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February  2016, 36(2): 953-969. doi: 10.3934/dcds.2016.36.953

Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment

Yuan Lou 1, , Dongmei Xiao 2, and  Peng Zhou 3,

1. 

Institute for Mathematical Sciences, Renmin University of China, Haidian District, Beijing, 100872
2. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240
3. 

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada

Received  July 2014 Published  August 2015

We study a two-species Lotka-Volterra competition model in an advective homogeneous environment. It is assumed that two species have the same population dynamics and diffusion rates but different advection rates. We show that if one competitor disperses by random diffusion only and the other assumes both random and directed movements, then the one without advection prevails. If two competitors are drifting along the same direction but with different advection rates, then the one with the smaller advection rate wins. Finally we prove that if the two competitors are drifting along the opposite direction, then two species will coexist. These results imply that the movement without advection in homogeneous environment is evolutionarily stable, as advection tends to move more individuals to the boundary of the habitat and thus cause the distribution of species mismatch with the resources which are evenly distributed in space.
Keywords: Reaction-diffusion-advection, stability, advective environment, co-existence steady state., Lotka-Volterra competition.
Mathematics Subject Classification: Primary: 35K57, 35K61; Secondary: 92D2.
Citation: Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953
References:
[1]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Ser. Math. Comput Biol., Wiley and Sons, 2003. doi: 10.1002/0470871296.

[2]

R. S. Cantrell, C. Cosner and Y. Lou, Movement toward better environments and the evolution of rapid diffusion, Math. Biosci., 204 (2006), 199-214. doi: 10.1016/j.mbs.2006.09.003.

[3]

R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh 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]

X. F. Chen, K.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst. A, 32 (2012), 3841-3859. doi: 10.3934/dcds.2012.32.3841.

[6]

X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204.

[7]

C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst. A, 34 (2014), 1701-1745. doi: 10.3934/dcds.2014.34.1701.

[8]

K. A. Dahmen, D. R. Nelson and N. M. Shnerb, Life and death near a windy oasis, J. Math. Biol., 41 (2000), 1-23. doi: 10.1007/s002850000025.

[9]

M. M. Desai and D. R. Nelson, A quasispecies on a moving oasis, Theor. Pop. Biol., 67 (2005), 33-45. doi: 10.1016/j.tpb.2004.07.005.

[10]

J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83. doi: 10.1007/s002850050120.

[11]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 24 (1983), 244-251. doi: 10.1016/0040-5809(83)90027-8.

[12]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk (N. S.), 3 (1948), 3-95.

[13]

K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differential Equations, 250 (2011), 161-181. doi: 10.1016/j.jde.2010.08.028.

[14]

K.-Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics II, SIAM J. Math. Anal., 44 (2012), 1808-1830. doi: 10.1137/100819758.

[15]

K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst. A, 28 (2010), 1051-1067. doi: 10.3934/dcds.2010.28.1051.

[16]

Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 159 (2015), 141-171. doi: 10.1016/j.jde.2015.02.004.

[17]

F. Lutscher, M. A. Lewis and E. McCauley, Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129-2160. doi: 10.1007/s11538-006-9100-1.

[18]

F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772. doi: 10.1137/050636152.

[19]

W.-M. Ni, The Mathematics of Diffusion, CBMS Reg. Conf. Ser. Appl. Math., 82, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971972.

[20]

A. Potapov, U. E. Schlägel and M. A. Lewis, Evolutionarily stable diffusive dispersal, Discrete Contin. Dyn. Syst. Series B, 19 (2014), 3319-3340. doi: 10.3934/dcdsb.2014.19.3319.

[21]

H. Smith, Monotone Dynamical System. An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., 41 Amer. Math. Soc., Providence, RI, 1995.

[22]

D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries, Ecology, 82 (2001), 1219-1237.

[23]

O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Quart., 18 (2010), 439-469.

show all references

References:
[1]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Ser. Math. Comput Biol., Wiley and Sons, 2003. doi: 10.1002/0470871296.

[2]

R. S. Cantrell, C. Cosner and Y. Lou, Movement toward better environments and the evolution of rapid diffusion, Math. Biosci., 204 (2006), 199-214. doi: 10.1016/j.mbs.2006.09.003.

[3]

R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh 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]

X. F. Chen, K.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst. A, 32 (2012), 3841-3859. doi: 10.3934/dcds.2012.32.3841.

[6]

X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204.

[7]

C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst. A, 34 (2014), 1701-1745. doi: 10.3934/dcds.2014.34.1701.

[8]

K. A. Dahmen, D. R. Nelson and N. M. Shnerb, Life and death near a windy oasis, J. Math. Biol., 41 (2000), 1-23. doi: 10.1007/s002850000025.

[9]

M. M. Desai and D. R. Nelson, A quasispecies on a moving oasis, Theor. Pop. Biol., 67 (2005), 33-45. doi: 10.1016/j.tpb.2004.07.005.

[10]

J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83. doi: 10.1007/s002850050120.

[11]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 24 (1983), 244-251. doi: 10.1016/0040-5809(83)90027-8.

[12]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk (N. S.), 3 (1948), 3-95.

[13]

K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differential Equations, 250 (2011), 161-181. doi: 10.1016/j.jde.2010.08.028.

[14]

K.-Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics II, SIAM J. Math. Anal., 44 (2012), 1808-1830. doi: 10.1137/100819758.

[15]

K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst. A, 28 (2010), 1051-1067. doi: 10.3934/dcds.2010.28.1051.

[16]

Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 159 (2015), 141-171. doi: 10.1016/j.jde.2015.02.004.

[17]

F. Lutscher, M. A. Lewis and E. McCauley, Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129-2160. doi: 10.1007/s11538-006-9100-1.

[18]

F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772. doi: 10.1137/050636152.

[19]

W.-M. Ni, The Mathematics of Diffusion, CBMS Reg. Conf. Ser. Appl. Math., 82, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971972.

[20]

A. Potapov, U. E. Schlägel and M. A. Lewis, Evolutionarily stable diffusive dispersal, Discrete Contin. Dyn. Syst. Series B, 19 (2014), 3319-3340. doi: 10.3934/dcdsb.2014.19.3319.

[21]

H. Smith, Monotone Dynamical System. An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., 41 Amer. Math. Soc., Providence, RI, 1995.

[22]

D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries, Ecology, 82 (2001), 1219-1237.

[23]

O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Quart., 18 (2010), 439-469.

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