# American Institute of Mathematical Sciences

December  2021, 29(6): 4297-4314. doi: 10.3934/era.2021086

## Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop

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

* Corresponding author: Liangchen Wang

Received  August 2021 Revised  September 2021 Published  December 2021 Early access  October 2021

This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop
 $\begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi_1\nabla\cdot(u\nabla v)+\mu_1 u(1-u-a_1w),\, x\in \Omega,\, t>0,\\ 0 = \Delta v-v+w,\,x\in\Omega,\, t>0,\\ w_t = \Delta w-\chi_2\nabla\cdot(w\nabla z)-\chi_3\nabla\cdot(w\nabla v)+\mu_2 w(1-w-a_2u), \,x\in \Omega,\,t>0,\\ 0 = \Delta z-z+u, \,x\in\Omega,\, t>0, \end{array} \right. \end{eqnarray*}$
under homogeneous Neumann boundary conditions in a bounded domain
 $\Omega\subset \mathbb{R}^n$
with
 $n\geq1$
, where the parameters
 $a_1,a_2$
,
 $\chi_1, \chi_2, \chi_3$
,
 $\mu_1, \mu_2$
are positive constants. We first showed some conditions between
 $\frac{\chi_1}{\mu_1}$
,
 $\frac{\chi_2}{\mu_2}$
,
 $\frac{\chi_3}{\mu_2}$
and other ingredients to guarantee boundedness. Moreover, the large time behavior and rates of convergence have also been investigated under some explicit conditions.
Citation: Rong Zhang, Liangchen Wang. Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop. Electronic Research Archive, 2021, 29 (6) : 4297-4314. doi: 10.3934/era.2021086
##### 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, 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 [4] 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 [5] E. Cruz, M. Negreanu and J. I. 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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] C. Huang, Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop, Elec. Res. Arch. doi: 10.3934/era.2021037.  Google Scholar [11] 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 [12] H. Knútsdóttir, E. Pálsson and L. Edelstein-Keshet, Mathematical model of macrophage-facilitated breast cancer cells invasion, J. Theoret. Biol., 357 (2014), 184-199.  doi: 10.1016/j.jtbi.2014.04.031.  Google Scholar [13] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I. 1968.  Google Scholar [14] 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 [15] K. Lin, C. Mu 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 [16] K. Lin and T. Xiang, On global solutions and blow-up for a short-ranged chemical signaling loop, J. Nonlinear Sci., 29 (2019), 551-591.  doi: 10.1007/s00332-018-9494-6.  Google Scholar [17] K. Lin and T. Xiang, On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop, Calc. Var. Partial Differential Equations, 59 (2020), 1-35.  doi: 10.1007/s00526-020-01777-7.  Google Scholar [18] A. Liu and B. Dai, Boundedness and stabilization in a two-species chemotaxis system with two chemicals, J. Math. Anal. Appl., 506 (2022), 125609.  doi: 10.1016/j.jmaa.2021.125609.  Google Scholar [19] 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 [20] 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 [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] J. D. Murray, Mathematical Biology, 2$^nd$ edition, Biomathematics, 19. Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.  Google Scholar [23] M. Negreanu and J. I. Tello, Global existence and asymptotic behavior of solutions to a predator-prey chemotaxis system with two chemicals, J. Math. Anal. Appl., 474 (2019), 1116-1131.  doi: 10.1016/j.jmaa.2019.02.007.  Google Scholar [24] 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 [25] X. Pan, L. Wang and J. Zhang, Boundedness in a three-dimensional two-species and two-stimuli chemotaxis system with chemical signalling loop, Math. Methods Appl. Sci., 43 (2020), 9529-9542.  doi: 10.1002/mma.6621.  Google Scholar [26] X. Pan, L. Wang, J. Zhang and J. Wang, Boundedness in a three-dimensional two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 71 (2020). doi: 10.1007/s00033-020-1248-2.  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] G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal. Real World Appl., 48 (2019), 288-325.  doi: 10.1016/j.nonrwa.2019.01.017.  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 Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar [32] 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 [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 and S. Qiu, Boundedness and convergence of constant equilibria in a two-species chemotaxis-competition system with loop, Nonlinear Anal., 198 (2020), 111923.  doi: 10.1016/j.na.2020.111923.  Google Scholar [35] X. Tu, C. Mu and S. Qiu, Global asymptotic stability in a parabolic-elliptic chemotaxis system with competitive kinetics and loop, Appl. Anal., 2020. doi: 10.1080/00036811.2020.1783536.  Google Scholar [36] 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 [37] 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.  doi: 10.1016/j.jmaa.2019.123705.  Google Scholar [38] L. Wang and C. Mu, A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 4585-4601.  doi: 10.3934/dcdsb.2020114.  Google Scholar [39] 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 [40] 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 [41] 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 [42] G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.  doi: 10.1017/S0956792501004843.  Google Scholar [43] L. Xie and Y. Wang, Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2717-2729.  doi: 10.3934/dcdsb.2017132.  Google Scholar [44] L. Xie and Y Wang, On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 471 (2019), 584-598.  doi: 10.1016/j.jmaa.2018.10.093.  Google Scholar [45] H. Yu, W. Wang and S. Zheng, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514.  doi: 10.1088/1361-6544/aa96c9.  Google Scholar [46] Q. Zhang, Competitive exclusion for a two-species chemotaxis system with two chemicals, Appl. Math. Lett., 83 (2018), 27-32.  doi: 10.1016/j.aml.2018.03.012.  Google Scholar [47] Q. Zhang, X. Liu and X. Yang, Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J. Math. Phys., 58 (2017), 111504.  doi: 10.1063/1.5011725.  Google Scholar

