# 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 and Continuous Dynamical Systems, 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), 336-374. 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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##### 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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