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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

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