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Dynamics of a reaction-diffusion-advection model for two competing species
1. | Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260 |
2. | School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States |
3. | Department of Mathematics, Mathematical Bioscience Institute, Ohio State University, Columbus, Ohio 43210 |
References:
[1] |
H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefiniteelliptic problems, J. Diff. Eqns., 146 (1998), 336-374.
doi: 10.1006/jdeq.1998.3440. |
[2] |
I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal, J. Biol. Dyn., in press. |
[3] |
F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics ofpopulations in heterogeneous environment, Canadian Appl. Math. Quarterly, 3 (1995), 379-397. |
[4] |
A. Bezuglyy and Y. Lou, Reaction-diffusion models with large advection coefficients, Appl. Anal., 89 (2010), 983-1004.
doi: 10.1080/00036810903479723. |
[5] |
S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Diff. Eqns., 178 (2002), 123-211.
doi: 10.1006/jdeq.2000.4003. |
[6] |
R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley Series in Mathematicaland Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. |
[7] |
R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolutionof rapid diffusion, Math. Biosciences, 204 (2006), 199-214.
doi: 10.1016/j.mbs.2006.09.003. |
[8] |
R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinb. A, 137 (2007), 497-518. |
[9] |
R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal inheterogeneous landscape, in "Spatial Ecology" (eds. R. S. Cantrell, C. Cosner and S. Ruan), Mathematical and Computational Biology Series, Chapman Hall/CRC Press, (2009), 213-229. |
[10] |
R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal freedistribution, Math. Biosci. Eng., 7 (2010), 17-36. |
[11] |
C. Cosner and Y. Lou, Does movement toward better environments always benefit a population?, J. Math. Anal. Appl., 277 (2003), 489-503.
doi: 10.1016/S0022-247X(02)00575-9. |
[12] |
X. 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. |
[13] |
X. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of anelliptic 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. |
[14] |
X. Chen and Y. Lou, Effects of diffusion and advection on the smallest eigenvalue of an elliptic operator and their applications, Indiana Univ. Math. J., in press. |
[15] |
E. N. Dancer, Positivity of maps and applications, in "Topological Nonlinear Analysis" (eds. Matzeu and Vignoli), Progr. Nonlinear Differential Equations Appl., 15, Birkhäuser Boston, Boston, MA, (1995), 303-340. |
[16] |
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. |
[17] |
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equation of Second Order," 2$^nd$ edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, Springer-Verlag, Berlin, 1983. |
[18] |
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. |
[19] |
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. |
[20] |
P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity," Pitman Research Notes in Mathematics Series, 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. |
[21] |
S. Hsu, H. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.
doi: 10.1090/S0002-9947-96-01724-2. |
[22] |
V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behavior for a competing species problem with diffusion, in "Dynamical Systems and Applications," World Sci. Ser. Appl. Anal., 4, River Edge, NJ, (1995), 343-358. |
[23] |
K.-Y. Lam, Concentration phenomena of a semilinear elliptic equationwith large advection in an ecological model, J. Diff. Eqs., 250 (2011), 161-181.
doi: 10.1016/j.jde.2010.08.028. |
[24] |
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. |
[25] |
K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equationswith large advection in population dynamics, Discrete Cont. Dyn. Sys., 28 (2010), 1051-1067.
doi: 10.3934/dcds.2010.28.1051. |
[26] |
J. López-Gómez and J. C. Sabina de Lis, Coexistence states and global attractiveness for someconvective diffusive competing species models, Trans. Amer. Math. Soc., 347 (1995), 3797-3833.
doi: 10.2307/2155205. |
[27] |
H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1984), 645-673. |
[28] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. |
[29] |
H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, Rhode Island, 1995. |
show all references
References:
[1] |
H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefiniteelliptic problems, J. Diff. Eqns., 146 (1998), 336-374.
doi: 10.1006/jdeq.1998.3440. |
[2] |
I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal, J. Biol. Dyn., in press. |
[3] |
F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics ofpopulations in heterogeneous environment, Canadian Appl. Math. Quarterly, 3 (1995), 379-397. |
[4] |
A. Bezuglyy and Y. Lou, Reaction-diffusion models with large advection coefficients, Appl. Anal., 89 (2010), 983-1004.
doi: 10.1080/00036810903479723. |
[5] |
S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Diff. Eqns., 178 (2002), 123-211.
doi: 10.1006/jdeq.2000.4003. |
[6] |
R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley Series in Mathematicaland Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. |
[7] |
R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolutionof rapid diffusion, Math. Biosciences, 204 (2006), 199-214.
doi: 10.1016/j.mbs.2006.09.003. |
[8] |
R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinb. A, 137 (2007), 497-518. |
[9] |
R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal inheterogeneous landscape, in "Spatial Ecology" (eds. R. S. Cantrell, C. Cosner and S. Ruan), Mathematical and Computational Biology Series, Chapman Hall/CRC Press, (2009), 213-229. |
[10] |
R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal freedistribution, Math. Biosci. Eng., 7 (2010), 17-36. |
[11] |
C. Cosner and Y. Lou, Does movement toward better environments always benefit a population?, J. Math. Anal. Appl., 277 (2003), 489-503.
doi: 10.1016/S0022-247X(02)00575-9. |
[12] |
X. 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. |
[13] |
X. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of anelliptic 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. |
[14] |
X. Chen and Y. Lou, Effects of diffusion and advection on the smallest eigenvalue of an elliptic operator and their applications, Indiana Univ. Math. J., in press. |
[15] |
E. N. Dancer, Positivity of maps and applications, in "Topological Nonlinear Analysis" (eds. Matzeu and Vignoli), Progr. Nonlinear Differential Equations Appl., 15, Birkhäuser Boston, Boston, MA, (1995), 303-340. |
[16] |
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. |
[17] |
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equation of Second Order," 2$^nd$ edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, Springer-Verlag, Berlin, 1983. |
[18] |
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. |
[19] |
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. |
[20] |
P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity," Pitman Research Notes in Mathematics Series, 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. |
[21] |
S. Hsu, H. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.
doi: 10.1090/S0002-9947-96-01724-2. |
[22] |
V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behavior for a competing species problem with diffusion, in "Dynamical Systems and Applications," World Sci. Ser. Appl. Anal., 4, River Edge, NJ, (1995), 343-358. |
[23] |
K.-Y. Lam, Concentration phenomena of a semilinear elliptic equationwith large advection in an ecological model, J. Diff. Eqs., 250 (2011), 161-181.
doi: 10.1016/j.jde.2010.08.028. |
[24] |
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. |
[25] |
K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equationswith large advection in population dynamics, Discrete Cont. Dyn. Sys., 28 (2010), 1051-1067.
doi: 10.3934/dcds.2010.28.1051. |
[26] |
J. López-Gómez and J. C. Sabina de Lis, Coexistence states and global attractiveness for someconvective diffusive competing species models, Trans. Amer. Math. Soc., 347 (1995), 3797-3833.
doi: 10.2307/2155205. |
[27] |
H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1984), 645-673. |
[28] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. |
[29] |
H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, Rhode Island, 1995. |
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