American Institute of Mathematical Sciences

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February  2018, 38(2): 791-821. doi: 10.3934/dcds.2018034

N-barrier maximum principle for degenerate elliptic systems and its application

Chiun-Chuan Chen 1, , Li-Chang Hung 2,, and  Hsiao-Feng Liu 3,

1. 

Department of Mathematics, National Taiwan University, National Center for Theoretical Sciences, Taipei, Taiwan
2. 

College of Engineering, National Taiwan University of Science and Technology, Department of Mathematics, National Taiwan University, Taipei, Taiwan
3. 

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan

* Corresponding author

Received  November 2016 Revised  September 2017 Published  February 2018

Fund Project: The research of C.-C. Chen is partly supported by the grant 102-2115-M-002-011-MY3 of Ministry of Science and Technology, Taiwan. The research of L.-C. Hung is partly supported by the grant 104EFA0101550 of Ministry of Science and Technology, Taiwan.

In this paper, we prove the N-barrier maximum principle, which extends the result in C.-C. Chen and L.-C. Hung (2016) from linear diffusion equations to nonlinear diffusion equations, for a wide class of degenerate elliptic systems of porous medium type. The N-barrier maximum principle provides a priori upper and lower bounds of the solutions to the above-mentioned degenerate nonlinear diffusion equations including the Shigesada-Kawasaki-Teramoto model as a special case. We also apply the N-barrier maximum principle to a coexistence problem in ecology, where we show the nonexistence of traveling waves in a three-species degenerate elliptic system.

Keywords: Maximum principles, traveling wave solutions, degenerate elliptic systems, reaction-diffusion equations, Lotka-Volterra.
Mathematics Subject Classification: Primary: 35B50; Secondary: 35C07, 35K57.
Citation: Chiun-Chuan Chen, Li-Chang Hung, Hsiao-Feng Liu. N-barrier maximum principle for degenerate elliptic systems and its application. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 791-821. doi: 10.3934/dcds.2018034
References:
[1]

M. W. Adamson and A. Y. Morozov, Revising the role of species mobility in maintaining biodiversity in communities with cyclic competition, Bull. Math. Biol., 74 (2012), 2004-2031.  doi: 10.1007/s11538-012-9743-z.  Google Scholar

[2]

R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170.  doi: 10.1086/283553.  Google Scholar

[3]

A. J. Baczkowski, D. N. Joanes and G. M. Shamia, Range of validity of $α$ and $β$ for a generalized diversity index $H(α,β)$ due to Good, Math. Biosci., 148 (1998), 115-128.  doi: 10.1016/S0025-5564(97)10013-X.  Google Scholar

[4]

R. S. Cantrell and J. R., Jr. Ward, On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.  doi: 10.1137/S0036139995292367.  Google Scholar

[5]

C.-C. Chen and L.-C. Hung, A maximum principle for diffusive lotka-volterra systems of two competing species, J. Differential Equations, 261 (2016), 4573-4592.  doi: 10.1016/j.jde.2016.07.001.  Google Scholar

[6]

C.-C. Chen and L.-C. Hung, Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species, Commun. Pure Appl. Anal., 15 (2016), 1451-1469.  doi: 10.3934/cpaa.2016.15.1451.  Google Scholar

[7]

C. -C. Chen, L. -C. Hung and C. -C. Lai, An N-barrier maximum principle for autonomous systems of n species and its application to problems arising from population dynamics, submitted. Google Scholar

[8]

C.-C. Chen, L.-C. Hung, M. Mimura and D. Ueyama, Exact travelling wave solutions of three-species competition-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2653-2669.  doi: 10.3934/dcdsb.2012.17.2653.  Google Scholar

[9]

P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234.  doi: 10.1137/S0036139995294767.  Google Scholar

[10]

S.-I. Ei, R. Ikota and M. Mimura, Segregating partition problem in competition-diffusion systems, Interfaces Free Bound, 1 (1999), 57-80.  doi: 10.4171/IFB/4.  Google Scholar

[11]

B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction Progress in Nonlinear Differential Equations and their Applications, 60, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4.  Google Scholar

[12]

I. J. Good, The population frequencies of species and the estimation of population parameters, Biometrika, 40 (1953), 237-264.  doi: 10.1093/biomet/40.3-4.237.  Google Scholar

[13]

S. Grossberg, Decisions, patterns, and oscillations in nonlinear competitve systems with applications to Volterra-Lotka systems, J. Theoret. Biol., 73 (1978), 101-130.  doi: 10.1016/0022-5193(78)90182-0.  Google Scholar

[14]

M. Guedda, R. Kersner, M. Klincsik and E. Logak, Exact wavefronts and periodic patterns in a competition system with nonlinear diffusion, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1589-1600.  doi: 10.3934/dcdsb.2014.19.1589.  Google Scholar

[15]

W. Gurney and R. Nisbet, The regulation of inhomogeneous populations, Journal of Theoretical Biology, 52 (1975), 441-457.  doi: 10.1016/0022-5193(75)90011-9.  Google Scholar

[16]

W. Gurney and R. Nisbet, A note on non-linear population transport, Journal of theoretical biology, 56 (1976), 249-251.  doi: 10.1016/S0022-5193(76)80056-2.  Google Scholar

[17]

M. Gyllenberg and P. Yan, On a conjecture for three-dimensional competitive Lotka-Volterra systems with a heteroclinic cycle, Differ. Equ. Appl., 1 (2009), 473-490.  doi: 10.7153/dea-01-26.  Google Scholar

[18]

T. G. Hallam, L. J. Svoboda and T. C. Gard, Persistence and extinction in three species Lotka-Volterra competitive systems, Math. Biosci., 46 (1979), 117-124.  doi: 10.1016/0025-5564(79)90018-X.  Google Scholar

[19]

S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617.  doi: 10.1137/070700784.  Google Scholar

[20]

S. B. Hsu, H. 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

[21]

L.-C. Hung, Exact traveling wave solutions for diffusive Lotka-Volterra systems of two competing species, Jpn. J. Ind. Appl. Math., 29 (2012), 237-251.  doi: 10.1007/s13160-012-0056-2.  Google Scholar

[22]

S. R.-J. Jang, Competitive exclusion and coexistence in a Leslie-Gower competition model with Allee effects, Appl. Anal., 92 (2013), 1527-1540.  doi: 10.1080/00036811.2012.692365.  Google Scholar

[23]

J. Kastendiek, Competitor-mediated coexistence: interactions among three species of benthic macroalgae, Journal of Experimental Marine Biology and Ecology, 62 (1982), 201-210.  doi: 10.1016/0022-0981(82)90201-5.  Google Scholar

[24]

W. Ko, K. Ryu and I. Ahn, Coexistence of three competing species with non-negative cross-diffusion rate, J. Dyn. Control Syst., 20 (2014), 229-240.  doi: 10.1007/s10883-014-9219-6.  Google Scholar

[25]

R. S. Maier, The integration of three-dimensional Lotka-Volterra systems Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 469 (2013), 20120693, 27pp. doi: 10.1098/rspa.2012.0693.  Google Scholar

[26]

R. McGehee and R. A. Armstrong, Some mathematical problems concerning the ecological principle of competitive exclusion, J. Differential Equations, 23 (1977), 30-52.  doi: 10.1016/0022-0396(77)90135-8.  Google Scholar

[27]

M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition--diffusion system, Ecological Complexity, 21 (2015), 215-232.  doi: 10.1016/j.ecocom.2014.05.004.  Google Scholar

[28]

J. D. Murray, Mathematical Biology Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[29]

A. de Pablo and A. Sánchez, Travelling wave behaviour for a porous-Fisher equation, European J. Appl. Math., 9 (1998), 285-304.  doi: 10.1017/S0956792598003465.  Google Scholar

[30]

A. de Pablo and J. L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equations, 93 (1991), 19-61.  doi: 10.1016/0022-0396(91)90021-Z.  Google Scholar

[31]

S. Petrovskii, K. Kawasaki, F. Takasu and N. Shigesada, Diffusive waves, dynamical stabilization and spatio-temporal chaos in a community of three competitive species, Japan J. Indust. Appl. Math., 18 (2001), 459-481.  doi: 10.1007/BF03168586.  Google Scholar

[32]

H. Ramezani and S. Holm, Sample based estimation of landscape metrics; accuracy of line intersect sampling for estimating edge density and Shannon's diversity index, Environ. Ecol. Stat., 18 (2011), 109-130.  doi: 10.1007/s10651-009-0123-2.  Google Scholar

[33]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270.   Google Scholar

[34]

M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations, Japan J. Indust. Appl. Math., 18 (2001), 657-696.  doi: 10.1007/BF03167410.  Google Scholar

[35]

F. Sánchez-Garduño and P. K. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations, J. Math. Biol., 33 (1994), 163-192.  doi: 10.1007/BF00160178.  Google Scholar

[36]

F. Sánchez-Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations, 117 (1995), 281-319.  doi: 10.1006/jdeq.1995.1055.  Google Scholar

[37]

J. A. Sherratt and B. P. Marchant, Nonsharp travelling wave fronts in the Fisher equation with degenerate nonlinear diffusion, Appl. Math. Lett., 9 (1996), 33-38.  doi: 10.1016/0893-9659(96)00069-9.  Google Scholar

[38]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.  Google Scholar

[39]

E. H. Simpson, Measurement of diversity Nature, 163 (1949), p688. doi: 10.1038/163688a0.  Google Scholar

[40]

H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: the variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.  doi: 10.1137/S0036139993245344.  Google Scholar

[41]

T. P. Witelski, Merging traveling waves for the porous-Fisher's equation, Appl. Math. Lett., 8 (1995), 57-62.  doi: 10.1016/0893-9659(95)00047-T.  Google Scholar

[42]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.  doi: 10.1080/02681119308806158.  Google Scholar

[43]

J. B. Zeldovič, G. I. Barenblatt, V. B. Librovič and G. M. Mahviladze, Matematicheskaya Teoriya Goreniya I Vzryva (Mathematical Theory of Combustion and Explosion) "Nauka", Moscow, 1980.  Google Scholar

[44]

Y. B. Zeldovich, Theory of Flame Propagation 1951. Google Scholar

show all references

References:
[1]

M. W. Adamson and A. Y. Morozov, Revising the role of species mobility in maintaining biodiversity in communities with cyclic competition, Bull. Math. Biol., 74 (2012), 2004-2031.  doi: 10.1007/s11538-012-9743-z.  Google Scholar

[2]

R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170.  doi: 10.1086/283553.  Google Scholar

[3]

A. J. Baczkowski, D. N. Joanes and G. M. Shamia, Range of validity of $α$ and $β$ for a generalized diversity index $H(α,β)$ due to Good, Math. Biosci., 148 (1998), 115-128.  doi: 10.1016/S0025-5564(97)10013-X.  Google Scholar

[4]

R. S. Cantrell and J. R., Jr. Ward, On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.  doi: 10.1137/S0036139995292367.  Google Scholar

[5]

C.-C. Chen and L.-C. Hung, A maximum principle for diffusive lotka-volterra systems of two competing species, J. Differential Equations, 261 (2016), 4573-4592.  doi: 10.1016/j.jde.2016.07.001.  Google Scholar

[6]

C.-C. Chen and L.-C. Hung, Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species, Commun. Pure Appl. Anal., 15 (2016), 1451-1469.  doi: 10.3934/cpaa.2016.15.1451.  Google Scholar

[7]

C. -C. Chen, L. -C. Hung and C. -C. Lai, An N-barrier maximum principle for autonomous systems of n species and its application to problems arising from population dynamics, submitted. Google Scholar

[8]

C.-C. Chen, L.-C. Hung, M. Mimura and D. Ueyama, Exact travelling wave solutions of three-species competition-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2653-2669.  doi: 10.3934/dcdsb.2012.17.2653.  Google Scholar

[9]

P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234.  doi: 10.1137/S0036139995294767.  Google Scholar

[10]

S.-I. Ei, R. Ikota and M. Mimura, Segregating partition problem in competition-diffusion systems, Interfaces Free Bound, 1 (1999), 57-80.  doi: 10.4171/IFB/4.  Google Scholar

[11]

B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction Progress in Nonlinear Differential Equations and their Applications, 60, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4.  Google Scholar

[12]

I. J. Good, The population frequencies of species and the estimation of population parameters, Biometrika, 40 (1953), 237-264.  doi: 10.1093/biomet/40.3-4.237.  Google Scholar

[13]

