American Institute of Mathematical Sciences

Advanced
February  2016, 36(2): 953-969. doi: 10.3934/dcds.2016.36.953

Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment

Yuan Lou 1, , Dongmei Xiao 2, and  Peng Zhou 3,

1. 

Institute for Mathematical Sciences, Renmin University of China, Haidian District, Beijing, 100872
2. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240
3. 

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada

Received  July 2014 Published  August 2015

We study a two-species Lotka-Volterra competition model in an advective homogeneous environment. It is assumed that two species have the same population dynamics and diffusion rates but different advection rates. We show that if one competitor disperses by random diffusion only and the other assumes both random and directed movements, then the one without advection prevails. If two competitors are drifting along the same direction but with different advection rates, then the one with the smaller advection rate wins. Finally we prove that if the two competitors are drifting along the opposite direction, then two species will coexist. These results imply that the movement without advection in homogeneous environment is evolutionarily stable, as advection tends to move more individuals to the boundary of the habitat and thus cause the distribution of species mismatch with the resources which are evenly distributed in space.
Keywords: Reaction-diffusion-advection, stability, advective environment, co-existence steady state., Lotka-Volterra competition.
Mathematics Subject Classification: Primary: 35K57, 35K61; Secondary: 92D2.
Citation: Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953
References:
[1]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Ser. Math. Comput Biol., Wiley and Sons, 2003. doi: 10.1002/0470871296.  Google Scholar

[2]

R. S. Cantrell, C. Cosner and Y. Lou, Movement toward better environments and the evolution of rapid diffusion, Math. Biosci., 204 (2006), 199-214. doi: 10.1016/j.mbs.2006.09.003.  Google Scholar

[3]

R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497-518. doi: 10.1017/S0308210506000047.  Google Scholar

[4]

X. F. 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.  Google Scholar

[5]

X. F. Chen, K.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst. A, 32 (2012), 3841-3859. doi: 10.3934/dcds.2012.32.3841.  Google Scholar

[6]

X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic 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.  Google Scholar

[7]

C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst. A, 34 (2014), 1701-1745. doi: 10.3934/dcds.2014.34.1701.  Google Scholar

[8]

K. A. Dahmen, D. R. Nelson and N. M. Shnerb, Life and death near a windy oasis, J. Math. Biol., 41 (2000), 1-23. doi: 10.1007/s002850000025.  Google Scholar

[9]

M. M. Desai and D. R. Nelson, A quasispecies on a moving oasis, Theor. Pop. Biol., 67 (2005), 33-45. doi: 10.1016/j.tpb.2004.07.005.  Google Scholar

[10]

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

[11]

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.  Google Scholar

[12]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk (N. S.), 3 (1948), 3-95.  Google Scholar

[13]

K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differential Equations, 250 (2011), 161-181. doi: 10.1016/j.jde.2010.08.028.  Google Scholar

[14]

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.  Google Scholar

[15]

K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst. A, 28 (2010), 1051-1067. doi: 10.3934/dcds.2010.28.1051.  Google Scholar

[16]

Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 159 (2015), 141-171. doi: 10.1016/j.jde.2015.02.004.  Google Scholar

[17]

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

[18]

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.  Google Scholar

[19]

W.-M. Ni, The Mathematics of Diffusion, CBMS Reg. Conf. Ser. Appl. Math., 82, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971972.  Google Scholar

[20]

A. Potapov, U. E. Schlägel and M. A. Lewis, Evolutionarily stable diffusive dispersal, Discrete Contin. Dyn. Syst. Series B, 19 (2014), 3319-3340. doi: 10.3934/dcdsb.2014.19.3319.  Google Scholar

[21]

H. Smith, Monotone Dynamical System. An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., 41 Amer. Math. Soc., Providence, RI, 1995.  Google Scholar

[22]

D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries, Ecology, 82 (2001), 1219-1237. Google Scholar

[23]

O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Quart., 18 (2010), 439-469.  Google Scholar

show all references

References:
[1]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Ser. Math. Comput Biol., Wiley and Sons, 2003. doi: 10.1002/0470871296.  Google Scholar

[2]

R. S. Cantrell, C. Cosner and Y. Lou, Movement toward better environments and the evolution of rapid diffusion, Math. Biosci., 204 (2006), 199-214. doi: 10.1016/j.mbs.2006.09.003.  Google Scholar

[3]

R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497-518. doi: 10.1017/S0308210506000047.  Google Scholar

[4]

X. F. 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.  Google Scholar

[5]

X. F. Chen, K.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst. A, 32 (2012), 3841-3859. doi: 10.3934/dcds.2012.32.3841.  Google Scholar

