December 2017, 22(10): 3839-3874. doi: 10.3934/dcdsb.2017193

Asymptotic dynamics in a two-species chemotaxis model with non-local terms

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

* Corresponding author: Tahir Bachar Issa

Received  January 2017 Revised  May 2017 Published  July 2017

In this study, we consider an extended attraction two species chemotaxis system of parabolic-parabolic-elliptic type with nonlocal terms under homogeneous Neuman boundary conditions in a bounded domain $Ω \subset \mathbb{R}^n(n≥1)$ with smooth boundary. We first prove the global existence of non-negative classical solutions for various explicit parameter regions. Next, under some further explicit conditions on the coefficients and on the chemotaxis sensitivities, we show that the system has a unique positive constant steady state solution which is globally asymptotically stable. Finally, we also find some explicit conditions on the coefficient and on the chemotaxis sensitivities for which the phenomenon of competitive exclusion occurs in the sense that as time goes to infinity, one of the species dies out and the other reaches its carrying capacity. The method of eventual comparison is used to study the asymptotic behavior.

Citation: 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
References:
[1]

E. E. E. ArenasA. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338. doi: 10.1524/anly.2009.1029.

[2]

N. BellomoA. BellouquidY. 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.

[3]

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.

[4]

T. BlackJ. 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.

[5]

C. ConcaE. Espejo and K. Vilches, Remarks on the blow-up and global existence for a two species chemotactic Keller-Segel system in $\mathbb{R} ^2$, European J. Appl. Math., 22 (2011), 553-580. doi: 10.1017/S0956792511000258.

[6]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 24 (1997), 633-683.

[7]

T. Hillen and K. J. Painter, A users guide to PDE models for chemotaxis, Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[8]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I. Jber. DMW, 105 (2003), 103-165.

[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 Sc., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.

[10]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363.

[11]

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.

[12]

M. Isenbach, Chemotaxis, Imperial College Press, London, 2004.

[13]

T. B. Issa and W. Shen, Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst., 16 (2017), 926-973. doi: 10.1137/16M1092428.

[14]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.

[15]

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.

[16]

E. F. Keller and L. A. Segel, A model for chemotaxis, J.Theoret. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.

[17]

M. J. Kennedy and J. G. Lawless, Role of Chemotaxis in the Ecology of Denitrifiers, Applied and Environmental Microbiology, 49 (1985), 109-114.

[18]

A. Kurganov and M. Lukáčová-Medviđová, Numerical study of two-species chemotaxis models, Discrete Contin.Dyn. Syst. Ser. B, 19 (2014), 131-152. doi: 10.3934/dcdsb.2014.19.131.

[19]

K. KutoK. OsakiT. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxisdiffusion-growth model, Physica D, 241 (2012), 1629-1639. doi: 10.1016/j.physd.2012.06.009.

[20]

D. A. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics, Microbial. Ecol., 22 (1991), 175-185. doi: 10.1007/BF02540222.

[21]

Y. LouY. Tao and M. Winkler, Approaching the ideal free distribution in two-species competition models with fitness-dependent dispersal, SIAM J. Math. Anal., 46 (2014), 1228-1262. doi: 10.1137/130934246.

[22]

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.

[23]

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.

[24]

M. Negreanu and J. I. Tello, On a competitive system under chemotaxis effects with non-local terms, Nonlinearity, 26 (2013), 1083-1103. doi: 10.1088/0951-7715/26/4/1083.

[25]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Eq., 258 (2015), 1592-1617. doi: 10.1016/j.jde.2014.11.009.

[26]

M. Negreanu and J. I. Tello, On a comparison method to reaction-diffusion systems and its applications to chemotaxis, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2669-2688. doi: 10.3934/dcdsb.2013.18.2669.

[27]

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.

[28]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469.

[29]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011.

[30]

K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, Journal of Theoretical Biology, 225 (2003), 327-339. doi: 10.1016/S0022-5193(03)00258-3.

[31]

R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $R^N$, J. Differential Equations, 262 (2017), 5635-5690. doi: 10.1016/j.jde.2017.02.011.

[32]

R. B. Salako and W. Shen, Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$. https://arxiv.org/pdf/1609.05387.pdf

[33]

C. StinnerJ. I. Tello and W. Winkler, Competive exclusion in a two-species chemotaxis, J.Math.Biol., 68 (2014), 1607-1626. doi: 10.1007/s00285-013-0681-7.

