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Bility and traveling wavefronts for a convolution model of mistletoes and birds with nonlocal diffusion
Convergence of global and bounded solutions of a twospecies chemotaxis model with a logistic source
College of Mathematics and Statistics, Chongqing University, Chongqing University, Chongqing 401331, China 
$\left\{\begin{array}{llll}u_t=\Delta u\chi_1\nabla\cdot( u\nabla w)+\mu_1u(1ua_1v),\quad &x\in \Omega,\quad t>0,\\v_t=\Delta v\chi_2\nabla\cdot( v\nabla w)+\mu_2v(1a_2uv),\quad &x\in\Omega,\quad t>0,\\w_t=\Delta w w+u+v,\quad &x\in\Omega,\quad t>0,\\\end{array}\right.$ 
References:
[1] 
S. Ahmad, On the nonautonomous VolterraLotka competition equations, Proc. Amer. Math. Soc, 117 (1993), 199204. Google Scholar 
[2] 
X. Bai and M. Winkler, Equilibration in a fully parabolic twospecies chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553583. Google Scholar 
[3] 
P. Biler, E. Espejo and I. Guerra, Blow up in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 8998. Google Scholar 
[4] 
P. Brown, Decay to uniform states in ecological interactions, SIAM J. Appl. Math., 38 (1980), 2237. Google Scholar 
[5] 
X. Cao, Global bounded solutions of the higherdimensional KellerSegel system under smallness conditions in optimal spaces, Discrete Cont. Dyns. SA., 35 (2015), 18911904. Google Scholar 
[6] 
T. Cieślak and C. Stinner, Finitetime blowup and globalintime unbounded solutions to a parabolicparabolic quasilinear KellerSegel system in higher dimensions, J. Differential Equations, 252 (2012), 58325851. Google Scholar 
[7] 
C. Conca, E. Espejo and K. Vilches, Remarks on the blow up and global existence for a two species chemotactic KellerSegel system in R^{2}, European J. Appl. Math., 22 (2011), 553580. Google Scholar 
[8] 
P. de Mottoni, Qualitative Analysis for Some Quasilinear Parabolic Systems, Inst Math Pol Acad Sci, 1979. Google Scholar 
[9] 
E. Espejo, A. Stevens and J. Velzquez, Simultaneous finite time blowup in a twospecies model for chemotaxis, Analysis (Munich), 29 (2009), 317338. Google Scholar 
[10] 
A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subjected to chemotaxis, Math. Mod Meth Appl. Sci., 14 (2004), 503533. 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), 138163. Google Scholar 
[12] 
C. Gai, Q. Wang and J. Yan, Qualitative analysis of a LotkaVolterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 12391284. Google Scholar 
[13] 
H. Gajewski and K. Zacharias, Global behavior of a reactiondiffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77114. Google Scholar 
[14] 
D. Horstmann, From 1970 until present: The KellerSegel model in chemotaxis and its consequences, Jahresber. Deutsch. Math.Verien., 105 (2003), 103165. Google Scholar 
[15] 
D. Horstemann, Generalizing the KellerSegel model: Lyapunov functionals, steady state analysis, and blowup results for multispecies chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231270. Google Scholar 
[16] 
D. Horstemann and G. Wang, Blowup in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159177. Google Scholar 
[17] 
D. Horstmann and M. Winkler, Boundedness vs. blowup in a chemotaxis system, J. Differential Equations, 215 (2005), 52107. Google Scholar 
[18] 
S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear KellerSegel systems of parabolicparabolic type on nonconvex bounded domains, J. Differential Equations, 256 (2014), 29933010. 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), 819824. Google Scholar 
[20] 
E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399415. Google Scholar 
[21] 
F. Kelly, K. Dapsis and D. Lauffenburger, Effects of bacterial chemotaxis on dynamics of microbial competition, Microb Ecol., 16 (1988), 115131. Google Scholar 
[22] 
R. Kowalczyk and Z. Szymánska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379398. Google Scholar 
[23] 
J. Lankeit, Eventual smoothness and asymptotics in a threedimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 11581191. Google Scholar 
[24] 
D. Lauffenburger, R. Aris and K. Keller, Effects of cell motility and chemotaxis on microbial population growth, Biophys J., 40 (1982), 209219. Google Scholar 
[25] 
