# American Institute of Mathematical Sciences

October  2020, 40(10): 5815-5830. doi: 10.3934/dcds.2020247

## Evolution of dispersal in advective homogeneous environments

 1 College of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, Jiangxi, China 2 School of Mathematics(Zhuhai), Sun Yat-sen University, Zhuhai 519082, Guangdong, China

* Corresponding author

Received  September 2019 Revised  February 2020 Published  June 2020

Fund Project: The first author is supported by the National Natural Science Foundation of China(Nos. 11801089, 11901110) and the second author is supported by the Postdoctoral Science Foundation of China(No.2018M643281), the Fundamental Research Funds for the Central Universities (No. 191gpy246) and National Natural Science Foundation of China(No. 11901596)

The effects of weak and strong advection on the dynamics of reaction-diffusion models have long been investigated. In contrast, the role of intermediate advection still remains poorly understood. This paper is devoted to studying a two-species competition model in a one-dimensional advective homogeneous environment, where the two species are identical except their diffusion rates and advection rates. Zhou (P. Zhou, On a Lotka-Volterra competition system: diffusion vs advection, Calc. Var. Partial Differential Equations, 55 (2016), Art. 137, 29 pp) considered the system under the no-flux boundary conditions. It is pointed that, in this paper, we focus on the case where the upstream end has the Neumann boundary condition and the downstream end has the hostile condition. By employing a new approach, we firstly determine necessary and sufficient conditions for the persistence of the corresponding single species model, in forms of the critical diffusion rate and critical advection rate. Furthermore, for the two-species model, we find that (i) the strategy of slower diffusion together with faster advection is always favorable; (ii) two species will also coexist when the faster advection with appropriate faster diffusion.

Citation: Li Ma, De Tang. Evolution of dispersal in advective homogeneous environments. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5815-5830. doi: 10.3934/dcds.2020247
##### References:
 [1] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, UK, 2003. doi: 10.1002/0470871296.  Google Scholar [2] 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 [3] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar [4] A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Popul. Biol., 24 (1983), 244-251.  doi: 10.1016/0040-5809(83)90027-8.  Google Scholar [5] X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity Ⅰ, Comm. Pure. Appl. Math., 69 (2016), 981-1014.  doi: 10.1002/cpa.21596.  Google Scholar [6] 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.  Google Scholar [7] M. G. Kre$\rm\check{i}$n and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspehi Matem. Nauk (N. S.), 3 (1948), 3-95.   Google Scholar [8] Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, Tutorials in Mathematical Biosciences â…£, Lecture Notes in Math., 1922, Springer, Berlin, (2008), 171â€"205. doi: 10.1007/978-3-540-74331-6_5.  Google Scholar [9] Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2.  Google Scholar [10] Y. Lou, D. Xiao and P. Zhou, Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment, Discrete Contin. Dyn. Syst., 36 (2016), 953-969.  doi: 10.3934/dcds.2016.36.953.  Google Scholar [11] Y. Lou, X.-Q. Zhao and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments, J. Math. Pures Appl., 121 (2019), 47-82.  doi: 10.1016/j.matpur.2018.06.010.  Google Scholar [12] Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 259 (2015), 141-171.  doi: 10.1016/j.jde.2015.02.004.  Google Scholar [13] 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 [14] W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philedelphia, 2011. doi: 10.1137/1.9781611971972.  Google Scholar [15] R. Peng and M. Zhou, Effects of large degenerate advection and boundary conditions on the principal eigenvalue and its eigenfunction of a linear second-order elliptic operator, Indiana Univ. Math. J., 67 (2018), 2523-2568.  doi: 10.1512/iumj.2018.67.7547.  Google Scholar [16] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., 41, Amer. Math. Soc., Providence, RI, 1995.  Google Scholar [17] D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries, Ecology, 82 (2001), 1219-1237.   Google Scholar [18] D. Tang and Y. Chen, Global dynamics of a Lotka-Volterra competition-diffusion system in advective homogeneous environments, J. Differential Equations, 269 (2020), 1465-1483.  doi: 10.1016/j.jde.2020.01.011.  Google Scholar [19] D. Tang and P. Zhou, On a Lotka-Volterra competition-diffusion-advection system: Homogeneity vs heterogeneity, J. Differential Equations, 268 (2020), 1570-1599.  doi: 10.1016/j.jde.2019.09.003.  Google Scholar [20] O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Q., 18 (2010), 439-469.   Google Scholar [21] F. Xu and W. Gan, On a Lotka-Volterra type competition model from river ecology, Nonlinear Anal. Real World Appl., 47 (2019), 373-384.  doi: 10.1016/j.nonrwa.2018.11.011.  Google Scholar [22] F. Xu, W. Gan and D. Tang, Population dynamics and evolution in river ecosystems, Nonlinear Anal. Real World Appl., 51 (2020), 102983, 16 pp. doi: 10.1016/j.nonrwa.2019.102983.  Google Scholar [23] X.-Q. Zhao and P. Zhou, On a Lotka-Volterra competition model: The effects of advection and spatial variation, Calc. Var. Partial Differential Equations, 55 (2016), Art. 73, 25 pp. doi: 10.1007/s00526-016-1021-8.  Google Scholar [24] P. Zhou, On a Lotka-Volterra competition system: Diffusion vs advection, Calc. Var. Partial Differential Equations, 55 (2016), Art. 137, 29 pp. doi: 10.1007/s00526-016-1082-8.  Google Scholar [25] P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275 (2018), 356-380.  doi: 10.1016/j.jfa.2018.03.006.  Google Scholar [26] P. Zhou and X.-Q. Zhao, Global dynamics of a two species competition model in open stream environments, J. Dynam. Differential Equations, 30 (2018), 613-636.  doi: 10.1007/s10884-016-9562-2.  Google Scholar [27] P. Zhou and X.-Q. Zhao, Evolution of passive movement in advective environments: General boundary condition, J. Differential Equations, 264 (2018), 4176-4198.  doi: 10.1016/j.jde.2017.12.005.  Google Scholar

