August 2018, 23(6): 2245-2263. doi: 10.3934/dcdsb.2018195

Boundedness and persistence of populations in advective Lotka-Volterra competition system

Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China

* Corresponding author.QW is partially supported by NSF-China (Grant No. 11501460) and the Fundamental Research Funds for the Central Universities (Grant No. JBK1801062)

Received  September 2016 Revised  April 2018 Published  June 2018

We are concerned with a two-component reaction-advection-diffusion Lotka-Volterra competition system with constant diffusion rates subject to homogeneous Neumann boundary conditions. We first prove the global existence and uniform boundedness of positive classical solutions to this system. This result complements some of the global existence results in [Y. Lou, M. Winkler and Y. Tao, SIAM J. Math. Anal., 46 (2014), 1228-1262.], where one diffusion rate is taken to be a linear function of the population density. Our second result proves that the total population of each species admits a positive lower bound, under some conditions of system parameters (e.g., when the intraspecific competition rates are large). This result of population persistence indicates that the two competing species coexist over the habitat in a long time.

Citation: Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195
References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, differential operators and nonlinear Analysis, Teubner, Stuttgart, Leipzig, 133 (1993), 9-126. doi: 10.1007/978-3-663-11336-2_1.

[2]

R. Cantrell and C. Cosner, On the uniqueness and stability of positive solutions in the Lotka-Volterra competition model with diffusion, Houston J. Math., 15 (1989), 341-361.

[3]

E. Conway and J. Smoller, A comparison technique for systems of reaction-diffusion equations, Comm. Partial Differential Equations, 2 (1977), 679-697. doi: 10.1080/03605307708820045.

[4]

C. Cosner and A. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion, SIAM J. Appl. Math., 44 (1984), 1112-1132. doi: 10.1137/0144080.

[5]

E. CrooksE. Dancer and D. Hilhorst, Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 39-44. doi: 10.3934/dcdsb.2007.8.39.

[6]

E. CrooksE. DancerD. HilhorstM. Mimura and H. Ninomiya, Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions, Nonlinear Anal. Real World Appl., 5 (2004), 645-665. doi: 10.1016/j.nonrwa.2004.01.004.

[7]

W. Feng, Competitive exclusion and persistence in models of resource and sexual competition, J. Math. Biol., 35 (1997), 683-694. doi: 10.1007/s002850050071.

[8]

M. IidaT. MuramatsuH. Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system, Japan J. Indust. Appl. Math., 15 (1998), 233-252. doi: 10.1007/BF03167402.

[9]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028.

[10]

C. Kahane, On the competition-diffusion equations for closely competing species, Funkcial. Ekvac., 35 (1992), 51-64.

[11]

Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hiroshima Math. Journal, 23 (1993), 193-221.

[12]

K. Kishimoto and H. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems in convex domains, J. Differential Equations, 58 (1985), 15-21. doi: 10.1016/0022-0396(85)90020-8.

[13]

K. Kuto and T. Tsujikawa, Limiting structure of steady-states to the Lotka-Volterra competition model with large diffusion and advection, J. Differential Equations, 258 (2015), 1801-1858. doi: 10.1016/j.jde.2014.11.016.

[14]

Y. LouM. Winkler and Y. Tao, Approaching the ideal free distribution in two-species competition models with fitness-dependent dispersal, SIAM J. Math. Anal., 46 (2014), 1228-1262. doi: 10.1137/130934246.

[15]

H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS, Kyoto Univ., 19 (1983), 1049-1079. doi: 10.2977/prims/1195182020.

[16]

M. MimuraS.-I. Ei and Q. Fang, Effect of domain-shape on coexistence problems in a competition-diffusion system, J. Math. Biol., 29 (1991), 219-237. doi: 10.1007/BF00160536.

[17]

H. Ninomiya, Separatrices of competition-diffusion equations, J. Math. Kyoto Univ., 35 (1995), 539-567. doi: 10.1215/kjm/1250518709.

[18]

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.

[19]

Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Diffential Equations, 259 (2015), 6142-6161. doi: 10.1016/j.jde.2015.07.019.

[20]

Q. WangC. Gai and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284. doi: 10.3934/dcds.2015.35.1239.

[21]

Q. Wang, J. Yand and F. Yu, Global existence and uniform boundedness in advective Lotka-Volterra competition system with nonlinear diffusion, preprint, arXiv: 1605.05308.

