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Some global dynamics of a Lotka-Volterra competition-diffusion-advection system
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China |
This paper studies some population dynamics of a diffusive Lotka-Volterra competition advection model under no-flux boundary condition. We establish the main results that determine the stability of semi-trivial steady states.
References:
| [1] |
I. Averill, K. Y. Lam and Y. Lou, The role of advection in a two-species competition model: a bifurcation approach, Mem. Amer. Math. Soc., 245(1161) (2017).
doi: 10.1090/memo/1161. |
| [2] |
R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Series in Mathematical and Computational Biology, Wiley, Chichester, UK, 2003.
doi: 10.1002/0470871296. |
| [3] |
R. S. Cantrell, C. Cosner and Y. Lou,
Advection-mediated coexistence of competing species, Proc. R. Soc. Edinb. Sect. A, 137 (2007), 497-518.
doi: 10.1017/S0308210506000047. |
| [4] |
X. F. Chen, R. Hambrock and Y. Lou,
Evolution of conditional dispersal:a reaction-diffusion-advection model, J. Math. Biol., 57 (2008), 361-386.
doi: 10.1007/s00285-008-0166-2. |
| [5] |
C. Cosner,
Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst., 34 (2014), 1701-1745.
doi: 10.3934/dcds.2014.34.1701. |
| [6] |
R. Hambrock and Y. Lou,
The evolution of conditional dispersal strategies in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 1793-1817.
doi: 10.1007/s11538-009-9425-7. |
| [7] |
X. He and W. M. Ni,
The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅰ: heterogeneity vs. homogeneity, J. Differ. Equ., 254 (2013), 528-546.
doi: 10.1016/j.jde.2012.08.032. |
| [8] |
X. He and W. M. Ni,
The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅱ: the general case, J. Differ. Equ., 254 (2013), 4088-4108.
doi: 10.1016/j.jde.2013.02.009. |
| [9] |
X. He and W. M. Ni,
Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity Ⅰ, Commun. Pure Appl. Math., 69 (2016), 981-1014.
doi: 10.1002/cpa.21596. |
| [10] |
X. He and W. M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources Ⅱ, Calc. Var. Partial Differ. Equ., 55 (2016), 20.
doi: 10.1007/s00526-016-0964-0. |
| [11] |
X. He and W. M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources Ⅲ, Calc. Var. Partial Differ. Equ., 56 (2017), 26.
doi: 10.1007/s00526-017-1234-5. |
| [12] |
P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser. 247, Longman, Harlow, 1991. |
| [13] |
M. Hirsch,
Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53.
doi: 10.1515/crll.1988.383.1. |
| [14] |
V. Hutson, Y. Lou and K. Mischaikow,
Convergence in competition models with small diffusion coeffcients, J. Differ. Equ., 211 (2005), 135-161.
doi: 10.1016/j.jde.2004.06.003. |
| [15] |
K. Y. Lam,
Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differ. Equ., 250 (2011), 161-181.
doi: 10.1016/j.jde.2010.08.028. |
| [16] |
K. Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics Ⅱ, emphSIAM J. Math. Anal., 44 (2012), 1808–1830.
doi: 10.1137/100819758. |
| [17] |
Y. Lou,
On the effects of migration and spatial heterogeneity on single and multiple species, J. Differ. Equ., 223 (2006), 400-426.
doi: 10.1016/j.jde.2005.05.010. |
| [18] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, 2$^nd$ edition, Springer, 1984.
doi: 10.1007/978-1-4612-5282-5. |
| [19] |
P. Zhou and D. Xiao,
Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275 (2018), 356-380.
doi: 10.1016/j.jfa.2018.03.006. |
show all references
References:
| [1] |
I. Averill, K. Y. Lam and Y. Lou, The role of advection in a two-species competition model: a bifurcation approach, Mem. Amer. Math. Soc., 245(1161) (2017).
doi: 10.1090/memo/1161. |
| [2] |
R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Series in Mathematical and Computational Biology, Wiley, Chichester, UK, 2003.
doi: 10.1002/0470871296. |
| [3] |
R. S. Cantrell, C. Cosner and Y. Lou,
Advection-mediated coexistence of competing species, Proc. R. Soc. Edinb. Sect. A, 137 (2007), 497-518.
doi: 10.1017/S0308210506000047. |
| [4] |
X. F. Chen, R. Hambrock and Y. Lou,
Evolution of conditional dispersal:a reaction-diffusion-advection model, J. Math. Biol., 57 (2008), 361-386.
doi: 10.1007/s00285-008-0166-2. |
| [5] |
C. Cosner,
Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst., 34 (2014), 1701-1745.
doi: 10.3934/dcds.2014.34.1701. |
| [6] |
R. Hambrock and Y. Lou,
The evolution of conditional dispersal strategies in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 1793-1817.
doi: 10.1007/s11538-009-9425-7. |
| [7] |
X. He and W. M. Ni,
The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅰ: heterogeneity vs. homogeneity, J. Differ. Equ., 254 (2013), 528-546.
doi: 10.1016/j.jde.2012.08.032. |
| [8] |
X. He and W. M. Ni,
The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅱ: the general case, J. Differ. Equ., 254 (2013), 4088-4108.
doi: 10.1016/j.jde.2013.02.009. |
| [9] |
X. He and W. M. Ni,
Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity Ⅰ, Commun. Pure Appl. Math., 69 (2016), 981-1014.
doi: 10.1002/cpa.21596. |
| [10] |
X. He and W. M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources Ⅱ, Calc. Var. Partial Differ. Equ., 55 (2016), 20.
doi: 10.1007/s00526-016-0964-0. |
| [11] |
X. He and W. M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources Ⅲ, Calc. Var. Partial Differ. Equ., 56 (2017), 26.
doi: 10.1007/s00526-017-1234-5. |
| [12] |
P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser. 247, Longman, Harlow, 1991. |
| [13] |
M. Hirsch,
Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53.
doi: 10.1515/crll.1988.383.1. |
| [14] |
V. Hutson, Y. Lou and K. Mischaikow,
Convergence in competition models with small diffusion coeffcients, J. Differ. Equ., 211 (2005), 135-161.
doi: 10.1016/j.jde.2004.06.003. |
| [15] |
K. Y. Lam,
Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differ. Equ., 250 (2011), 161-181.
doi: 10.1016/j.jde.2010.08.028. |
| [16] |
K. Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics Ⅱ, emphSIAM J. Math. Anal., 44 (2012), 1808–1830.
doi: 10.1137/100819758. |
| [17] |
Y. Lou,
On the effects of migration and spatial heterogeneity on single and multiple species, J. Differ. Equ., 223 (2006), 400-426.
doi: 10.1016/j.jde.2005.05.010. |
| [18] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, 2$^nd$ edition, Springer, 1984.
doi: 10.1007/978-1-4612-5282-5. |
| [19] |
P. Zhou and D. Xiao,
Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275 (2018), 356-380.
doi: 10.1016/j.jfa.2018.03.006. |
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