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##### 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, 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 [4] 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 [5] E. Cruz, M. Negreanu and J. I. Tello, Asymptotic behavior and global existence of solutions to a two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 69 (2018), 20pp. doi: 10.1007/s00033-018-1002-1.  Google Scholar [6] T. Hillen and K. 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] C. Huang, Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop, Elec. Res. Arch. doi: 10.3934/era.2021037.  Google Scholar [11] 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 [12] H. Knútsdóttir, E. Pálsson and L. Edelstein-Keshet, Mathematical model of macrophage-facilitated breast cancer cells invasion, J. Theoret. Biol., 357 (2014), 184-199.  doi: 10.1016/j.jtbi.2014.04.031.  Google Scholar [13] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I. 1968.  Google Scholar [14] 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 [15] K. Lin, C. Mu 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 [16] K. Lin and T. Xiang, On global solutions and blow-up for a short-ranged chemical signaling loop, J. Nonlinear Sci., 29 (2019), 551-591.  doi: 10.1007/s00332-018-9494-6.  Google Scholar [17] K. Lin and T. Xiang, On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop, Calc. Var. Partial Differential Equations, 59 (2020), 1-35.  doi: 10.1007/s00526-020-01777-7.  Google Scholar [18] A. Liu and B. Dai, Boundedness and stabilization in a two-species chemotaxis system with two chemicals, J. Math. Anal. Appl., 506 (2022), 125609.  doi: 10.1016/j.jmaa.2021.125609.  Google Scholar [19] 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 [20] 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 [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] J. D. Murray, Mathematical Biology, 2$^nd$ edition, Biomathematics, 19. Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.  Google Scholar [23] M. Negreanu and J. I. Tello, Global existence and asymptotic behavior of solutions to a predator-prey chemotaxis system with two chemicals, J. Math. Anal. Appl., 474 (2019), 1116-1131.  doi: 10.1016/j.jmaa.2019.02.007.  Google Scholar [24] 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 [25] X. Pan, L. Wang and J. Zhang, Boundedness in a three-dimensional two-species and two-stimuli chemotaxis system with chemical signalling loop, Math. Methods Appl. Sci., 43 (2020), 9529-9542.  doi: 10.1002/mma.6621.  Google Scholar [26] X. Pan, L. Wang, J. Zhang and J. Wang, Boundedness in a three-dimensional two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 71 (2020). doi: 10.1007/s00033-020-1248-2.  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] G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal. Real World Appl., 48 (2019), 288-325.  doi: 10.1016/j.nonrwa.2019.01.017.  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 Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar [32] 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 [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 and S. Qiu, Boundedness and convergence of constant equilibria in a two-species chemotaxis-competition system with loop, Nonlinear Anal., 198 (2020), 111923.  doi: 10.1016/j.na.2020.111923.  Google Scholar [35] X. Tu, C. Mu and S. Qiu, Global asymptotic stability in a parabolic-elliptic chemotaxis system with competitive kinetics and loop, Appl. Anal., 2020. doi: 10.1080/00036811.2020.1783536.  Google Scholar [36] 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 [37] 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.  doi: 10.1016/j.jmaa.2019.123705.  Google Scholar [38] L. Wang and C. Mu, A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 4585-4601.  doi: 10.3934/dcdsb.2020114.  Google Scholar [39] 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 [40] 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. 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