# American Institute of Mathematical Sciences

April  2019, 24(4): 1569-1587. doi: 10.3934/dcdsb.2018220

## Global existence and stability in a two-species chemotaxis system

 College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China

* Corresponding author: S. Guo

Received  December 2017 Revised  March 2018 Published  April 2019 Early access  June 2018

Fund Project: The second author is supported by NSF of China (Grants No. 11671123).

This paper deals with the following two-species chemotaxis system
 $\left\{ \begin{array}{*{35}{l}} \ \ {{u}_{t}}=\Delta u-{{\chi }_{1}}\nabla \cdot (u\nabla v)+{{\mu }_{1}}u(1-u-{{a}_{1}}w), & x\in \Omega ,t>0, & \\ \ \ {{v}_{t}}=\Delta v-v+h(w), & x\in \Omega ,t>0, & \\ \ \ {{w}_{t}}=\Delta w-{{\chi }_{2}}\nabla \cdot (w\nabla z)+{{\mu }_{2}}w(1-w-{{a}_{2}}u), & x\in \Omega ,t>0, & \\ \ \ {{z}_{t}}=\Delta z-z+h(u),& x\in \Omega ,t>0, & \\\end{array} \right.$
under homogeneous Neumann boundary conditions in a bounded domain
 $Ω\subset\mathbb{R}^{n}$
with smooth boundary. The parameters in the system are positive and the signal production function h is a prescribed C1-regular function. The main objectives of this paper are two-fold: One is the existence and boundedness of global solutions, the other is the large time behavior of the global bounded solutions in three competition cases (i.e., a weak competition case, a partially strong competition case and a fully strong competition case). It is shown that the unique positive spatially homogeneous equilibrium
 $(u_{*}, v_{*}, w_{*}, z_{*})$
may be globally attractive in the weak competition case (i.e.,
 $0 < a_{1}, a_{2} < 1$
), while the constant stationary solution (0, h(1), 1, 0) may be globally attractive and globally stable in the partially strong competition case (i.e.,
 $a_{1}>1>a_{2}>0$
). In the fully strong competition case (i.e.
 $a_{1}, a_{2}>1$
), however, we can only obtain the local stability of the two semi-trivial stationary solutions (0, h(1), 1, 0) and (1, 0, 0, h(1)) and the instability of the positive spatially homogeneous
 $(u_{*}, v_{*}, w_{*}, z_{*})$
. The matter which species ultimately wins out depends crucially on the starting advantage each species has.
Citation: Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220
##### 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, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061. [3] M. A. J. Chaplain and J. I. Tello, On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Applied Mathematics Letters, 57 (2016), 1-6.  doi: 10.1016/j.aml.2015.12.001. [4] X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, Stability of spiky solution of Keller-Segel's minimal chemotaxis model, Journal of Differential Equations, 257 (2014), 3102-3134.  doi: 10.1016/j.jde.2014.06.008. [5] A.-K. Drangeid, The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Analysis: Theory, Methods & Applications, 13 (1989), 1091-1113.  doi: 10.1016/0362-546X(89)90097-7. [6] S. Guo, Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Analysis: Real World Applications, 42 (2018), 448-477.  doi: 10.1016/j.nonrwa.2018.01.011. [7] S. Guo and S. Yan, Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, Journal of Differential Equations, 260 (2016), 781-817.  doi: 10.1016/j.jde.2015.09.031. [8] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3. [9] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences ⅱ, Jahresber Deutsch. Math.-Verein., 106 (2004), 51-69. [10] K. Kang and A. Stevens, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017. [11] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [12] K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D: Nonlinear Phenomena, 241 (2012), 1629-1639.  