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

Yu Ma; Chunlai Mu; Shuyan Qiu

doi:10.3934/dcdsb.2021218

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doi: 10.3934/dcdsb.2021218

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

Yu Ma ^1,, , Chunlai Mu ^1, and Shuyan Qiu ^2,

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 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 $

Keywords: chemotaxis, predator-prey, boundedness, logistic source, asymptotic stability.

Mathematics Subject Classification: Primary: 35A01, 35B40; Secondary: 35Q92, 92C17.

Citation: Yu Ma, Chunlai Mu, Shuyan Qiu. Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021218

References:

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[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. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[28]	T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. Google Scholar
[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. Google Scholar
[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. Google Scholar
[31]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[37]	J. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. Google Scholar
[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. Google Scholar
[8]	H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. 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]	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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. Google Scholar
[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. Google Scholar
[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. Google Scholar
[31]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[37]	J. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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