November  2020, 40(11): 6547-6573. doi: 10.3934/dcds.2020290

Global dynamics of a general Lotka-Volterra competition-diffusion system in heterogeneous environments

1. 

School of Mathematical Sciences, East China Normal University, Shanghai 200241, China

2. 

School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

3. 

Chinese University of Hong Kong – Shenzhen, Shenzhen, China

4. 

School of Mathematics, University of Minnesota, MN 55455, USA

* Corresponding author: Xiaoqing He

Received  May 2020 Revised  May 2020 Published  July 2020

Previously in [14], we considered a diffusive logistic equation with two parameters, $ r(x) $ – intrinsic growth rate and $ K(x) $ – carrying capacity. We investigated and compared two special cases of the way in which $ r(x) $ and $ K(x) $ are related for both the logistic equations and the corresponding Lotka-Volterra competition-diffusion systems. In this paper, we continue to study the Lotka-Volterra competition-diffusion system with general intrinsic growth rates and carrying capacities for two competing species in heterogeneous environments. We establish the main result that determines the global dynamics of the system under a general criterion. Furthermore, when the ratios of the intrinsic growth rate to the carrying capacity for each species are proportional — such ratios can also be interpreted as the competition coefficients — this criterion reduces to what we obtained in [18]. We also study the detailed dynamics in terms of dispersal rates for such general case. On the other hand, when the two ratios are not proportional, our results in [14] show that the criterion in [18] cannot be fully recovered as counterexamples exist. This indicates the importance and subtleties of the roles of heterogeneous competition coefficients in the dynamics of the Lotka-Volterra competition-diffusion systems. Our results apply to competition-diffusion-advection systems as well. (See Corollary 5.1 in the last section.)

Citation: Qian Guo, Xiaoqing He, Wei-Ming Ni. Global dynamics of a general Lotka-Volterra competition-diffusion system in heterogeneous environments. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6547-6573. doi: 10.3934/dcds.2020290
References:
[1]

I. Averill, K.-Y. Lam and Y. Lou, The role of advection in a two-species competition model: A bifurcation approach, Mem. Amer. Math. Soc., 245 (2017) doi: 10.1090/memo/1161.  Google Scholar

[2]

I. AverillD. Munther and Y. Lou, On several conjectures from evolution of dispersal, J. Biol. Dyn., 6 (2012), 117-130.  doi: 10.1080/17513758.2010.529169.  Google Scholar

[3]

X. Bai and F. Li, Classification of global dynamics of competition models with nonlocal dispersals I: Symmetric kernels, Calc. Var. Partial Differential Equations, 57 (2018), 35pp. doi: 10.1007/s00526-018-1419-6.  Google Scholar

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K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120.  doi: 10.1016/0022-247X(80)90309-1.  Google Scholar

[5]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318.  doi: 10.1017/S030821050001876X.  Google Scholar

[6]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments. II, SIAM J. Math. Anal., 22 (1991), 1043-1064.  doi: 10.1137/0522068.  Google Scholar

[7]

R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the persistence of interacting species, J. Math. Biol., 37 (1998), 103-145.  doi: 10.1007/s002850050122.  Google Scholar

[8]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[9]

R. S. CantrellC. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution, Math. Biosci. Eng., 7 (2010), 17-36.  doi: 10.3934/mbe.2010.7.17.  Google Scholar

[10]

X. ChenK.-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

[11]

D. DeAngelisW.-M. Ni and B. Zhang, Dispersal and spatial heterogeneity: Single species, J. Math. Biol., 72 (2016), 239-254.  doi: 10.1007/s00285-015-0879-y.  Google Scholar

[12]

D. DeAngelisW.-M. Ni and B. Zhang, Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theoretical Ecology, 9 (2016), 443-453.  doi: 10.1007/s12080-016-0302-3.  Google Scholar

[13]

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

[14]

Q. Guo, X. He and W.-M. Ni, On the effects of carrying capacity and intrinsic growth rate on single and multiple species in spatially heterogeneous environments, J. Math. Biol., (2020). doi: 10.1007/s00285-020-01507-9.  Google Scholar

[15]

X. HeK.-Y. LamY. Lou and W.-M. Ni, Dynamics of a consumer-resource reaction-diffusion model, J. Math. Biol., 78 (2019), 1605-1636.  doi: 10.1007/s00285-018-1321-z.  Google Scholar

[16]

X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system I: Heterogeneity vs. homogeneity, J. Differential Equations, 254 (2013), 528-546.  doi: 10.1016/j.jde.2012.08.032.  Google Scholar

[17]

X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system II: The general case, J. Differential Equations, 254 (2013), 4088-4108.  doi: 10.1016/j.jde.2013.02.009.  Google Scholar

[18]

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

[19]

X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, II, Calc. Var. Partial Differential Equations, 55 (2016), 20pp. doi: 10.1007/s00526-016-0964-0.  Google Scholar

[20]

X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, III, Calc. Var. Partial Differential Equations, 56 (2017), 26pp. doi: 10.1007/s00526-017-1234-5.  Google Scholar

