# American Institute of Mathematical Sciences

November  2012, 32(11): 3841-3859. doi: 10.3934/dcds.2012.32.3841

## 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

Received  June 2011 Revised  November 2011 Published  June 2012

We study the dynamics of a reaction-diffusion-advection model for two competing species in a spatially heterogeneous environment. The two species are assumed to have the same population dynamics but different dispersal strategies: both species disperse by random diffusion and advection along the environmental gradient, but with different random dispersal and/or advection rates. Given any advection rates, we show that three scenarios can occur: (i) If one random dispersal rate is small and the other is large, two competing species coexist; (ii) If both random dispersal rates are large, the species with much larger random dispersal rate is driven to extinction; (iii) If both random dispersal rates are small, the species with much smaller random dispersal rate goes to extinction. Our results suggest that if both advection rates are positive and equal, an intermediate random dispersal rate may evolve. This is in contrast to the case when both advection rates are zero, where the species with larger random dispersal rate is always driven to extinction.
Citation: Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841
##### References:
 [1] H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefiniteelliptic problems,, J. Diff. Eqns., 146 (1998), 336.  doi: 10.1006/jdeq.1998.3440.  Google Scholar [2] I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, J. Biol. Dyn., ().   Google Scholar [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.   Google Scholar [4] A. Bezuglyy and Y. Lou, Reaction-diffusion models with large advection coefficients,, Appl. Anal., 89 (2010), 983.  doi: 10.1080/00036810903479723.  Google Scholar [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.  doi: 10.1006/jdeq.2000.4003.  Google Scholar [6] R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,", Wiley Series in Mathematicaland Computational Biology, (2003).   Google Scholar [7] R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolutionof rapid diffusion,, Math. Biosciences, 204 (2006), 199.  doi: 10.1016/j.mbs.2006.09.003.  Google Scholar [8] R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species,, Proc. Roy. Soc. Edinb. A, 137 (2007), 497.   Google Scholar [9] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal inheterogeneous landscape,, in, (2009), 213.   Google Scholar [10] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal freedistribution,, Math. Biosci. Eng., 7 (2010), 17.   Google Scholar [11] C. Cosner and Y. Lou, Does movement toward better environments always benefit a population?,, J. Math. Anal. Appl., 277 (2003), 489.  doi: 10.1016/S0022-247X(02)00575-9.  Google Scholar [12] X. Chen, R. Hambrock and Y. Lou, Evolution of conditional dispersal: A reaction-diffusion-advection model,, J. Math. Biol., 57 (2008), 361.  doi: 10.1007/s00285-008-0166-2.  Google Scholar [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.  doi: 10.1512/iumj.2008.57.3204.  Google Scholar [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., ().   Google Scholar [15] E. N. Dancer, Positivity of maps and applications,, in, 15 (1995), 303.   Google Scholar [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.  doi: 10.1007/s002850050120.  Google Scholar [17] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equation of Second Order,", 2$^{nd}$ edition, 224 (1983).   Google Scholar [18] R. Hambrock and Y. Lou, The evolution of conditional dispersal strategies in spatially heterogeneous habitats,, Bull. Math. Biol., 71 (2009), 1793.  doi: 10.1007/s11538-009-9425-7.  Google Scholar [19] A. Hastings, Can spatial variation alone lead to selection for dispersal?,, Theor. Pop. Biol., 24 (1983), 244.  doi: 10.1016/0040-5809(83)90027-8.  Google Scholar [20] P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity,", Pitman Research Notes in Mathematics Series, 247 (1991).   Google Scholar [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.  doi: 10.1090/S0002-9947-96-01724-2.  Google Scholar [22] V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behavior for a competing species problem with diffusion,, in, 4 (1995), 343.   Google Scholar [23] K.-Y. Lam, Concentration phenomena of a semilinear elliptic equationwith large advection in an ecological model,, J. Diff. Eqs., 250 (2011), 161.  doi: 10.1016/j.jde.2010.08.028.  Google Scholar [24] K. Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics. II,, SIAM J. Math. Anal., 44 (2012), 1808.  doi: 10.1137/100819758.  Google Scholar [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.  doi: 10.3934/dcds.2010.28.1051.  Google Scholar [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.  doi: 10.2307/2155205.  Google Scholar [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.   Google Scholar [28] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Corrected reprint of the 1967 original, (1967).   Google Scholar [29] H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,", Mathematical Surveys and Monographs, 41 (1995).   Google Scholar