[6]

X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic 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.  Google Scholar

[7]

C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst. A, 34 (2014), 1701-1745. doi: 10.3934/dcds.2014.34.1701.  Google Scholar

[8]

K. A. Dahmen, D. R. Nelson and N. M. Shnerb, Life and death near a windy oasis, J. Math. Biol., 41 (2000), 1-23. doi: 10.1007/s002850000025.  Google Scholar

[9]

M. M. Desai and D. R. Nelson, A quasispecies on a moving oasis, Theor. Pop. Biol., 67 (2005), 33-45. doi: 10.1016/j.tpb.2004.07.005.  Google Scholar

[10]

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

[11]

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.  Google Scholar

[12]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk (N. S.), 3 (1948), 3-95.  Google Scholar

[13]

K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differential Equations, 250 (2011), 161-181. doi: 10.1016/j.jde.2010.08.028.  Google Scholar

[14]

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.  Google Scholar

[15]

K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst. A, 28 (2010), 1051-1067. doi: 10.3934/dcds.2010.28.1051.  Google Scholar

[16]

Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 159 (2015), 141-171. doi: 10.1016/j.jde.2015.02.004.  Google Scholar

[17]

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

[18]

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.  Google Scholar

[19]

W.-M. Ni, The Mathematics of Diffusion, CBMS Reg. Conf. Ser. Appl. Math., 82, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971972.  Google Scholar

[20]

A. Potapov, U. E. Schlägel and M. A. Lewis, Evolutionarily stable diffusive dispersal, Discrete Contin. Dyn. Syst. Series B, 19 (2014), 3319-3340. doi: 10.3934/dcdsb.2014.19.3319.  Google Scholar

[21]

H. Smith, Monotone Dynamical System. An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., 41 Amer. Math. Soc., Providence, RI, 1995.  Google Scholar

[22]

D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries, Ecology, 82 (2001), 1219-1237. Google Scholar

[23]

O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Quart., 18 (2010), 439-469.  Google Scholar

[1]

Qi Wang. On steady state of some Lotka-Volterra competition-diffusion-advection model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 859-875. doi: 10.3934/dcdsb.2019193
[2]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197
[3]

De Tang. Dynamical behavior for a Lotka-Volterra weak competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4913-4928. doi: 10.3934/dcdsb.2019037
[4]

Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195
[5]

Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021067
[6]

Qi Wang. Some global dynamics of a Lotka-Volterra competition-diffusion-advection system. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3245-3255. doi: 10.3934/cpaa.2020142
[7]

Qi Wang, Chunyi Gai, Jingda Yan. Qualitative analysis of a Lotka-Volterra competition system with advection. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1239-1284. doi: 10.3934/dcds.2015.35.1239
[8]

Zhi-Cheng Wang, Hui-Ling Niu, Shigui Ruan. On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1111-1144. doi: 10.3934/dcdsb.2017055
[9]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083
[10]

Danhua Jiang, Zhi-Cheng Wang, Liang Zhang. A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4557-4578. doi: 10.3934/dcdsb.2018176
[11]

Bo Duan, Zhengce Zhang. A two-species weak competition system of reaction-diffusion-advection with double free boundaries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 801-829. doi: 10.3934/dcdsb.2018208
[12]

Fang Li, Liping Wang, Yang Wang. On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 669-686. doi: 10.3934/dcdsb.2011.15.669
[13]

Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135
[14]

Shuling Yan, Shangjiang Guo. Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1559-1579. doi: 10.3934/dcdsb.2018059
[15]

Li-Jun Du, Wan-Tong Li, Jia-Bing Wang. Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1187-1213. doi: 10.3934/mbe.2017061
[16]

Yuan Lou, Salomé Martínez, Wei-Ming Ni. On $3\times 3$ Lotka-Volterra competition systems with cross-diffusion. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 175-190. doi: 10.3934/dcds.2000.6.175
[17]

Qian Guo, Xiaoqing He, Wei-Ming Ni. Global dynamics of a general Lotka-Volterra competition-diffusion system in heterogeneous environments. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6547-6573. doi: 10.3934/dcds.2020290
[18]

De-han Chen, Daijun jiang. Convergence rates of Tikhonov regularization for recovering growth rates in a Lotka-Volterra competition model with diffusion. Inverse Problems & Imaging, 2021, 15 (5) : 951-974. doi: 10.3934/ipi.2021023
[19]

Wenzhang Huang. Co-existence of traveling waves for a model of microbial growth and competition in a flow reactor. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 883-896. doi: 10.3934/dcds.2009.24.883
[20]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (233)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences