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June  2020, 19(6): 3245-3255. doi: 10.3934/cpaa.2020142

Some global dynamics of a Lotka-Volterra competition-diffusion-advection system

Qi Wang

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received  July 2019 Revised  December 2019 Published  March 2020

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.

Keywords: Lotka-Volterra, competition-diffusion-advection system, steady state; stability.
Mathematics Subject Classification: 35K51, 35K57, 35B09, 35B35, 92D25.
Citation: Qi Wang. Some global dynamics of a Lotka-Volterra competition-diffusion-advection system. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3245-3255. doi: 10.3934/cpaa.2020142
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser. 247, Longman, Harlow, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser. 247, Longman, Harlow, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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