[22]

Q. Wang and L. Zhang, On the multi-dimensional advective Lotka-Volterra competition systems, Nonlinear Anal. Real World Appl., 37 (2017), 329-349. doi: 10.1016/j.nonrwa.2017.02.011.

[23]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Diffential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.

[24]

Y. Zhang and L. Xia, Stationary solutions and spatial-temporal dynamics of a shadow system of LV competition models, Adv. Difference Equ., (2017), Paper No. 25, 16 pp. doi: 10.1186/s13662-017-1308-x.

show all references

References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, differential operators and nonlinear Analysis, Teubner, Stuttgart, Leipzig, 133 (1993), 9-126. doi: 10.1007/978-3-663-11336-2_1.

[2]

R. Cantrell and C. Cosner, On the uniqueness and stability of positive solutions in the Lotka-Volterra competition model with diffusion, Houston J. Math., 15 (1989), 341-361.

[3]

E. Conway and J. Smoller, A comparison technique for systems of reaction-diffusion equations, Comm. Partial Differential Equations, 2 (1977), 679-697. doi: 10.1080/03605307708820045.

[4]

C. Cosner and A. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion, SIAM J. Appl. Math., 44 (1984), 1112-1132. doi: 10.1137/0144080.

[5]

E. CrooksE. Dancer and D. Hilhorst, Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 39-44. doi: 10.3934/dcdsb.2007.8.39.

[6]

E. CrooksE. DancerD. HilhorstM. Mimura and H. Ninomiya, Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions, Nonlinear Anal. Real World Appl., 5 (2004), 645-665. doi: 10.1016/j.nonrwa.2004.01.004.

[7]

W. Feng, Competitive exclusion and persistence in models of resource and sexual competition, J. Math. Biol., 35 (1997), 683-694. doi: 10.1007/s002850050071.

[8]

M. IidaT. MuramatsuH. Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system, Japan J. Indust. Appl. Math., 15 (1998), 233-252. doi: 10.1007/BF03167402.

[9]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028.

[10]

C. Kahane, On the competition-diffusion equations for closely competing species, Funkcial. Ekvac., 35 (1992), 51-64.

[11]

Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hiroshima Math. Journal, 23 (1993), 193-221.

[12]

K. Kishimoto and H. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems in convex domains, J. Differential Equations, 58 (1985), 15-21. doi: 10.1016/0022-0396(85)90020-8.

[13]

K. Kuto and T. Tsujikawa, Limiting structure of steady-states to the Lotka-Volterra competition model with large diffusion and advection, J. Differential Equations, 258 (2015), 1801-1858. doi: 10.1016/j.jde.2014.11.016.

[14]

Y. LouM. Winkler and Y. Tao, Approaching the ideal free distribution in two-species competition models with fitness-dependent dispersal, SIAM J. Math. Anal., 46 (2014), 1228-1262. doi: 10.1137/130934246.

[15]

H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS, Kyoto Univ., 19 (1983), 1049-1079. doi: 10.2977/prims/1195182020.

[16]

M. MimuraS.-I. Ei and Q. Fang, Effect of domain-shape on coexistence problems in a competition-diffusion system, J. Math. Biol., 29 (1991), 219-237. doi: 10.1007/BF00160536.

[17]

H. Ninomiya, Separatrices of competition-diffusion equations, J. Math. Kyoto Univ., 35 (1995), 539-567. doi: 10.1215/kjm/1250518709.

[18]

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.

[19]

Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Diffential Equations, 259 (2015), 6142-6161. doi: 10.1016/j.jde.2015.07.019.

[20]

Q. WangC. Gai and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284. doi: 10.3934/dcds.2015.35.1239.

[21]

Q. Wang, J. Yand and F. Yu, Global existence and uniform boundedness in advective Lotka-Volterra competition system with nonlinear diffusion, preprint, arXiv: 1605.05308.

[22]

Q. Wang and L. Zhang, On the multi-dimensional advective Lotka-Volterra competition systems, Nonlinear Anal. Real World Appl., 37 (2017), 329-349. doi: 10.1016/j.nonrwa.2017.02.011.

[23]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Diffential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.

[24]

Y. Zhang and L. Xia, Stationary solutions and spatial-temporal dynamics of a shadow system of LV competition models, Adv. Difference Equ., (2017), Paper No. 25, 16 pp. doi: 10.1186/s13662-017-1308-x.

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