Y. Li and Y. Li, Finitetime blowup in higher dimensional fullyparabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 7284. 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), 50255046. Google Scholar 
[27] 
K. Lin, C. Mu and L. Wang, Boundedness in a twospecies chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 50855096. Google Scholar 
[28] 
Y. Lou and W. Ni, Diffusion, selfdiffusion and crossdiffusion, J. Differential Equations, 131 (1996), 79131. Google Scholar 
[29] 
Y. Lou and M. Winkler, Global existence and uniform boundedness of smooth solutions to a crossdiffusion system with equal diffusion rates, Comm. Partial Differential Equations, 40 (2015), 19051941. Google Scholar 
[30] 
N. Mizoguchi and P. Souplet, Nondegeneracy of blowup points for the parabolic KellerSegel system, Ann. Inst. H. Poincare Anal. Non Lineaire, 31 (2014), 851875. 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), 26502669. Google Scholar 
[32] 
J. Murray, Mathematical Biology, 2nd edition, Biomathematics series, Springer, Berlin, 1993. Google Scholar 
[33] 
T. Nagai, Blowup of nonradial solutions to parabolicelliptic systems modeling chemotaxis in twodimensional domains, J. Inequal. Appl., 6 (2001), 3755. Google Scholar 
[34] 
T. Nagai, T. Senba and K. Yoshida, Application of the TrudingerMoser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj. Ser. Int., 40 (1997), 411433. 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), 37613781. Google Scholar 
[36] 
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxisgrowth system of equations, Nonlinear Anal., 51 (2002), 119144. Google Scholar 
[37] 
K. Osaki and A. Yagi, Finite dimensional attractors for onedimensional KellerSegel equations, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441469. Google Scholar 
[38] 
S. Shim, Uniform boundedness and convergence of solutions to crossdiffusion systems, J. Differential Equations, 185 (2002), 281305. Google Scholar 
[39] 
C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDEODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 19692007. Google Scholar 
[40] 
C. Stinner, C. Stinner, J. Tello and M. Winkler, Competitive exclusion in a twospecies chemotaxis model, J. Math. Biol., 68 (2014), 16071626. Google Scholar 
[41] 
Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 136. Google Scholar 
[42] 
Y. Tao and M. Winkler, Boundedness in a quasilinear parabolicparabolic KellerSegel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692715. Google Scholar 
[43] 
Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a threedimensional chemotaxisfluid system, Z. Angew. Math. Phys., 66 (2015), 25552573. Google Scholar 
[44] 
J. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849877. Google Scholar 
[45] 
J. Tello and M. Winkler, Stabilization in a twospecies chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 14131425. 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 twocompetingspecies KellerSegel chemotaxis model effect of cellular growth, preprint, arXiv: 1505.06463. Google Scholar 
[48] 
Q. Wang, L. Zhang, J. Yang and J. Hu, Global existence and steady states of a two competing species KellerSegel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777807. Google Scholar 
[49] 
M. Winkler, Chemotaxis with logistic source: Very weak global solutions and boundedness properties, J. Math. Anal Appl, 348 (2008), 708729. Google Scholar 
[50] 
M. Winkler, Boundedness in the higherdimensional parabolicparabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 15161537. Google Scholar 
[51] 
M. Winkler, Aggregation versus global diffusive behavior in the higherdimensional KellerSegel model, J. Differential Equations, 248 (2010), 28892905. Google Scholar 
[52] 
M. Winkler, Finitetime blowup in the higherdimensional parabolicparabolic KellerSegel system, J. Math. Pures Appl., 100 (2013), 748767. 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), 10561077. Google Scholar 
[54] 
M. Winkler, Boundedness and large time behavior in a threedimensional chemotaxisStokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 37893828. Google Scholar 
[55] 
G. Wolansky, Multicomponents chemotaxis system in absence of confict, European J. Appl. Math., 13 (2002), 641661. Google Scholar 
