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  April 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (4) : 2273-2297. doi: 10.3934/dcdsb.2021006
References:
[1]

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

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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