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##### References:
 [1] H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefiniteelliptic problems,, J. Diff. Eqns., 146 (1998), 336.  doi: 10.1006/jdeq.1998.3440.  Google Scholar [2] I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, J. Biol. Dyn., ().   Google Scholar [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.   Google Scholar [4] A. Bezuglyy and Y. Lou, Reaction-diffusion models with large advection coefficients,, Appl. Anal., 89 (2010), 983.  doi: 10.1080/00036810903479723.  Google Scholar [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.  doi: 10.1006/jdeq.2000.4003.  Google Scholar [6] R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,", Wiley Series in Mathematicaland Computational Biology, (2003).   Google Scholar [7] R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolutionof rapid diffusion,, Math. Biosciences, 204 (2006), 199.  doi: 10.1016/j.mbs.2006.09.003.  Google Scholar [8] R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species,, Proc. Roy. Soc. Edinb. A, 137 (2007), 497.   Google Scholar [9] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal inheterogeneous landscape,, in, (2009), 213.   Google Scholar [10] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal freedistribution,, Math. Biosci. Eng., 7 (2010), 17.   Google Scholar [11] C. Cosner and Y. Lou, Does movement toward better environments always benefit a population?,, J. Math. Anal. Appl., 277 (2003), 489.  doi: 10.1016/S0022-247X(02)00575-9.  Google Scholar [12] X. Chen, R. Hambrock and Y. Lou, Evolution of conditional dispersal: A reaction-diffusion-advection model,, J. Math. Biol., 57 (2008), 361.  doi: 10.1007/s00285-008-0166-2.  Google Scholar [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.  doi: 10.1512/iumj.2008.57.3204.  Google Scholar [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., ().   Google Scholar [15] E. N. Dancer, Positivity of maps and applications,, in, 15 (1995), 303.   Google Scholar [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.  doi: 10.1007/s002850050120.  Google Scholar [17] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equation of Second Order,", 2$^{nd}$ edition, 224 (1983).   Google Scholar [18] R. Hambrock and Y. Lou, The evolution of conditional dispersal strategies in spatially heterogeneous habitats,, Bull. Math. Biol., 71 (2009), 1793.  doi: 10.1007/s11538-009-9425-7.  Google Scholar [19] A. Hastings, Can spatial variation alone lead to selection for dispersal?,, Theor. Pop. Biol., 24 (1983), 244.  doi: 10.1016/0040-5809(83)90027-8.  Google Scholar [20] P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity,", Pitman Research Notes in Mathematics Series, 247 (1991).   Google Scholar [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.  doi: 10.1090/S0002-9947-96-01724-2.  Google Scholar [22] V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behavior for a competing species problem with diffusion,, in, 4 (1995), 343.   Google Scholar [23] K.-Y. Lam, Concentration phenomena of a semilinear elliptic equationwith large advection in an ecological model,, J. Diff. Eqs., 250 (2011), 161.  doi: 10.1016/j.jde.2010.08.028.  Google Scholar [24] K. Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics. II,, SIAM J. Math. Anal., 44 (2012), 1808.  doi: 10.1137/100819758.  Google Scholar [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.  doi: 10.3934/dcds.2010.28.1051.  Google Scholar [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.  doi: 10.2307/2155205.  Google Scholar [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.   Google Scholar [28] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Corrected reprint of the 1967 original, (1967).   Google Scholar [29] H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,", Mathematical Surveys and Monographs, 41 (1995).   Google Scholar
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