[21]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow; copublished in the US with John Wiley & Sons, Inc., New York), 1991.  Google Scholar

[22]

P. Hess and S. Senn, Another approach to elliptic eigenvalue problems with respect to indefinite weight functions, in Nonlinear Analysis and Optimization, Lecture Notes in Math., 1107, Springer, Berlin, 1984,106–114. doi: 10.1007/BFb0101496.  Google Scholar

[23]

M. W. Hirsch and H. L. Smith, Asymptotically stable equilibria for monotone semiflows, Discrete Contin. Dyn. Syst., 14 (2006), 385-398.  doi: 10.3934/dcds.2006.14.385.  Google Scholar

[24]

V. HutsonY. Lou and and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136.  doi: 10.1006/jdeq.2001.4157.  Google Scholar

[25]

M. G. Kre${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$n and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Matem. Nauk (N. S.), 3 (1948), 3-95.   Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutorials in Mathematical Biosciences, Lecture Notes in Math., 1922, Springer, Berlin, 2008,171–205. doi: 10.1007/978-3-540-74331-6_5.  Google Scholar

[31]

Y. Lou and E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics, Japan J. Indust. Appl. Math., 23 (2006), 275-292.  doi: 10.1007/BF03167595.  Google Scholar

[32]

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

[33]

Y. Lou, X.-Q. Zhao and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments, J. Math. Pures Appl. (9), 121 (2019) 47–82. doi: 10.1016/j.matpur.2018.06.010.  Google Scholar

[34]

J. Mallet, The struggle for existence: How the notion of carrying capacity, K, obscures the links between demography, Darwinian evolution,and speciation, Evolutionary Ecology Research, 14 (2012), 627-665.   Google Scholar

[35]

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

[36]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 979-1000.  doi: 10.1512/iumj.1972.21.21079.  Google Scholar

[37]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann., 258 (1981/82), 459-470.  doi: 10.1007/BF01453979.  Google Scholar

[38]

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

[39]

D. Tang and P. Zhou, On a Lotka-Volterra competition-diffusion-advection system: Homogeneity vs heterogeneity, J. Differential Equations, 268 (2020), 1570-1599.  doi: 10.1016/j.jde.2019.09.003.  Google Scholar

[40]

B. Zhang, D. DeAngelis, W.-M. Ni, Y. Wang, L. Zhai, A. Kula, S. Xu and J. D. Van Dyken, Effect of stressors on the carrying capacity of spatially-distributed metapopulations, Amer. Naturalist, (2020). doi: 10.1086/709293.  Google Scholar

[41]

B. ZhangA. KulaK. MackL. ZhaiA. RyceW.-M. NiD. DeAngelis and J. D. Van Dyken, Carrying capacity in a heterogeneous environment with habitat connectivity, Ecology Lett., 20 (2017), 1118-1128.  doi: 10.1111/ele.12807.  Google Scholar

[42]

B. ZhangX. LiuD. L. DeAngelisW.-M. Ni and G. Wang, Effects of dispersal on total biomass in a patchy heterogeneous environment: Analysis and experiment, Math. Biosci., 264 (2015), 54-62.  doi: 10.1016/j.mbs.2015.03.005.  Google Scholar

[43]

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

show all references

References:
[1]

I. Averill, K.-Y. Lam and Y. Lou, The role of advection in a two-species competition model: A bifurcation approach, Mem. Amer. Math. Soc., 245 (2017) doi: 10.1090/memo/1161.  Google Scholar

[2]

I. AverillD. Munther and Y. Lou, On several conjectures from evolution of dispersal, J. Biol. Dyn., 6 (2012), 117-130.  doi: 10.1080/17513758.2010.529169.  Google Scholar

[3]

X. Bai and F. Li, Classification of global dynamics of competition models with nonlocal dispersals I: Symmetric kernels, Calc. Var. Partial Differential Equations, 57 (2018), 35pp. doi: 10.1007/s00526-018-1419-6.  Google Scholar

[4]

K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120.  doi: 10.1016/0022-247X(80)90309-1.  Google Scholar

[5]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318.  doi: 10.1017/S030821050001876X.  Google Scholar

[6]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments. II, SIAM J. Math. Anal., 22 (1991), 1043-1064.  doi: 10.1137/0522068.  Google Scholar

[7]

R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the persistence of interacting species, J. Math. Biol., 37 (1998), 103-145.  doi: 10.1007/s002850050122.  Google Scholar

[8]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[9]

R. S. CantrellC. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution, Math. Biosci. Eng., 7 (2010), 17-36.  doi: 10.3934/mbe.2010.7.17.  Google Scholar

[10]

X. ChenK.-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

[11]

D. DeAngelisW.-M. Ni and B. Zhang, Dispersal and spatial heterogeneity: Single species, J. Math. Biol., 72 (2016), 239-254.  doi: 10.1007/s00285-015-0879-y.  Google Scholar

[12]

D. DeAngelisW.-M. Ni and B. Zhang, Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theoretical Ecology, 9 (2016), 443-453.  doi: 10.1007/s12080-016-0302-3.  Google Scholar

