September  2019, 24(9): 4913-4928. doi: 10.3934/dcdsb.2019037

Dynamical behavior for a Lotka-Volterra weak competition system in advective homogeneous environment

School of Mathematics(Zhuhai), Sun Yat-sen University, Zhuhai 519082, Guangdong, China

* Corresponding author

Received  April 2018 Revised  July 2018 Published  February 2019

Fund Project: The author is supported by NSF grant 11801089, Postdoctoral Science Foundation of China(N0.2018M643281)

We consider a two-species Lotka-Volterra weak competition model in a one-dimensional advective homogeneous environment, where individuals are exposed to unidirectional flow. It is assumed that two species have the same population dynamics but different diffusion rates, advection rates and intensities of competition. We study the following useful scenarios: (1) if one species disperses by random diffusion only and the other assumes both random and unidirectional movements, two species will coexist; (2) if two species are drifting along the different direction, two species will coexist; (3) if the intensities of inter-specific competition are small enough, two species will coexist; (4) if the intensities of inter-specific competition are close to 1, the competitive exclusion principle holds. These results provide a new mechanism for the coexistence of competing species. Finally, we apply a perturbation argument to illustrate that two species will converge to the unique coexistence steady state.

Citation: 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
References:
[1]

L. Altenberg, Resolvent positive linear operators exhibit the reduction phenomenon, Proc. Natl. Acad. Sci. USA, 109 (2012), 3705-3710. doi: 10.1073/pnas.1113833109. Google Scholar

[2]

F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environments, Canad. Appl. Math. Quart., 3 (1995), 379-397. Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296. Google Scholar

[4]

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

[5]

X. F. Chen and Y. Lou, Effects of diffusion and advection on the smallest eigenvalue of an elliptic operator and their applications, Indiana Univ. Math. J., 61 (2012), 45-80. doi: 10.1512/iumj.2012.61.4518. Google Scholar

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

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J. DockeryV. HutsonK. 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

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L. Evans, Partial Differential Equations, 2nd Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. Google Scholar

[9] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.
[10] P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, Longman Scientific and Technical, Harlow, UK, 1991.
[11]

M. G. Krein 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

[12]

K. Y. Lam and W. M. Ni, Uniqueness and complete dynamics in heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712. doi: 10.1137/120869481. Google Scholar

[13]

Y. Lou, Some Challenging Mathematical Problems in Evolution of Dispersal and Population Dynamics, Tutorials in Mathematical Biosciences IV, Lecture Notes in Math., 1922, Springer, Berlin, 2008,171–205. doi: 10.1007/978-3-540-74331-6_5. Google Scholar

[14]

Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection in population genetics, J. Differential Equations, 204 (2004), 292-322. doi: 10.1016/j.jde.2004.01.009. Google Scholar

[15]

F. LutscherM. 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

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F. LutscherE. 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

[17]

Y. LouD. 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

[18]

Y. LouX.-Q. Zhao and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments, J. Math. Pures Appl., 275 (2018), 356-380. doi: 10.1016/j.jfa.2018.03.006. Google Scholar

[19]

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

[20]

W. M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972. Google Scholar

[21]

H. Richard, Evolution of Conditional Dispersal: A Reaction-Diffusion-Advection Approach, Ph.D thesis, The Ohio State University, 2007.Google Scholar

[22]

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

[23]

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

[24]

D. Tang and L. Ma, Dynamical behavior of a general reaction-diffusion-advection model for two competing species, Computers and Mathematics with Applications, 75 (2018), 1128-1142. doi: 10.1016/j.camwa.2017.10.026. Google Scholar

[25]

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

[26] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.
[27]

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

[28]

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

[29]

P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. of Functional Analysis, 275 (2018), 356-380. doi: 10.1016/j.jfa.2018.03.006. Google Scholar

[30]

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

[31]

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

show all references

References:
[1]

L. Altenberg, Resolvent positive linear operators exhibit the reduction phenomenon, Proc. Natl. Acad. Sci. USA, 109 (2012), 3705-3710. doi: 10.1073/pnas.1113833109. Google Scholar

[2]

F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environments, Canad. Appl. Math. Quart., 3 (1995), 379-397. Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296. Google Scholar

[4]

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

[5]

X. F. Chen and Y. Lou, Effects of diffusion and advection on the smallest eigenvalue of an elliptic operator and their applications, Indiana Univ. Math. J., 61 (2012), 45-80. doi: 10.1512/iumj.2012.61.4518. Google Scholar

[6]

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

[7]

J. DockeryV. HutsonK. 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

[8]

L. Evans, Partial Differential Equations, 2nd Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. Google Scholar

[9] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.
[10] P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, Longman Scientific and Technical, Harlow, UK, 1991.
[11]

M. G. Krein 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

[12]

K. Y. Lam and W. M. Ni, Uniqueness and complete dynamics in heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712. doi: 10.1137/120869481. Google Scholar

[13]

Y. Lou, Some Challenging Mathematical Problems in Evolution of Dispersal and Population Dynamics, Tutorials in Mathematical Biosciences IV, Lecture Notes in Math., 1922, Springer, Berlin, 2008,171–205. doi: 10.1007/978-3-540-74331-6_5. Google Scholar

[14]

Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection in population genetics, J. Differential Equations, 204 (2004), 292-322. doi: 10.1016/j.jde.2004.01.009. Google Scholar

[15]

F. LutscherM. 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

[16]

F. LutscherE. 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

[17]

Y. LouD. 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

[18]

Y. LouX.-Q. Zhao and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments, J. Math. Pures Appl., 275 (2018), 356-380. doi: 10.1016/j.jfa.2018.03.006. Google Scholar

[19]

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

[20]

W. M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972. Google Scholar

[21]

H. Richard, Evolution of Conditional Dispersal: A Reaction-Diffusion-Advection Approach, Ph.D thesis, The Ohio State University, 2007.Google Scholar

[22]

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

[23]

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

[24]

D. Tang and L. Ma, Dynamical behavior of a general reaction-diffusion-advection model for two competing species, Computers and Mathematics with Applications, 75 (2018), 1128-1142. doi: 10.1016/j.camwa.2017.10.026. Google Scholar

[25]

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

[26] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.
[27]

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

[28]

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

[29]

P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. of Functional Analysis, 275 (2018), 356-380. doi: 10.1016/j.jfa.2018.03.006. Google Scholar

[30]

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

[31]

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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