April  2021, 26(4): 2273-2297. doi: 10.3934/dcdsb.2021006

Positive solution branches of two-species competition model in open advective environments

1. 

School of Computer Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China

2. 

School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China

* Corresponding author

Received  August 2020 Revised  December 2020 Published  December 2020

Fund Project: The work is supported by the National Natural Science Foundation of China (12071270, 61907030), the Fundamental Research Funds for the Central Universities (GK201903088)

The effect of competition is an important topic in spatial ecology. This paper deals with a general two-species competition system in open advective and inhomogeneous environments. At first, the critical values on the interspecific competition coefficients are established, which determine the stability of semi-trivial steady states. Secondly, by analyzing the nonexistence of coexistence steady states and using the theory of monotone dynamical system, we find that the competitive exclusion principle holds if one of the interspecific competition coefficients is large and the other is in a certain range. Thirdly, in terms of these critical values, the structure and direction of bifurcating branches of positive equilibria arising from two semi-trivial steady states are given by means of the bifurcation theory and stability analysis. Finally, we show that multiple coexistence occurs under certain regimes.

Citation: Yan'e Wang, Nana Tian, Hua Nie. Positive solution branches of two-species competition model in open advective environments. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2273-2297. doi: 10.3934/dcdsb.2021006
References:
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J. E. Bailey and D. F. Ollis, Biochemical Engineering Fundamentals, McGraw-Hill, New York, 1986. Google Scholar

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R. S. CantrellC. CosnerS. Martínez and N. Torres, On a competitive system with ideal free dispersal, J. Differential Equations, 265 (2018), 3464-3493.  doi: 10.1016/j.jde.2018.05.008.  Google Scholar

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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

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K.-Y. LamY. Lou and F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), 188-212.  doi: 10.1080/17513758.2014.969336.  Google Scholar

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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

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Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

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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

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Y. LouH. Nie and Y. E. Wang, Coexistence and bistability of a competition model in open advective environments, Math. Biosci., 306 (2018), 10-19.  doi: 10.1016/j.mbs.2018.09.013.  Google Scholar

[15]

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

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Y. LouX.-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

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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

[18]

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

[19]

H. NieS.-B. Hsu and J. Wu, Coexistence solutions of a competition model with two species in a water column, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2691-2714.  doi: 10.3934/dcdsb.2015.20.2691.  Google Scholar

[20]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[21]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.  Google Scholar

[22]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[23]

D. Tang, Dynamical behavior for a Lotka-Volterra weak competition system in advective homogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4913-4928.  doi: 10.3934/dcdsb.2019037.  Google Scholar

[24]

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

[25]

Y. E. Wang, H. Nie and J. Wu, Coexistence and bistability of a competition model with mixed dispersal strategy, Nonlinear Anal. Real World Appl., 56 (2020), 103175, 19 pp. doi: 10.1016/j.nonrwa.2020.103175.  Google Scholar

[26]

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

[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. Funct. Anal., 275 (2018), 356-380.  doi: 10.1016/j.jfa.2018.03.006.  Google Scholar

[30]

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

[31]

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

show all references

References:
[1]

J. E. Bailey and D. F. Ollis, Biochemical Engineering Fundamentals, McGraw-Hill, New York, 1986. Google Scholar

[2]

R. S. CantrellC. CosnerS. Martínez and N. Torres, On a competitive system with ideal free dispersal, J. Differential Equations, 265 (2018), 3464-3493.  doi: 10.1016/j.jde.2018.05.008.  Google Scholar

[3]

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Interscience Publishers, New York, 1953.  Google Scholar

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

S.-B. Hsu and Y. Lou, Single phytoplankton species growth with light and advection in a water column, SIAM J. Appl. Math., 70 (2010), 2942-2974.  doi: 10.1137/100782358.  Google Scholar

[9]

S.-B. HsuH. L. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.  doi: 10.1090/S0002-9947-96-01724-2.  Google Scholar

[10]

K.-Y. LamY. Lou and F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), 188-212.  doi: 10.1080/17513758.2014.969336.  Google Scholar

[11]

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

[12]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[13]

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

[14]

Y. LouH. Nie and Y. E. Wang, Coexistence and bistability of a competition model in open advective environments, Math. Biosci., 306 (2018), 10-19.  doi: 10.1016/j.mbs.2018.09.013.  Google Scholar

[15]

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

[16]

Y. LouX.-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

[17]

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

[18]

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

[19]

H. NieS.-B. Hsu and J. Wu, Coexistence solutions of a competition model with two species in a water column, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2691-2714.  doi: 10.3934/dcdsb.2015.20.2691.  Google Scholar

[20]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[21]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.  Google Scholar

[22]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[23]

D. Tang, Dynamical behavior for a Lotka-Volterra weak competition system in advective homogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4913-4928.  doi: 10.3934/dcdsb.2019037.  Google Scholar

[24]

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

[25]

Y. E. Wang, H. Nie and J. Wu, Coexistence and bistability of a competition model with mixed dispersal strategy, Nonlinear Anal. Real World Appl., 56 (2020), 103175, 19 pp. doi: 10.1016/j.nonrwa.2020.103175.  Google Scholar

[26]

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

[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. Funct. Anal., 275 (2018), 356-380.  doi: 10.1016/j.jfa.2018.03.006.  Google Scholar

[30]

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

[31]

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

Figure 1.1.  The schematic diagrams of the curves $ \mathcal{S}^+ = \{(\|u\|_\infty, b): (u,v,b)\in \mathcal{B}^+\} $ with $ \mathcal{B}^+ $ defined in Section 4.3. Here $ b_0<b^* $ in (ⅰ)-(ⅱ), and $ b_0>b^* $ in (ⅲ)-(ⅳ)
Figure 1.2.  The schematic diagrams of the curves $ \mathcal{S}^+ = \{(\|u\|_\infty, b): (u,v,b)\in \mathcal{B}^+\} $ with $ \mathcal{B}^+ $ is defined in Section 4.3
Figure 1.3.  The schematic diagrams of the curves $ \mathcal{S}^+ = \{(\|u\|_\infty, b): (u,v,b)\in \mathcal{B}^+\} $ with $ \mathcal{B}^+ $ defined in Section 4.3
Figure 5.1.  The graphs of $ b_0-b^* $ vs. $ d_1 $ in (ⅰ) and vs. $ d_2 $ in (ⅱ) with other parameters fixed as (5.1)
Figure 5.2.  The graphs of $ b_0-b^* $ vs. $ d_1 $ in (ⅰ) and vs. $ d_2 $ in (ⅱ) with $ q_1 = q_2 = 0 $ and other parameters fixed as (5.1)
Figure 5.3.  Schematic diagram of the global dynamics on system (1.3) in $ b-c $ plane
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