doi: 10.1016/j.physd.2012.06.009. [13] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, Journal of Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016. [14] J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete and Continuous Dynamical Systems - Series B, 20 (2017), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499. [15] D. Li and S. Guo, Bifurcation and stability of a Mimura-Tsujikawa model with nonlocal delay effect, Mathematical Methods in the Applied Sciences, 40 (2017), 2219-2247. [16] D. Li and S. Guo, Stability and Hopf bifurcation in a reaction-diffusion model with chemotaxis and nonlocal delay effect, International Journal of Bifurcation and Chaos, 28 (2018), 1850046.  doi: 10.1142/S0218127418500463. [17] P. L. Lions, Résolution de problemes elliptiques quasilinéaires, Archive for Rational Mechanics and Analysis, 74 (1980), 335-353.  doi: 10.1007/BF00249679. [18] D. Liu and Y. Tao, Global boundedness in a fully parabolic attractionrepulsion chemotaxis model, Mathematical Methods in the Applied Sciences, 38 (2015), 2537-2546.  doi: 10.1002/mma.3240. [19] A. Lunardi, Asymptotic exponential stability in quasilinear parabolic equations, Nonlinear Analysis: Theory, Methods & Applications, 9 (1985), 563-586.  doi: 10.1016/0362-546X(85)90041-0. [20] J. D. Murray, Mathematical Biology. Ⅱ Spatial Models and Biomedical Applications {Interdisciplinary Applied Mathematics V. 18}. Springer-Verlag, New York, 2003. [21] E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete and Continuous Dynamical Systems - Series B, 18 (2014), 2627-2646.  doi: 10.3934/dcdsb.2013.18.2627. [22] E. Nakaguchi and K. Osaki, Lp-estimates of solutions to n-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcialaj Ekvacioj, 59 (2016), 51-66.  doi: 10.1619/fesi.59.51. [23] E. Nakaguchi and K. Osaki, et al., Global existence of solutions to an n-dimensional parabolicparabolic system for chemotaxis with logistic-type growth and superlinear production, Osaka Journal of Mathematics, 55 (2018), 51-70. [24] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis: Theory, Methods & Applications, 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X. [25] K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D: Nonlinear Phenomena, 240 (2011), 363-375.  doi: 10.1016/j.physd.2010.09.011. [26] C. G. Simader, The weak Dirichlet and Neumann problem for the Laplacian in Lq for bounded and exterior domains. applications, In Nonlinear Analysis, Function Spaces and Applications Vol. 4, Springer, 119 (1990), 180-223. [27] C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, Journal of Mathematical Biology, 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7. [28] Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165. [29] Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Zeitschrift für angewandte Mathematik und Physik, 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1. [30] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [31] 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, Journal of Differential Equations, 264 (2018), 3369-3401.  doi: 10.1016/j.jde.2017.11.019. [32] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008. [33] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Communications in Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [34] M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis & Applications, 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057. [35] M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, Journal of Nonlinear Science, 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x. [36] M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Zeitschrift für angewandte Mathematik und Physik, 69 (2018), Art. 69, 40 pp. doi: 10.1007/s00033-018-0935-8. [37] S. Yan and S. Guo, Bifurcation phenomena in a Lotka-Volterra model with cross-diffusion and delay effect, International Journal of Bifurcation and Chaos 27 (2017), 1750105, 24pp. doi: 10.1142/S021812741750105X. [38] P. Zheng, C. Mu, R. Willie and X. Hu, Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity, Computers & Mathematics with Applications, 75 (2018), 1667-1675.  doi: 10.1016/j.camwa.2017.11.032. [39] R. Zou and S. Guo, Bifurcation of reaction cross-diffusion systems, International Journal of Bifurcation and Chaos, 27 (2017), 1750049, 22pp.  doi: 10.1142/S0218127417500493.

