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: |
[1] |
S. Ahmad, On the nonautonomous Volterra-Lotka competition equations, Proc. Amer. Math. Soc, 117 (1993), 199-204. |
[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. |
[3] |
P. Biler, E. Espejo and I. Guerra, Blow up in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98. |
[4] |
P. Brown, Decay to uniform states in ecological interactions, SIAM J. Appl. Math., 38 (1980), 22-37. |
[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. |
[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. |
[7] |
C. Conca, E. Espejo and K. Vilches, Remarks on the blow up and global existence for a two species chemotactic Keller-Segel system in R^{2}, European J. Appl. Math., 22 (2011), 553-580. |
[8] | P. de Mottoni, Qualitative Analysis for Some Quasilinear Parabolic Systems, Inst Math Pol Acad Sci, 1979. |
[9] |
E. Espejo, A. Stevens and J. Velzquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338. |
[10] |
A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subjected to chemotaxis, Math. Mod Meth Appl. Sci., 14 (2004), 503-533. |
[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. |
[12] |
C. Gai, Q. Wang and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284. |
[13] |
H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. |
[14] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Deutsch. Math.-Verien., 105 (2003), 103-165. |
[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. |
[16] |
D. Horstemann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177. |
[17] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. |
[18] |
S. Ishida, K. 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. |
[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. |
[20] |
E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. |
[21] |
F. Kelly, K. Dapsis and D. Lauffenburger, Effects of bacterial chemotaxis on dynamics of microbial competition, Microb Ecol., 16 (1988), 115-131. |
[22] |
R. Kowalczyk and Z. Szymánska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. |
[23] |
J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191. |
[24] |
D. Lauffenburger, R. Aris and K. Keller, Effects of cell motility and chemotaxis on microbial population growth, Biophys J., 40 (1982), 209-219. |
[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. |
[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. |
[27] |
K. Lin, C. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096. |
[28] |
Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. |
[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. |
[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. |
[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. |
[32] | J. Murray, Mathematical Biology, 2nd edition, Biomathematics series, Springer, Berlin, 1993. |
[33] |
T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. |
[34] |
T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj. Ser. Int., 40 (1997), 411-433. |
[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. |
[36] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxisgrowth system of equations, Nonlinear Anal., 51 (2002), 119-144. |
[37] |
K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441-469. |
[38] |
S. Shim, Uniform boundedness and convergence of solutions to cross-diffusion systems, J. Differential Equations, 185 (2002), 281-305. |
[39] |
C. Stinner, C. 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. |
[40] |
C. Stinner, C. Stinner, J. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626. |
[41] |
Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. |
[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. |
[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. |
[44] |
J. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. |
[45] |
J. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425. |
[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. |
[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. |
[48] |
Q. Wang, L. Zhang, J. 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. |
[49] |
M. Winkler, Chemotaxis with logistic source: Very weak global solutions and boundedness properties, J. Math. Anal Appl, 348 (2008), 708-729. |
[50] |
M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. |
[51] |
M. Winkler, Aggregation versus global diffusive behavior in the higher-dimensional KellerSegel model, J. Differential Equations, 248 (2010), 2889-2905. |
[52] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. |
[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. |
[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. |
[55] |
G. Wolansky, Multi-components chemotaxis system in absence of confict, European J. Appl. Math., 13 (2002), 641-661. |
[56] |
M. Zeeman, Extinction in competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 123 (1995), 87-96. |
[57] |
Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2002), 83-93. |