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The effects of spatial heterogeneities on some multiplicity results
Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment
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 |
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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