Two-patch models are used to mimic the unidirectional movement of organisms in continuous, advective environments. We assume that species can move between two patches, with patch 1 as the upper stream patch and patch 2 as the downstream patch. Species disperse between two patches with the same rate, and species in patch 1 is transported to patch 2 by drift, but not vice versa. We also mimic no-flux boundary conditions at the upstream and zero Dirichlet boundary conditions at the downstream. The criteria for the persistence of a single species is established. For two competing species model, we show that there is an intermediate dispersal rate which is evolutionarily stable. These results support the conjecture in [
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[1] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, in Series in Mathematical and Computational Biology John Wiley and Sons, Chichester, UK, 2003.
doi: 10.1002/0470871296.![]() ![]() ![]() |
[2] |
C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discre. Contin. Dyn. Syst., 34 (2014), 1701-1745.
doi: 10.3934/dcds.2014.34.1701.![]() ![]() ![]() |
[3] |
U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes, J. Math. Biol., 34 (1996), 579-612.
doi: 10.1007/BF02409751.![]() ![]() ![]() |
[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.![]() ![]() |
[5] |
K.-Y. Lam, Y. Lou and F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), 244-251.
doi: 10.1080/17513758.2014.969336.![]() ![]() ![]() |
[6] |
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.![]() ![]() ![]() |
[7] |
Y. Lou, D. M. Xiao and P. Zhou, Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment, Discre. Contin. Dyn. Syst. A, 36 (2016), 953-969.
doi: 10.3934/dcds.2016.36.953.![]() ![]() ![]() |
[8] |
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.![]() ![]() ![]() |
[9] |
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.![]() ![]() ![]() |
[10] |
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.![]() ![]() ![]() |
[11] |
H. L. Smith, Monotone Dynamical System. An Introduction to the Theory of Competitive and Cooperative Systems, in Math. Surveys Monogr., 41, Amer. Math. Soc., Providence, RI, 1995.
![]() ![]() |
[12] |
J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, 1982.
![]() |
[13] |
D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries, Ecology, 82 (2001), 1219-1237.
![]() |
[14] |
O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Quart., 18 (2010), 439-469.
![]() ![]() |
[15] |
O. Vasilyeva and F. Lutscher, Competition in advective environments, Bull. Math. Biol., 74 (2012), 2935-2958.
doi: 10.1007/s11538-012-9792-3.![]() ![]() ![]() |
[16] |
B. L. Xu and N. Liu, Optimal diffusion rate of species in flowing habitat, Advances in Difference Equations, 2017 (2017), Paper No. 266, 10 pp.
doi: 10.1186/s13662-017-1326-8.![]() ![]() ![]() |
[17] |
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.![]() ![]() ![]() |