Q. Wang and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 1239-1284. doi: 10.3934/dcds.2015.35.1239.
[8]	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.
[9]	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.
[10]	D. Horstmann, Generaizing 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.
[11]	M. W. Htwe and Y.F Wang, Boundedness in a full parabolic two-species chemotaxis system, C. R. Acad. Sci. Ser. I., 355 (2017), 80-83. doi: 10.1016/j.crma.2016.10.024.
[12]	J. Hu, Q. Wang, J. Yang and L. Zhang, Globale 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.
[13]	H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478. doi: 10.1016/j.jmaa.2014.09.049.
[14]	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.
[15]	R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005.
[16]	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.
[17]	Y. Li, Global bounded solutions and their asymptotic properties under small initial data condition in a two-dimensional chemotaxis system for two species, J. Math. Anal. Appl., 429 (2015), 1291-1304. doi: 10.1016/j.jmaa.2015.04.052.
[18]	K. Lin and C. L. 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.
[19]	K. Lin, C. L. Mu and L. C. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096. doi: 10.1002/mma.3429.
[20]	K. Lin, C. L. Mu and L. C. Wang, Large time behavior for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124. doi: 10.1016/j.jmaa.2014.12.052.
[21]	J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41. doi: 10.1080/17513758.2011.571722.
[22]	P. Liu, J. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597.
[23]	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.
[24]	M. Mizukami, Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system, AIMS Mathematics, 1 (2016), 156-164.
[25]	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.
[26]	M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669. doi: 10.1016/j.jde.2016.05.008.
[27]	T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.
[28]	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.
[29]	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.
[30]	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.
[31]	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.
[32]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441-469.
[33]	K. Osaki, T. Tsujikawa, T. A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Real World Appl., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.
[34]	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.
[35]	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.
[36]	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.
[37]	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.
[38]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. Partial Diff. Eqns., 32 (2007), 849-877. doi: 10.1080/03605300701319003.
[39]	Q. Wang, J. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3547-3574. doi: 10.3934/dcdsb.2017179.
[40]	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.
[41]	M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Diff. Eqns., 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.
[42]	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.
[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.
[44]	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.
[45]	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.
[46]	Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93. doi: 10.1007/s00033-013-0383-4.
[47]	Q. Zhang and Y. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63. doi: 10.1016/j.jmaa.2014.03.084.
[48]	J. Zheng, Boundedness in a two-species quasilinear chemotaxis system with two chemicals, Topol. Methods Nonl. Anal., 49 (2017), 463-480.
[49]	P. Zheng and C. L. 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.
[50]	P. Zheng, C. L. Mu and X. Hu, Persistence property in a two-species chemotaxis system with two signals J. Math. Phys. 58 (2017), 111501, 17pp. doi: 10.1063/1.5010681.
[51]	P. Zheng, C. L. Mu and Y. Mi, Global stability in a two-competing-species chemotaxis system with two chemicals, Diff. Integ. Equa., 31 (2018), 547-558.

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.
[2]	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.
[3]	J. Cao, W. Wang and H. Yu, Asymptotic behavior of solutions to two-dimensional chemotaxis system with logistic source and singular sensitivity, J. Math. Anal. Appl., 436 (2016), 382-392. doi: 10.1016/j.jmaa.2015.11.058.
[4]	E. E. Espejo and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34. doi: 10.1016/j.aml.2014.04.007.
[5]	A. Friedman, Partoal Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que-London, 1969.
[6]	K. Fujie, A. Ito, M. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169. doi: 10.3934/dcds.2016.36.151.
[7]	C. Gai, Q. Wang and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 1239-1284. doi: 10.3934/dcds.2015.35.1239.
[8]	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.
[9]	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.
[10]	D. Horstmann, Generaizing 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.
[11]	M. W. Htwe and Y.F Wang, Boundedness in a full parabolic two-species chemotaxis system, C. R. Acad. Sci. Ser. I., 355 (2017), 80-83. doi: 10.1016/j.crma.2016.10.024.
[12]	J. Hu, Q. Wang, J. Yang and L. Zhang, Globale 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.
[13]	H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478. doi: 10.1016/j.jmaa.2014.09.049.
[14]	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.
[15]	R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005.
[16]	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.
[17]	Y. Li, Global bounded solutions and their asymptotic properties under small initial data condition in a two-dimensional chemotaxis system for two species, J. Math. Anal. Appl., 429 (2015), 1291-1304. doi: 10.1016/j.jmaa.2015.04.052.
[18]	K. Lin and C. L. 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.
[19]	K. Lin, C. L. Mu and L. C. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096. doi: 10.1002/mma.3429.
[20]	K. Lin, C. L. Mu and L. C. Wang, Large time behavior for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124. doi: 10.1016/j.jmaa.2014.12.052.
[21]	J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41. doi: 10.1080/17513758.2011.571722.
[22]	P. Liu, J. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597.
[23]	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.
[24]	M. Mizukami, Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system, AIMS Mathematics, 1 (2016), 156-164.
[25]	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.
[26]	M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669. doi: 10.1016/j.jde.2016.05.008.
[27]	T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.
[28]	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.
[29]	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.
[30]	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.
[31]	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.
[32]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441-469.
[33]	K. Osaki, T. Tsujikawa, T. A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Real World Appl., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.
[34]	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.
[35]	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.
[36]	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.
[37]	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.
[38]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. Partial Diff. Eqns., 32 (2007), 849-877. doi: 10.1080/03605300701319003.
[39]	Q. Wang, J. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3547-3574. doi: 10.3934/dcdsb.2017179.
[40]	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.
[41]	M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Diff. Eqns., 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.
[42]	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.
[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.
[44]	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.
[45]	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.
[46]	Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93. doi: 10.1007/s00033-013-0383-4.
[47]	Q. Zhang and Y. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63. doi: 10.1016/j.jmaa.2014.03.084.
[48]	J. Zheng, Boundedness in a two-species quasilinear chemotaxis system with two chemicals, Topol. Methods Nonl. Anal., 49 (2017), 463-480.
[49]	P. Zheng and C. L. 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.
[50]	P. Zheng, C. L. Mu and X. Hu, Persistence property in a two-species chemotaxis system with two signals J. Math. Phys. 58 (2017), 111501, 17pp. doi: 10.1063/1.5010681.
[51]	P. Zheng, C. L. Mu and Y. Mi, Global stability in a two-competing-species chemotaxis system with two chemicals, Diff. Integ. Equa., 31 (2018), 547-558.

[1]	Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195
[2]	Liangchen Wang, Yuhuan Li, Chunlai Mu. Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 789-802. doi: 10.3934/dcds.2014.34.789
[3]	Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097
[4]	Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324
[5]	Guoqiang Ren, Bin Liu. Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3843-3883. doi: 10.3934/cpaa.2020170
[6]	Pan Zheng, Chunlai Mu, Xuegang Hu. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299
[7]	Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic and Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034
[8]	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
[9]	Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5141-5164. doi: 10.3934/dcds.2021071
[10]	Chunhua Jin. Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3547-3566. doi: 10.3934/dcds.2018150
[11]	Tomomi Yokota, Noriaki Yoshino. Existence of solutions to chemotaxis dynamics with logistic source. Conference Publications, 2015, 2015 (special) : 1125-1133. doi: 10.3934/proc.2015.1125
[12]	Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2211-2236. doi: 10.3934/cpaa.2021064
[13]	Yu Ma, Chunlai Mu, Shuyan Qiu. Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 4077-4095. doi: 10.3934/dcdsb.2021218
[14]	Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503
[15]	Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[16]	Lih-Ing W. Roeger, Razvan Gelca. Dynamically consistent discrete-time Lotka-Volterra competition models. Conference Publications, 2009, 2009 (Special) : 650-658. doi: 10.3934/proc.2009.2009.650
[17]	Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435
[18]	Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953
[19]	Bang-Sheng Han, Zhi-Cheng Wang, Zengji Du. Traveling waves for nonlocal Lotka-Volterra competition systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1959-1983. doi: 10.3934/dcdsb.2020011
[20]	Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197

American Institute of Mathematical Sciences

Global dynamics in a two-species chemotaxis-competition system with two signals

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Global dynamics in a two-species chemotaxis-competition system with two signals

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors