August  2017, 22(6): 2233-2260. doi: 10.3934/dcdsb.2017094

Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source

College of Mathematics and Statistics, Chongqing University, Chongqing University, Chongqing 401331, China

Ke Lin, E-mail address: shuxuelk@126.com

Received  February 2016 Revised  June 2017 Published  March 2017

Fund Project: The first author is partially supported by Chongqing graduate student research innovation project (Grant No. CYB15042) and Chongqing Nova program, and the second author is partially supported by NSFC (Grant No. 11371384 and 11571062) and the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007)

In this paper, we consider a system of three parabolic equations in high-dimensional smoothly bounded domain
$\left\{\begin{array}{llll}u_t=\Delta u-\chi_1\nabla\cdot( u\nabla w)+\mu_1u(1-u-a_1v),\quad &x\in \Omega,\quad t>0,\\v_t=\Delta v-\chi_2\nabla\cdot( v\nabla w)+\mu_2v(1-a_2u-v),\quad &x\in\Omega,\quad t>0,\\w_t=\Delta w- w+u+v,\quad &x\in\Omega,\quad t>0,\\\end{array}\right.$
which describes the mutual competition between two populations on account of the Lotka-Volterra dynamics.
For any cross-diffusivities $\chi_1>0$ and $\chi_2>0$ and the rates $a_1>0$ and $a_2>0$, it is proved that the global classical bounded solutions exist for sufficiently regular initial data when the parameters $\mu_1$ and $\mu_2$ are sufficiently large. In deriving the convergence of solutions to this system, we need to distinguish two cases $a_1, a_2\in[0, 1)$ and $a_1>1$ and $0\leq a_2 < 1$ to prove globally asymptotic stability.
Citation: Ke Lin, Chunlai Mu. Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2233-2260. doi: 10.3934/dcdsb.2017094
References:
[1]

S. Ahmad, On the nonautonomous Volterra-Lotka competition equations, Proc. Amer. Math. Soc, 117 (1993), 199-204.   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.   Google Scholar

[3]

P. BilerE. Espejo and I. Guerra, Blow up in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.   Google Scholar

[4]

P. Brown, Decay to uniform states in ecological interactions, SIAM J. Appl. Math., 38 (1980), 22-37.   Google Scholar

[5]

X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Cont. Dyns. S-A., 35 (2015), 1891-1904.   Google Scholar

[6]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.   Google Scholar

[7]

C. ConcaE. Espejo and K. Vilches, Remarks on the blow up and global existence for a two species chemotactic Keller-Segel system in R2, European J. Appl. Math., 22 (2011), 553-580.   Google Scholar

[8]

P. de Mottoni, Qualitative Analysis for Some Quasilinear Parabolic Systems, Inst Math Pol Acad Sci, 1979. Google Scholar

[9]

E. EspejoA. Stevens and J. Velzquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.   Google Scholar

[10]

A. FasanoA. Mancini and M. Primicerio, Equilibrium of two populations subjected to chemotaxis, Math. Mod Meth Appl. Sci., 14 (2004), 503-533.   Google Scholar

[11]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163.   Google Scholar

[12]

C. GaiQ. Wang and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284.   Google Scholar

[13]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.   Google Scholar

[14]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Deutsch. Math.-Verien., 105 (2003), 103-165.   Google Scholar

[15]

D. Horstemann, 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.   Google Scholar

[16]

D. Horstemann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.   Google Scholar

[17]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.   Google Scholar

[18]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolicparabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.   Google Scholar

[19]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Am. Math. Soc., 329 (1992), 819-824.   Google Scholar

[20]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.   Google Scholar

[21]

F. KellyK. Dapsis and D. Lauffenburger, Effects of bacterial chemotaxis on dynamics of microbial competition, Microb Ecol., 16 (1988), 115-131.   Google Scholar

[22]

R. Kowalczyk and Z. Szymánska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.   Google Scholar

[23]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.   Google Scholar

[24]

D. LauffenburgerR. Aris and K. Keller, Effects of cell motility and chemotaxis on microbial population growth, Biophys J., 40 (1982), 209-219.   Google Scholar

[25]

Y. Li and Y. Li, Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84.   Google Scholar

[26]

K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.   Google Scholar

[27]

K. LinC. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.   Google Scholar

[28]

Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.   Google Scholar

[29]

Y. Lou and M. Winkler, Global existence and uniform boundedness of smooth solutions to a cross-diffusion system with equal diffusion rates, Comm. Partial Differential Equations, 40 (2015), 1905-1941.   Google Scholar

[30]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincare Anal. Non Lineaire, 31 (2014), 851-875.   Google Scholar

[31]

M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669.   Google Scholar

[32]

J. Murray, Mathematical Biology, 2nd edition, Biomathematics series, Springer, Berlin, 1993. Google Scholar

[33]

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

[34]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj. Ser. Int., 40 (1997), 411-433.   Google Scholar

[35]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. MAth. Anal., 46 (2014), 3761-3781.   Google Scholar

[36]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxisgrowth system of equations, Nonlinear Anal., 51 (2002), 119-144.   Google Scholar

[37]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441-469.   Google Scholar

[38]

S. Shim, Uniform boundedness and convergence of solutions to cross-diffusion systems, J. Differential Equations, 185 (2002), 281-305.   Google Scholar

[39]

C. StinnerC. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.   Google Scholar

[40]

C. StinnerC. StinnerJ. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.   Google Scholar

[41]

Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.   Google Scholar

[42]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.   Google Scholar

[43]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.   Google Scholar

[44]

J. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.   Google Scholar

[45]

J. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.   Google Scholar

[46]

Q. Wang, J. Yang and F. Yu, Global existence and uniform boundedness in advective LotkaVolterra competition system with nonlinear diffusion, preprint, arXiv: 1605.05308. Google Scholar

[47]

Q. Wang, J. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model effect of cellular growth, preprint, arXiv: 1505.06463. Google Scholar

[48]

Q. WangL. ZhangJ. Yang and J. Hu, Global existence and steady states of a two competing species Keller-Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807.   Google Scholar

[49]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and boundedness properties, J. Math. Anal Appl, 348 (2008), 708-729.   Google Scholar

[50]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.   Google Scholar

[51]

M. Winkler, Aggregation versus global diffusive behavior in the higher-dimensional KellerSegel model, J. Differential Equations, 248 (2010), 2889-2905.   Google Scholar

[52]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.   Google Scholar

[53]

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.   Google Scholar

[54]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828.   Google Scholar

[55]

G. Wolansky, Multi-components chemotaxis system in absence of confict, European J. Appl. Math., 13 (2002), 641-661.   Google Scholar

[56]

M. Zeeman, Extinction in competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 123 (1995), 87-96.   Google Scholar

[57]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2002), 83-93.   Google Scholar

show all references

References:
[1]

S. Ahmad, On the nonautonomous Volterra-Lotka competition equations, Proc. Amer. Math. Soc, 117 (1993), 199-204.   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.   Google Scholar

[3]

P. BilerE. Espejo and I. Guerra, Blow up in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.   Google Scholar

[4]

P. Brown, Decay to uniform states in ecological interactions, SIAM J. Appl. Math., 38 (1980), 22-37.   Google Scholar

[5]

X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Cont. Dyns. S-A., 35 (2015), 1891-1904.   Google Scholar

[6]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.   Google Scholar

[7]

C. ConcaE. Espejo and K. Vilches, Remarks on the blow up and global existence for a two species chemotactic Keller-Segel system in R2, European J. Appl. Math., 22 (2011), 553-580.   Google Scholar

[8]

P. de Mottoni, Qualitative Analysis for Some Quasilinear Parabolic Systems, Inst Math Pol Acad Sci, 1979. Google Scholar

[9]

E. EspejoA. Stevens and J. Velzquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.   Google Scholar

[10]

A. FasanoA. Mancini and M. Primicerio, Equilibrium of two populations subjected to chemotaxis, Math. Mod Meth Appl. Sci., 14 (2004), 503-533.   Google Scholar

[11]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163.   Google Scholar

[12]

C. GaiQ. Wang and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284.   Google Scholar

[13]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.   Google Scholar

[14]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Deutsch. Math.-Verien., 105 (2003), 103-165.   Google Scholar

[15]

D. Horstemann, 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.   Google Scholar

[16]

D. Horstemann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.   Google Scholar

[17]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.   Google Scholar

[18]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolicparabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.   Google Scholar

[19]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Am. Math. Soc., 329 (1992), 819-824.   Google Scholar

[20]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.   Google Scholar

[21]

F. KellyK. Dapsis and D. Lauffenburger, Effects of bacterial chemotaxis on dynamics of microbial competition, Microb Ecol., 16 (1988), 115-131.   Google Scholar

[22]

R. Kowalczyk and Z. Szymánska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.   Google Scholar

[23]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.   Google Scholar

[24]

D. LauffenburgerR. Aris and K. Keller, Effects of cell motility and chemotaxis on microbial population growth, Biophys J., 40 (1982), 209-219.   Google Scholar

[25]

Y. Li and Y. Li, Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84.   Google Scholar

[26]

K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.   Google Scholar

[27]

K. LinC. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.   Google Scholar

[28]

Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.   Google Scholar

[29]

Y. Lou and M. Winkler, Global existence and uniform boundedness of smooth solutions to a cross-diffusion system with equal diffusion rates, Comm. Partial Differential Equations, 40 (2015), 1905-1941.   Google Scholar

[30]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincare Anal. Non Lineaire, 31 (2014), 851-875.   Google Scholar

[31]

M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669.   Google Scholar

[32]

J. Murray, Mathematical Biology, 2nd edition, Biomathematics series, Springer, Berlin, 1993. Google Scholar

[33]

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

[34]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj. Ser. Int., 40 (1997), 411-433.   Google Scholar

[35]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. MAth. Anal., 46 (2014), 3761-3781.   Google Scholar

[36]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxisgrowth system of equations, Nonlinear Anal., 51 (2002), 119-144.   Google Scholar

[37]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441-469.   Google Scholar

[38]

S. Shim, Uniform boundedness and convergence of solutions to cross-diffusion systems, J. Differential Equations, 185 (2002), 281-305.   Google Scholar

[39]

C. StinnerC. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.   Google Scholar

[40]

C. StinnerC. StinnerJ. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.   Google Scholar

[41]

Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.   Google Scholar

[42]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.   Google Scholar

[43]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.   Google Scholar

[44]

J. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.   Google Scholar

[45]

J. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.   Google Scholar

[46]

Q. Wang, J. Yang and F. Yu, Global existence and uniform boundedness in advective LotkaVolterra competition system with nonlinear diffusion, preprint, arXiv: 1605.05308. Google Scholar

[47]

Q. Wang, J. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model effect of cellular growth, preprint, arXiv: 1505.06463. Google Scholar

[48]

Q. WangL. ZhangJ. Yang and J. Hu, Global existence and steady states of a two competing species Keller-Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807.   Google Scholar

[49]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and boundedness properties, J. Math. Anal Appl, 348 (2008), 708-729.   Google Scholar

[50]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.   Google Scholar

[51]

M. Winkler, Aggregation versus global diffusive behavior in the higher-dimensional KellerSegel model, J. Differential Equations, 248 (2010), 2889-2905.   Google Scholar

[52]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.   Google Scholar

[53]

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.   Google Scholar

[54]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828.   Google Scholar

[55]

G. Wolansky, Multi-components chemotaxis system in absence of confict, European J. Appl. Math., 13 (2002), 641-661.   Google Scholar

[56]

M. Zeeman, Extinction in competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 123 (1995), 87-96.   Google Scholar

[57]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2002), 83-93.   Google Scholar

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