# American Institute of Mathematical Sciences

July  2022, 27(7): 4077-4095. doi: 10.3934/dcdsb.2021218

## Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model

 1 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 2 School of Sciences, Southwest Petroleum University, Chengdu 610500, China

* Corresponding author: Yu Ma

Received  May 2021 Revised  July 2021 Published  July 2022 Early access  September 2021

This work deals with a Neumann initial-boundary value problem for a two-species predator-prey chemotaxis system
 $\begin{eqnarray*} \left\{ \begin{array}{llll} u_t = d_1\Delta u-\chi\nabla\cdot(u\nabla w)+u(\lambda-u+av),\quad &x\in \Omega,\quad t>0,\\ v_t = d_2\Delta v+\xi\nabla\cdot(v\nabla w)+v(\mu-v-bu),\quad &x\in \Omega,\quad t>0,\\ 0 = d_3\Delta w-\alpha w+\beta_1 u+ \beta_2 v,\quad &x\in\Omega,\quad t>0,\\ \end{array} \right. \end{eqnarray*}$
in a bounded domain
 $\Omega\subset \mathbb{R}^n \,\,(n = 2,3)$
with smooth boundary
 $\partial\Omega$
, where the parameters
 $d_1, d_2, d_3,\chi, \xi,\lambda,\mu,\alpha,\beta_1,\beta_2, a, b$
are positive. It is shown that for any appropriate regular initial date
 $u_0$
,
 $v_0$
, the corresponding system possesses a global bounded classical solution in
 $n = 2$
, and also in
 $n = 3$
for
 $\chi$
being sufficiently small. Moreover, by constructing some suitable functionals, it is proved that if
 $b\lambda<\mu$
and the parameters
 $\chi$
and
 $\xi$
are sufficiently small, then the solution
 $(u,v,w)$
of this system converges to
 $(\frac{\lambda+a\mu}{1+ab}, \frac{\mu-b\lambda}{1+ab}, \frac{\beta_1(\lambda+a\mu)+\beta_2(\mu-b\lambda)}{\alpha(1+ab)})$
exponentially as
 $t\rightarrow \infty$
; if
 $b\lambda\geq \mu$
and
 $\chi$
is sufficiently small and
 $\xi$
is arbitrary, then the solution
 $(u,v,w)$
converges to
 $(\lambda,0,\frac{\beta_1\lambda}{\alpha})$
with exponential decay when
 $b\lambda> \mu$
, and with algebraic decay when
 $b\lambda = \mu$
.
Citation: 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
##### References:
 [1] 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. [2] 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. [3] X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891. [4] T. Cie$\acute{s}$lak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonliearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009. [5] E. Espejo, A. Stevens and J. L Velzquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.  doi: 10.1524/anly.2009.1029. [6] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. [7] S. Fu and L. Miao, Global existence and asymptotic stability in a predator-prey chemotaxis model, Nonlinear Anal. Real World Appl., 54 (2020), 103079.  doi: 10.1016/j.nonrwa.2019.103079. [8] H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. [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. [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 chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022. [12] 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. [13] M. Herrero and J. Vel$\acute{a}$zquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. [14] E. Keller and L. 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. [15] 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. [16] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016. [17] Y. Li, Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetic, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5461-5480.  doi: 10.3934/dcdsb.2019066. [18] X. Li 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. [19] 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. [20] K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.  doi: 10.3934/dcds.2016018. [21] 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. [22] K. Lin, C. Mu and H. Zhong, A new approach toward stabilization in a two-species chemotaxis model with logistic source, Comput. Math. Appl., 75 (2018), 837-849.  doi: 10.1016/j.camwa.2017.10.007. [23] G. Li, Y. Tao and M. Winkler, Large time behavior in a predator-prey system with indirect pursuit-evasion interaction, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4383-4396.  doi: 10.3934/dcdsb.2020102. [24] 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. [25] 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. [26] 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. [27] L. Miao, H. Yang and S. Fu, Global boundedness in a two-species predator-prey chemotaxis model, Appl. Math. Lett., 111 (2021), 106639.  doi: 10.1016/j.aml.2020.106639. [28] T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. [29] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [30] M. Negreanu and J. 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. [31] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. [32] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X. [33] C. Stinner, J. 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. [34] S. Qiu, C. Mu and X. Tu, Global dynamics of a two-species chemotaxis-consumption system with signal-dependent motilities, Nonlinear Anal. Real World Appl., 57 (2021), 103190.  doi: 10.1016/j.nonrwa.2020.103190. [35] X. Tu, C. Mu, S. Qiu and L. Yang, Boundedness in the higher-dimensional fully parabolic chemotaxis-competition system with loop, Z. Angew. Math. Phys., 71 (2020), 18pp. doi: 10.1007/s00033-020-01413-6. [36] 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. [37] J. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [38] J. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonliearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413. [39] Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonliearity, 21 (2008), 2221-2238.  doi: 10.1088/0951-7715/21/10/002. [40] Y. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. Real World Appl., 11 (2010), 2056-2064.  doi: 10.1016/j.nonrwa.2009.05.005. [41] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019. [42] 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. [43] Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), 6142-6161.  doi: 10.1016/j.jde.2015.07.019. [44] 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. [45] 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. [46] 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. [47] M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057. [48] 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. [49] 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. [50] M. Winkler, How far can chemotaxis can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x.

