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April  2019, 24(4): 1875-1887. doi: 10.3934/dcdsb.2018245

Evolutionarily stable dispersal strategies in a two-patch advective environment

Jing-Jing Xiang 1, and  Yihao Fang 2,,

1. 

School of Science, Xi'an University of Architecture and Technology, Xi'an 710055, China
2. 

Institute for Mathematical Science, Renmin University of China, Beijing 100872, China

* Corresponding author: Yihao Fang

Received  December 2017 Revised  March 2018 Published  April 2019 Early access  August 2018

Fund Project: Jing-jing Xiang is partially supported by the Research Foundation of Education Bureau of Shaanxi Province (15JK1433), "The mathematical modeling and analysis of disease spreading in media". Yihao Fang is partially supported by the National Natural Science Foundation of China(11571364).

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 [6], initially proposed for reaction-diffusion models with continuous advective environments.

Keywords: Evolution of dispersal, patch model, advective environment, evolutionarily stable strategy.
Mathematics Subject Classification: Primary: 37X75; Secondary: 92D40.
Citation: Jing-Jing Xiang, Yihao Fang. Evolutionarily stable dispersal strategies in a two-patch advective environment. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1875-1887. doi: 10.3934/dcdsb.2018245
References:
[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.

show all references

References:
[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.

Figure 1.  Illustration of Lemma 3.6 for $0\leq q < 1$
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Figure 2.  Illustration of Lemma 3.7 for $1\leq q < 5/4$
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Figure 3.  PIP for $0<q < 1$ with the sign of $\lambda_1$
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Figure 4.  PIP for $1\leq q < 5/4$ with the sign of $\lambda_1$
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