[56] 
M. Zeeman, Extinction in competitive LotkaVolterra systems, Proc. Amer. Math. Soc., 123 (1995), 8796. Google Scholar 
[57] 
Q. Zhang and Y. Li, Global boundedness of solutions to a twospecies chemotaxis system, Z. Angew. Math. Phys., 66 (2002), 8393. Google Scholar 
show all references
References:
[1] 
S. Ahmad, On the nonautonomous VolterraLotka competition equations, Proc. Amer. Math. Soc, 117 (1993), 199204. Google Scholar 
[2] 
X. Bai and M. Winkler, Equilibration in a fully parabolic twospecies chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553583. Google Scholar 
[3] 
P. Biler, E. Espejo and I. Guerra, Blow up in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 8998. Google Scholar 
[4] 
P. Brown, Decay to uniform states in ecological interactions, SIAM J. Appl. Math., 38 (1980), 2237. Google Scholar 
[5] 
X. Cao, Global bounded solutions of the higherdimensional KellerSegel system under smallness conditions in optimal spaces, Discrete Cont. Dyns. SA., 35 (2015), 18911904. Google Scholar 
[6] 
T. Cieślak and C. Stinner, Finitetime blowup and globalintime unbounded solutions to a parabolicparabolic quasilinear KellerSegel system in higher dimensions, J. Differential Equations, 252 (2012), 58325851. Google Scholar 
[7] 
C. Conca, E. Espejo and K. Vilches, Remarks on the blow up and global existence for a two species chemotactic KellerSegel system in R^{2}, European J. Appl. Math., 22 (2011), 553580. Google Scholar 
[8] 
P. de Mottoni, Qualitative Analysis for Some Quasilinear Parabolic Systems, Inst Math Pol Acad Sci, 1979. Google Scholar 
[9] 
E. Espejo, A. Stevens and J. Velzquez, Simultaneous finite time blowup in a twospecies model for chemotaxis, Analysis (Munich), 29 (2009), 317338. Google Scholar 
[10] 
A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subjected to chemotaxis, Math. Mod Meth Appl. Sci., 14 (2004), 503533. 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), 138163. Google Scholar 
[12] 
C. Gai, Q. Wang and J. Yan, Qualitative analysis of a LotkaVolterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 12391284. Google Scholar 
[13] 
H. Gajewski and K. Zacharias, Global behavior of a reactiondiffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77114. Google Scholar 
[14] 
D. Horstmann, From 1970 until present: The KellerSegel model in chemotaxis and its consequences, Jahresber. Deutsch. Math.Verien., 105 (2003), 103165. Google Scholar 
[15] 
D. Horstemann, Generalizing the KellerSegel model: Lyapunov functionals, steady state analysis, and blowup results for multispecies chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231270. Google Scholar 
[16] 
D. Horstemann and G. Wang, Blowup in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159177. Google Scholar 
[17] 
D. Horstmann and M. Winkler, Boundedness vs. blowup in a chemotaxis system, J. Differential Equations, 215 (2005), 52107. Google Scholar 
[18] 
S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear KellerSegel systems of parabolicparabolic type on nonconvex bounded domains, J. Differential Equations, 256 (2014), 29933010. 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), 819824. Google Scholar 
[20] 
E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399415. Google Scholar 
[21] 
F. Kelly, K. Dapsis and D. Lauffenburger, Effects of bacterial chemotaxis on dynamics of microbial competition, Microb Ecol., 16 (1988), 115131. Google Scholar 
[22] 
R. Kowalczyk and Z. Szymánska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379398. Google Scholar 
[23] 
J. Lankeit, Eventual smoothness and asymptotics in a threedimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 11581191. Google Scholar 
[24] 
D. Lauffenburger, R. Aris and K. Keller, Effects of cell motility and chemotaxis on microbial population growth, Biophys J., 40 (1982), 209219. Google Scholar 
[25] 
Y. Li and Y. Li, Finitetime blowup in higher dimensional fullyparabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 7284. 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), 50255046. Google Scholar 
[27] 
K. Lin, C. Mu and L. Wang, Boundedness in a twospecies chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 50855096. Google Scholar 
[28] 
Y. Lou and W. Ni, Diffusion, selfdiffusion and crossdiffusion, J. Differential Equations, 131 (1996), 79131. Google Scholar 
[29] 
Y. Lou and M. Winkler, Global existence and uniform boundedness of smooth solutions to a crossdiffusion system with equal diffusion rates, Comm. Partial Differential Equations, 40 (2015), 19051941. Google Scholar 
[30] 
N. Mizoguchi and P. Souplet, Nondegeneracy of blowup points for the parabolic KellerSegel system, Ann. Inst. H. Poincare Anal. Non Lineaire, 31 (2014), 851875. 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), 26502669. Google Scholar 
[32] 
J. Murray, Mathematical Biology, 2nd edition, Biomathematics series, Springer, Berlin, 1993. Google Scholar 
[33] 
T. Nagai, Blowup of nonradial solutions to parabolicelliptic systems modeling chemotaxis in twodimensional domains, J. Inequal. Appl., 6 (2001), 3755. Google Scholar 
[34] 
T. Nagai, T. Senba and K. Yoshida, Application of the TrudingerMoser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj. Ser. Int., 40 (1997), 411433. 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), 37613781. Google Scholar 
[36] 
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxisgrowth system of equations, Nonlinear Anal., 51 (2002), 119144. Google Scholar 
[37] 
K. Osaki and A. Yagi, Finite dimensional attractors for onedimensional KellerSegel equations, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441469. Google Scholar 
[38] 
S. Shim, Uniform boundedness and convergence of solutions to crossdiffusion systems, J. Differential Equations, 185 (2002), 281305. Google Scholar 
[39] 
C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDEODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 19692007. Google Scholar 
[40] 
C. Stinner, C. Stinner, J. Tello and M. Winkler, Competitive exclusion in a twospecies chemotaxis model, J. Math. Biol., 68 (2014), 16071626. Google Scholar 
[41] 
Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 136. Google Scholar 
[42] 
Y. Tao and M. Winkler, Boundedness in a quasilinear parabolicparabolic KellerSegel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692715. Google Scholar 
[43] 
Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a threedimensional chemotaxisfluid system, Z. Angew. Math. Phys., 66 (2015), 25552573. Google Scholar 
[44] 
J. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849877. Google Scholar 
[45] 
J. Tello and M. Winkler, Stabilization in a twospecies chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 14131425. 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 twocompetingspecies KellerSegel chemotaxis model effect of cellular growth, preprint, arXiv: 1505.06463. Google Scholar 
[48] 
Q. Wang, L. Zhang, J. Yang and J. Hu, Global existence and steady states of a two competing species KellerSegel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777807. Google Scholar 
[49] 
M. Winkler, Chemotaxis with logistic source: Very weak global solutions and boundedness properties, J. Math. Anal Appl, 348 (2008), 708729. Google Scholar 
[50] 
M. Winkler, Boundedness in the higherdimensional parabolicparabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 15161537. Google Scholar 
[51] 
M. Winkler, Aggregation versus global diffusive behavior in the higherdimensional KellerSegel model, J. Differential Equations, 248 (2010), 28892905. Google Scholar 
[52] 
M. Winkler, Finitetime blowup in the higherdimensional parabolicparabolic KellerSegel system, J. Math. Pures Appl., 100 (2013), 748767. 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), 10561077. Google Scholar 
[54] 
M. Winkler, Boundedness and large time behavior in a threedimensional chemotaxisStokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 37893828. Google Scholar 
[55] 
G. Wolansky, Multicomponents chemotaxis system in absence of confict, European J. Appl. Math., 13 (2002), 641661. Google Scholar 
[56] 
M. Zeeman, Extinction in competitive LotkaVolterra systems, Proc. Amer. Math. Soc., 123 (1995), 8796. Google Scholar 
[57] 
Q. Zhang and Y. Li, Global boundedness of solutions to a twospecies chemotaxis system, Z. Angew. Math. Phys., 66 (2002), 8393. Google Scholar 
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