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##### References:
 [1] 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. [2] 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. [3] X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891. [4] T. Cie$\acute{s}$lak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonliearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009. [5] E. Espejo, A. Stevens and J. L Velzquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.  doi: 10.1524/anly.2009.1029. [6] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. [7] S. Fu and L. Miao, Global existence and asymptotic stability in a predator-prey chemotaxis model, Nonlinear Anal. Real World Appl., 54 (2020), 103079.  doi: 10.1016/j.nonrwa.2019.103079. [8] H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. [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. [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 chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022. [12] 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. [13] M. Herrero and J. Vel$\acute{a}$zquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. [14] E. Keller and L. 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. [15] 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. [16] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016. [17] Y. Li, Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetic, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5461-5480.  doi: 10.3934/dcdsb.2019066. [18] X. Li 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. [19] 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. [20] K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.  doi: 10.3934/dcds.2016018. [21] 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. [22] K. Lin, C. Mu and H. Zhong, A new approach toward stabilization in a two-species chemotaxis model with logistic source, Comput. Math. Appl., 75 (2018), 837-849.  doi: 10.1016/j.camwa.2017.10.007. [23] G. Li, Y. Tao and M. Winkler, Large time behavior in a predator-prey system with indirect pursuit-evasion interaction, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4383-4396.  doi: 10.3934/dcdsb.2020102. [24] 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. [25] 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. [26] 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. [27] L. Miao, H. Yang and S. Fu, Global boundedness in a two-species predator-prey chemotaxis model, Appl. Math. Lett., 111 (2021), 106639.  doi: 10.1016/j.aml.2020.106639. [28] T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. [29] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [30] M. Negreanu and J. 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. [31] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. [32] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X. [33] C. Stinner, J. 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. [34] S. Qiu, C. Mu and X. Tu, Global dynamics of a two-species chemotaxis-consumption system with signal-dependent motilities, Nonlinear Anal. Real World Appl., 57 (2021), 103190.  doi: 10.1016/j.nonrwa.2020.103190. [35] X. Tu, C. Mu, S. Qiu and L. Yang, Boundedness in the higher-dimensional fully parabolic chemotaxis-competition system with loop, Z. Angew. Math. Phys., 71 (2020), 18pp. doi: 10.1007/s00033-020-01413-6. [36] 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. [37] J. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [38] J. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonliearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413. [39] Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonliearity, 21 (2008), 2221-2238.  doi: 10.1088/0951-7715/21/10/002. [40] Y. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. Real World Appl., 11 (2010), 2056-2064.  doi: 10.1016/j.nonrwa.2009.05.005. [41] Y. Tao and M. 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