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##### References:
 [1] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, UK, 2003. doi: 10.1002/0470871296.  Google Scholar [2] 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 [3] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar [4] A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Popul. Biol., 24 (1983), 244-251.  doi: 10.1016/0040-5809(83)90027-8.  Google Scholar [5] X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity Ⅰ, Comm. Pure. Appl. Math., 69 (2016), 981-1014.  doi: 10.1002/cpa.21596.  Google Scholar [6] 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.  Google Scholar [7] M. G. Kre$\rm\check{i}$n and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspehi Matem. Nauk (N. S.), 3 (1948), 3-95.   Google Scholar [8] Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, Tutorials in Mathematical Biosciences â…£, Lecture Notes in Math., 1922, Springer, Berlin, (2008), 171â€"205. doi: 10.1007/978-3-540-74331-6_5.  Google Scholar [9] Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2.  Google Scholar [10] Y. Lou, D. Xiao and P. Zhou, Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment, Discrete Contin. Dyn. Syst., 36 (2016), 953-969.  doi: 10.3934/dcds.2016.36.953.  Google Scholar [11] Y. Lou, X.-Q. Zhao and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments, J. Math. Pures Appl., 121 (2019), 47-82.  doi: 10.1016/j.matpur.2018.06.010.  Google Scholar [12] Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 259 (2015), 141-171.  doi: 10.1016/j.jde.2015.02.004.  Google Scholar [13] 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 [14] W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philedelphia, 2011. doi: 10.1137/1.9781611971972.  Google Scholar [15] R. Peng and M. Zhou, Effects of large degenerate advection and boundary conditions on the principal eigenvalue and its eigenfunction of a linear second-order elliptic operator, Indiana Univ. Math. J., 67 (2018), 2523-2568.  doi: 10.1512/iumj.2018.67.7547.  Google Scholar [16] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., 41, Amer. Math. Soc., Providence, RI, 1995.  Google Scholar [17] D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries, Ecology, 82 (2001), 1219-1237.   Google Scholar [18] D. Tang and Y. Chen, Global dynamics of a Lotka-Volterra competition-diffusion system in advective homogeneous environments, J. Differential Equations, 269 (2020), 1465-1483.  doi: 10.1016/j.jde.2020.01.011.  Google Scholar [19] D. Tang and P. Zhou, On a Lotka-Volterra competition-diffusion-advection system: Homogeneity vs heterogeneity, J. Differential Equations, 268 (2020), 1570-1599.  doi: 10.1016/j.jde.2019.09.003.  Google Scholar [20] O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Q., 18 (2010), 439-469.   Google Scholar [21] F. Xu and W. Gan, On a Lotka-Volterra type competition model from river ecology, Nonlinear Anal. Real World Appl., 47 (2019), 373-384.  doi: 10.1016/j.nonrwa.2018.11.011.  Google Scholar [22] F. Xu, W. Gan and D. Tang, Population dynamics and evolution in river ecosystems, Nonlinear Anal. Real World Appl., 51 (2020), 102983, 16 pp. doi: 10.1016/j.nonrwa.2019.102983.  Google Scholar [23] X.-Q. Zhao and P. Zhou, On a Lotka-Volterra competition model: The effects of advection and spatial variation, Calc. Var. Partial Differential Equations, 55 (2016), Art. 73, 25 pp. doi: 10.1007/s00526-016-1021-8.  Google Scholar [24] P. Zhou, On a Lotka-Volterra competition system: Diffusion vs advection, Calc. Var. Partial Differential Equations, 55 (2016), Art. 137, 29 pp. doi: 10.1007/s00526-016-1082-8.  Google Scholar [25] P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275 (2018), 356-380.  doi: 10.1016/j.jfa.2018.03.006.  Google Scholar [26] P. Zhou and X.-Q. Zhao, Global dynamics of a two species competition model in open stream environments, J. Dynam. Differential Equations, 30 (2018), 613-636.  doi: 10.1007/s10884-016-9562-2.  Google Scholar [27] P. Zhou and X.-Q. Zhao, Evolution of passive movement in advective environments: General boundary condition, J. Differential Equations, 264 (2018), 4176-4198.  doi: 10.1016/j.jde.2017.12.005.  Google Scholar

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