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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. [2] T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061. [3] M. A. J. Chaplain and J. I. Tello, On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Applied Mathematics Letters, 57 (2016), 1-6.  doi: 10.1016/j.aml.2015.12.001. [4] X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, Stability of spiky solution of Keller-Segel's minimal chemotaxis model, Journal of Differential Equations, 257 (2014), 3102-3134.  doi: 10.1016/j.jde.2014.06.008. [5] A.-K. Drangeid, The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Analysis: Theory, Methods & Applications, 13 (1989), 1091-1113.  doi: 10.1016/0362-546X(89)90097-7. [6] S. Guo, Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Analysis: Real World Applications, 42 (2018), 448-477.  doi: 10.1016/j.nonrwa.2018.01.011. [7] S. Guo and S. Yan, Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, Journal of Differential Equations, 260 (2016), 781-817.  doi: 10.1016/j.jde.2015.09.031. [8] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3. [9] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences ⅱ, Jahresber Deutsch. Math.-Verein., 106 (2004), 51-69. [10] K. Kang and A. Stevens, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017. [11] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [12] K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D: Nonlinear Phenomena, 241 (2012), 1629-1639.  doi: 10.1016/j.physd.2012.06.009. [13] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, Journal of Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016. [14] J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete and Continuous Dynamical Systems - Series B, 20 (2017), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499. [15] D. Li and S. Guo, Bifurcation and stability of a Mimura-Tsujikawa model with nonlocal delay effect, Mathematical Methods in the Applied Sciences, 40 (2017), 2219-2247. [16] D. Li and S. Guo, Stability and Hopf bifurcation in a reaction-diffusion model with chemotaxis and nonlocal delay effect, International Journal of Bifurcation and Chaos, 28 (2018), 1850046.  doi: 10.1142/S0218127418500463. [17] P. L. Lions, Résolution de problemes elliptiques quasilinéaires, Archive for Rational Mechanics and Analysis, 74 (1980), 335-353.  doi: 10.1007/BF00249679. [18] D. Liu and Y. Tao, Global boundedness in a fully parabolic attractionrepulsion chemotaxis model, Mathematical Methods in the Applied Sciences, 38 (2015), 2537-2546.  doi: 10.1002/mma.3240. [19] A. Lunardi, Asymptotic exponential stability in quasilinear parabolic equations, Nonlinear Analysis: Theory, Methods & Applications, 9 (1985), 563-586.  doi: 10.1016/0362-546X(85)90041-0. [20] J. D. Murray, Mathematical Biology. Ⅱ Spatial Models and Biomedical Applications {Interdisciplinary Applied Mathematics V. 18}. Springer-Verlag, New York, 2003. [21] E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete and Continuous Dynamical Systems - Series B, 18 (2014), 2627-2646.  doi: 10.3934/dcdsb.2013.18.2627. [22] E. Nakaguchi and K. Osaki, Lp-estimates of solutions to n-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcialaj Ekvacioj, 59 (2016), 51-66.  doi: 10.1619/fesi.59.51. [23] E. Nakaguchi and K. Osaki, et al., Global existence of solutions to an n-dimensional parabolicparabolic system for chemotaxis with logistic-type growth and superlinear production, Osaka Journal of Mathematics, 55 (2018), 51-70. [24] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis: Theory, Methods & Applications, 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X. [25] K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D: Nonlinear Phenomena, 240 (2011), 363-375.  doi: 10.1016/j.physd.2010.09.011. [26] C. G. Simader, The weak Dirichlet and Neumann problem for the Laplacian in Lq for bounded and exterior domains. applications, In Nonlinear Analysis, Function Spaces and Applications Vol. 4, Springer, 119 (1990), 180-223. [27] C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, Journal of Mathematical Biology, 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7. [28] Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165. [29] Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Zeitschrift für angewandte Mathematik und Physik, 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1. [30] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [31] 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, Journal of Differential Equations, 264 (2018), 3369-3401.  doi: 10.1016/j.jde.2017.11.019. [32] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008. [33] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Communications in Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [34] M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis & Applications, 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057. [35] M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, Journal of Nonlinear Science, 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x. [36] M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Zeitschrift für angewandte Mathematik und Physik, 69 (2018), Art. 69, 40 pp. doi: 10.1007/s00033-018-0935-8. [37] S. Yan and S. Guo, Bifurcation phenomena in a Lotka-Volterra model with cross-diffusion and delay effect, International Journal of Bifurcation and Chaos 27 (2017), 1750105, 24pp. doi: 10.1142/S021812741750105X. [38] P. Zheng, C. Mu, R. Willie and X. Hu, Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity, Computers & Mathematics with Applications, 75 (2018), 1667-1675.  doi: 10.1016/j.camwa.2017.11.032. [39] R. Zou and S. Guo, Bifurcation of reaction cross-diffusion systems, International Journal of Bifurcation and Chaos, 27 (2017), 1750049, 22pp.  doi: 10.1142/S0218127417500493.
Solutions of model (3) tend to a positive steady state with parameters (53) and initial condition (54)
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