### Electronic Research Archive

2021, Issue 6: 4297-4314. doi: 10.3934/era.2021086

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

• Received: 01 August 2021 Revised: 01 September 2021 Published: 26 October 2021
• Primary: 92C17, 35K35; Secondary: 35B35, 35A01

• 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[J]. Electronic Research Archive, 2021, 29(6): 4297-4314. doi: 10.3934/era.2021086

### Related Papers:

• 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.

 [1] Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics. Indiana Univ. Math. J. (2016) 65: 553-583. [2] Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. (2015) 25: 1663-1763. [3] Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete Contin. Dyn. Syst. Ser. B (2017) 22: 1253-1272. [4] On the weakly competitive case in a two-species chemotaxis model. IMA J. Appl. Math. (2016) 81: 860-876. [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 [6] A user's guide to PDE models for chemotaxis. J. Math. Biol. (2009) 58: 183-217. [7] Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics. J. Differential Equations (2017) 263: 470-490. [8] From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein. (2003) 105: 103-165. [9] 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. (2011) 21: 231-270. [10] C. Huang, Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop, Elec. Res. Arch. doi: 10.3934/era.2021037 [11] Initiation of slime mold aggregation viewed as an instability. J. Theoret. Biol. (1970) 26: 399-415. [12] Mathematical model of macrophage-facilitated breast cancer cells invasion. J. Theoret. Biol. (2014) 357: 184-199. [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. [14] Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source. Discrete Contin. Dyn. Syst. Ser. B (2017) 22: 2233-2260. [15] Boundedness in a two-species chemotaxis system. Math. Methods Appl. Sci. (2015) 38: 5085-5096. [16] On global solutions and blow-up for a short-ranged chemical signaling loop. J. Nonlinear Sci. (2019) 29: 551-591. [17] On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop. Calc. Var. Partial Differential Equations (2020) 59: 1-35. [18] Boundedness and stabilization in a two-species chemotaxis system with two chemicals. J. Math. Anal. Appl. (2022) 506: 125609. [19] Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete Contin. Dyn. Syst. Ser. B (2017) 22: 2301-2319. [20] Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type. Math. Methods Appl. Sci. (2018) 41: 234-249. [21] Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete Contin. Dyn. Syst. Ser. S (2020) 13: 269-278. [22] J. D. Murray, Mathematical Biology, 2$^nd$ edition, Biomathematics, 19. Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869 [23] Global existence and asymptotic behavior of solutions to a predator-prey chemotaxis system with two chemicals. J. Math. Anal. Appl. (2019) 474: 1116-1131. [24] Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis. Bull. Math. Biol. (2009) 71: 1117-1147. [25] Boundedness in a three-dimensional two-species and two-stimuli chemotaxis system with chemical signalling loop. Math. Methods Appl. Sci. (2020) 43: 9529-9542. [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 [27] Global existence and stability in a two-species chemotaxis system. Discrete Contin. Dyn. Syst. Ser. B (2019) 24: 1569-1587. [28] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2007. [29] Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals. Nonlinear Anal. Real World Appl. (2019) 48: 288-325. [30] Competitive exclusion in a two-species chemotaxis model. J. Math. Biol. (2014) 68: 1607-1626. [31] Competing effects of attraction vs. repulsion in chemotaxis. Math. Models Methods Appl. Sci. (2013) 23: 1-36. [32] Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals. Discrete Contin. Dyn. Syst.-Ser. B (2015) 20: 3165-3183. [33] Stabilization in a two-species chemotaxis system with a logistic source. Nonlinearity (2012) 25: 1413-1425. [34] Boundedness and convergence of constant equilibria in a two-species chemotaxis-competition system with loop. Nonlinear Anal. (2020) 198: 111923. [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 [36] Global dynamics in a two-species chemotaxis-competition system with two signals. Discrete Contin. Dyn. Syst. (2018) 38: 3617-3636. [37] Improvement of conditions for boundedness in a two-species chemotaxis competition system of parabolic-parabolic-elliptic type. J. Math. Anal. Appl. (2020) 484: 123705. [38] A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete Contin. Dyn. Syst. Ser. B. (2020) 25: 4585-4601. [39] Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant. J. Differential Equations (2018) 264: 3369-3401. [40] Boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete Contin. Dyn. Syst. Ser. B. (2020) 25: 191-221. [41] Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Comm. Partial Differential Equations (2010) 35: 1516-1537. [42] Multi-components chemotactic system in the absence of conflicts. European J. Appl. Math. (2002) 13: 641-661. [43] Boundedness in a two-species chemotaxis parabolic system with two chemicals. Discrete Contin. Dyn. Syst. Ser. B (2017) 22: 2717-2729. [44] On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics. J. Math. Anal. Appl. (2019) 471: 584-598. [45] Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals. Nonlinearity (2018) 31: 502-514. [46] Competitive exclusion for a two-species chemotaxis system with two chemicals. Appl. Math. Lett. (2018) 83: 27-32. [47] Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals. J. Math. Phys. (2017) 58: 111504.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.604 0.8