[13]

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

[14]

Q. Guo, X. He and W.-M. Ni, On the effects of carrying capacity and intrinsic growth rate on single and multiple species in spatially heterogeneous environments, J. Math. Biol., (2020). doi: 10.1007/s00285-020-01507-9.  Google Scholar

[15]

X. HeK.-Y. LamY. Lou and W.-M. Ni, Dynamics of a consumer-resource reaction-diffusion model, J. Math. Biol., 78 (2019), 1605-1636.  doi: 10.1007/s00285-018-1321-z.  Google Scholar

[16]

X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system I: Heterogeneity vs. homogeneity, J. Differential Equations, 254 (2013), 528-546.  doi: 10.1016/j.jde.2012.08.032.  Google Scholar

[17]

X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system II: The general case, J. Differential Equations, 254 (2013), 4088-4108.  doi: 10.1016/j.jde.2013.02.009.  Google Scholar

[18]

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

[19]

X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, II, Calc. Var. Partial Differential Equations, 55 (2016), 20pp. doi: 10.1007/s00526-016-0964-0.  Google Scholar

[20]

X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, III, Calc. Var. Partial Differential Equations, 56 (2017), 26pp. doi: 10.1007/s00526-017-1234-5.  Google Scholar

[21]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow; copublished in the US with John Wiley & Sons, Inc., New York), 1991.  Google Scholar

[22]

P. Hess and S. Senn, Another approach to elliptic eigenvalue problems with respect to indefinite weight functions, in Nonlinear Analysis and Optimization, Lecture Notes in Math., 1107, Springer, Berlin, 1984,106–114. doi: 10.1007/BFb0101496.  Google Scholar

[23]

M. W. Hirsch and H. L. Smith, Asymptotically stable equilibria for monotone semiflows, Discrete Contin. Dyn. Syst., 14 (2006), 385-398.  doi: 10.3934/dcds.2006.14.385.  Google Scholar

[24]

V. HutsonY. Lou and and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136.  doi: 10.1006/jdeq.2001.4157.  Google Scholar

[25]

M. G. Kre${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$n and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Matem. Nauk (N. S.), 3 (1948), 3-95.   Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutorials in Mathematical Biosciences, Lecture Notes in Math., 1922, Springer, Berlin, 2008,171–205. doi: 10.1007/978-3-540-74331-6_5.  Google Scholar

[31]

Y. Lou and E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics, Japan J. Indust. Appl. Math., 23 (2006), 275-292.  doi: 10.1007/BF03167595.  Google Scholar

[32]

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

[33]

Y. Lou, X.-Q. Zhao and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments, J. Math. Pures Appl. (9), 121 (2019) 47–82. doi: 10.1016/j.matpur.2018.06.010.  Google Scholar

[34]

J. Mallet, The struggle for existence: How the notion of carrying capacity, K, obscures the links between demography, Darwinian evolution,and speciation, Evolutionary Ecology Research, 14 (2012), 627-665.   Google Scholar

[35]

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

[36]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 979-1000.  doi: 10.1512/iumj.1972.21.21079.  Google Scholar

[37]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann., 258 (1981/82), 459-470.  doi: 10.1007/BF01453979.  Google Scholar

[38]

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

[39]

D. Tang and P. Zhou, On a Lotka-Volterra competition-diffusion-advection system: Homogeneity vs heterogeneity, J. Differential Equations, 268 (2020), 1570-1599.  doi: 10.1016/j.jde.2019.09.003.  Google Scholar

[40]

B. Zhang, D. DeAngelis, W.-M. Ni, Y. Wang, L. Zhai, A. Kula, S. Xu and J. D. Van Dyken, Effect of stressors on the carrying capacity of spatially-distributed metapopulations, Amer. Naturalist, (2020). doi: 10.1086/709293.  Google Scholar

[41]

B. ZhangA. KulaK. MackL. ZhaiA. RyceW.-M. NiD. DeAngelis and J. D. Van Dyken, Carrying capacity in a heterogeneous environment with habitat connectivity, Ecology Lett., 20 (2017), 1118-1128.  doi: 10.1111/ele.12807.  Google Scholar

[42]

B. ZhangX. LiuD. L. DeAngelisW.-M. Ni and G. Wang, Effects of dispersal on total biomass in a patchy heterogeneous environment: Analysis and experiment, Math. Biosci., 264 (2015), 54-62.  doi: 10.1016/j.mbs.2015.03.005.  Google Scholar

[43]

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

Figure 1.  Global dynamics of system (4) with $ r_i/\xi_i\not\equiv{\rm{const}} $, $ i = 1, 2 $. See Theorem 1.3
Figure 2.  Shapes of $ \Sigma_U $ and $ \Sigma_- $ for Theorem 1.4 as illustration
Figure 3.  Another scenario for Theorem 1.4. See also Theorem 4.2(ⅰ) for details
Figure 4.  Shapes of $ \Sigma_U $, $ \Sigma_V $ and $ \Sigma_- $ for Theorem 1.5 as illustration
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