[34]

J. I. Tello and W. Winkler, A chemotaxis system with logistic source, Common Partial Diff. Eq., 32 (2007), 849-877. doi: 10.1080/03605300701319003.

[35]

J. I. Tello and M. Winkler, Stabilization in two-species chemotaxis with a logistic source, Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413.

[36]

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, ArXiv preprints, May 2015.

[37]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057.

[38]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, Journal of Differential Equations, 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023.

show all references

References:
[1]

E. E. E. ArenasA. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338. doi: 10.1524/anly.2009.1029.

[2]

N. BellomoA. BellouquidY. 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.

[3]

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.

[4]

T. BlackJ. 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.

[5]

C. ConcaE. Espejo and K. Vilches, Remarks on the blow-up and global existence for a two species chemotactic Keller-Segel system in $\mathbb{R} ^2$, European J. Appl. Math., 22 (2011), 553-580. doi: 10.1017/S0956792511000258.

[6]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 24 (1997), 633-683.

[7]

T. Hillen and K. J. Painter, A users guide to PDE models for chemotaxis, Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[8]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I. Jber. DMW, 105 (2003), 103-165.

[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 Sc., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.

[10]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363.

[11]

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.

[12]

M. Isenbach, Chemotaxis, Imperial College Press, London, 2004.

[13]

T. B. Issa and W. Shen, Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst., 16 (2017), 926-973. doi: 10.1137/16M1092428.

[14]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.

[15]

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.

[16]

E. F. Keller and L. A. Segel, A model for chemotaxis, J.Theoret. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.

[17]

M. J. Kennedy and J. G. Lawless, Role of Chemotaxis in the Ecology of Denitrifiers, Applied and Environmental Microbiology, 49 (1985), 109-114.

[18]

A. Kurganov and M. Lukáčová-Medviđová, Numerical study of two-species chemotaxis models, Discrete Contin.Dyn. Syst. Ser. B, 19 (2014), 131-152. doi: 10.3934/dcdsb.2014.19.131.

[19]

K. KutoK. OsakiT. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxisdiffusion-growth model, Physica D, 241 (2012), 1629-1639. doi: 10.1016/j.physd.2012.06.009.

[20]

D. A. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics, Microbial. Ecol., 22 (1991), 175-185. doi: 10.1007/BF02540222.

[21]

Y. LouY. Tao and M. Winkler, Approaching the ideal free distribution in two-species competition models with fitness-dependent dispersal, SIAM J. Math. Anal., 46 (2014), 1228-1262. doi: 10.1137/130934246.

[22]

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.

[23]

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.

[24]

M. Negreanu and J. I. Tello, On a competitive system under chemotaxis effects with non-local terms, Nonlinearity, 26 (2013), 1083-1103. doi: 10.1088/0951-7715/26/4/1083.

[25]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Eq., 258 (2015), 1592-1617. doi: 10.1016/j.jde.2014.11.009.

[26]

M. Negreanu and J. I. Tello, On a comparison method to reaction-diffusion systems and its applications to chemotaxis, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2669-2688. doi: 10.3934/dcdsb.2013.18.2669.

[27]

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.

[28]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469.

[29]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011.

[30]

K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, Journal of Theoretical Biology, 225 (2003), 327-339. doi: 10.1016/S0022-5193(03)00258-3.

[31]

R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $R^N$, J. Differential Equations, 262 (2017), 5635-5690. doi: 10.1016/j.jde.2017.02.011.

[32]

R. B. Salako and W. Shen, Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$. https://arxiv.org/pdf/1609.05387.pdf

[33]

C. StinnerJ. I. Tello and W. Winkler, Competive exclusion in a two-species chemotaxis, J.Math.Biol., 68 (2014), 1607-1626. doi: 10.1007/s00285-013-0681-7.

[34]

J. I. Tello and W. Winkler, A chemotaxis system with logistic source, Common Partial Diff. Eq., 32 (2007), 849-877. doi: 10.1080/03605300701319003.

[35]

J. I. Tello and M. Winkler, Stabilization in two-species chemotaxis with a logistic source, Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413.

[36]

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, ArXiv preprints, May 2015.

[37]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057.

[38]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, Journal of Differential Equations, 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023.

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