S. Grossberg, Decisions, patterns, and oscillations in nonlinear competitve systems with applications to Volterra-Lotka systems, J. Theoret. Biol., 73 (1978), 101-130.  doi: 10.1016/0022-5193(78)90182-0.  Google Scholar

[14]

M. Guedda, R. Kersner, M. Klincsik and E. Logak, Exact wavefronts and periodic patterns in a competition system with nonlinear diffusion, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1589-1600.  doi: 10.3934/dcdsb.2014.19.1589.  Google Scholar

[15]

W. Gurney and R. Nisbet, The regulation of inhomogeneous populations, Journal of Theoretical Biology, 52 (1975), 441-457.  doi: 10.1016/0022-5193(75)90011-9.  Google Scholar

[16]

W. Gurney and R. Nisbet, A note on non-linear population transport, Journal of theoretical biology, 56 (1976), 249-251.  doi: 10.1016/S0022-5193(76)80056-2.  Google Scholar

[17]

M. Gyllenberg and P. Yan, On a conjecture for three-dimensional competitive Lotka-Volterra systems with a heteroclinic cycle, Differ. Equ. Appl., 1 (2009), 473-490.  doi: 10.7153/dea-01-26.  Google Scholar

[18]

T. G. Hallam, L. J. Svoboda and T. C. Gard, Persistence and extinction in three species Lotka-Volterra competitive systems, Math. Biosci., 46 (1979), 117-124.  doi: 10.1016/0025-5564(79)90018-X.  Google Scholar

[19]

S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617.  doi: 10.1137/070700784.  Google Scholar

[20]

S. B. Hsu, H. 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

[21]

L.-C. Hung, Exact traveling wave solutions for diffusive Lotka-Volterra systems of two competing species, Jpn. J. Ind. Appl. Math., 29 (2012), 237-251.  doi: 10.1007/s13160-012-0056-2.  Google Scholar

[22]

S. R.-J. Jang, Competitive exclusion and coexistence in a Leslie-Gower competition model with Allee effects, Appl. Anal., 92 (2013), 1527-1540.  doi: 10.1080/00036811.2012.692365.  Google Scholar

[23]

J. Kastendiek, Competitor-mediated coexistence: interactions among three species of benthic macroalgae, Journal of Experimental Marine Biology and Ecology, 62 (1982), 201-210.  doi: 10.1016/0022-0981(82)90201-5.  Google Scholar

[24]

W. Ko, K. Ryu and I. Ahn, Coexistence of three competing species with non-negative cross-diffusion rate, J. Dyn. Control Syst., 20 (2014), 229-240.  doi: 10.1007/s10883-014-9219-6.  Google Scholar

[25]

R. S. Maier, The integration of three-dimensional Lotka-Volterra systems Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 469 (2013), 20120693, 27pp. doi: 10.1098/rspa.2012.0693.  Google Scholar

[26]

R. McGehee and R. A. Armstrong, Some mathematical problems concerning the ecological principle of competitive exclusion, J. Differential Equations, 23 (1977), 30-52.  doi: 10.1016/0022-0396(77)90135-8.  Google Scholar

[27]

M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition--diffusion system, Ecological Complexity, 21 (2015), 215-232.  doi: 10.1016/j.ecocom.2014.05.004.  Google Scholar

[28]

J. D. Murray, Mathematical Biology Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[29]

A. de Pablo and A. Sánchez, Travelling wave behaviour for a porous-Fisher equation, European J. Appl. Math., 9 (1998), 285-304.  doi: 10.1017/S0956792598003465.  Google Scholar

[30]

A. de Pablo and J. L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equations, 93 (1991), 19-61.  doi: 10.1016/0022-0396(91)90021-Z.  Google Scholar

[31]

S. Petrovskii, K. Kawasaki, F. Takasu and N. Shigesada, Diffusive waves, dynamical stabilization and spatio-temporal chaos in a community of three competitive species, Japan J. Indust. Appl. Math., 18 (2001), 459-481.  doi: 10.1007/BF03168586.  Google Scholar

[32]

H. Ramezani and S. Holm, Sample based estimation of landscape metrics; accuracy of line intersect sampling for estimating edge density and Shannon's diversity index, Environ. Ecol. Stat., 18 (2011), 109-130.  doi: 10.1007/s10651-009-0123-2.  Google Scholar

[33]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270.   Google Scholar

[34]

M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations, Japan J. Indust. Appl. Math., 18 (2001), 657-696.  doi: 10.1007/BF03167410.  Google Scholar

[35]

F. Sánchez-Garduño and P. K. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations, J. Math. Biol., 33 (1994), 163-192.  doi: 10.1007/BF00160178.  Google Scholar

[36]

F. Sánchez-Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations, 117 (1995), 281-319.  doi: 10.1006/jdeq.1995.1055.  Google Scholar

[37]

J. A. Sherratt and B. P. Marchant, Nonsharp travelling wave fronts in the Fisher equation with degenerate nonlinear diffusion, Appl. Math. Lett., 9 (1996), 33-38.  doi: 10.1016/0893-9659(96)00069-9.  Google Scholar

[38]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.  Google Scholar

[39]

E. H. Simpson, Measurement of diversity Nature, 163 (1949), p688. doi: 10.1038/163688a0.  Google Scholar

[40]

H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: the variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.  doi: 10.1137/S0036139993245344.  Google Scholar

[41]

T. P. Witelski, Merging traveling waves for the porous-Fisher's equation, Appl. Math. Lett., 8 (1995), 57-62.  doi: 10.1016/0893-9659(95)00047-T.  Google Scholar

[42]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.  doi: 10.1080/02681119308806158.  Google Scholar

[43]

J. B. Zeldovič, G. I. Barenblatt, V. B. Librovič and G. M. Mahviladze, Matematicheskaya Teoriya Goreniya I Vzryva (Mathematical Theory of Combustion and Explosion) "Nauka", Moscow, 1980.  Google Scholar

[44]

Y. B. Zeldovich, Theory of Flame Propagation 1951. Google Scholar

Figure 1.  Red line: $1-u-a_1\, v=0$; blue line: $1-a_2\, u-v=0$; green curve: $F(u, v):=\alpha\, u\, (1-u-a_1\, v)+\beta\, v\, (1-a_2\, u-v)=0$; brown line: $\displaystyle\frac{u}{\underline{u}}+\frac{v}{\underline{v}}=1$, where $\underline{u}$ and $\underline{v}$ are given by (55) and (56); magenta ellipse (above): $\alpha\, d_1\, u^2+\beta\, d_2\, v^2=\lambda_1$, where $\lambda_1$ is given by (61); magenta ellipse (below): $\alpha\, d_1\, u^2+\beta\, d_2\, v^2=\lambda_2$, where $\lambda_2$ is given by (70); yellow line (above): $\alpha\, u+\beta\, v=\eta_1$, where $\eta_1$ is given by (62); yellow line (below): $\alpha\, u+\beta\, v=\eta_2$, where $\eta_2$ is given by (71); dashed orange curve: the solution $(u(x), v(x))$; dotted line (above): $\displaystyle\frac{u}{\sqrt{\frac{\lambda_1}{\alpha\, d_1}}}+\displaystyle\frac{v}{\sqrt{\frac{\lambda_1}{\beta\, d_2}}}=1$; dotted line (below): $\displaystyle\frac{u}{\sqrt{\frac{\lambda_2}{\alpha\, d_1}}}+\displaystyle\frac{v}{\sqrt{\frac{\lambda_2}{\beta\, d_2}}}=1$
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Figure 2.  Red line: $1-u-a_1\, v=0$; blue line: $1-a_2\, u-v=0$; green curve: $F(u, v)=0$; brown line: $\displaystyle\frac{u}{\overline{u}}+\frac{v}{\overline{v}}=1$, where $\overline{u}$ and $\overline{v}$ are given by (55) and (56); magenta ellipses : $\alpha\, d_1\, u^2+\beta\, d_2\, v^2=\lambda_1, \lambda_2$, where $\lambda_1$ (below) is given by (96) and $\lambda_2$ (above) by (103); yellow lines: $\alpha\, u+\beta\, v=\eta_1, \eta_2$, where $\eta_1$ (below) is given by (102) and $\eta_2$ (above) by (109); dashed orange curve: the solution $(u(x), v(x))$; dotted lines: $\displaystyle\sqrt{\alpha\, d_1}\, u+\sqrt{\beta\, d_2}\, v=\sqrt{\lambda_1}$ (below), $\displaystyle\sqrt{\lambda_2}$ (above); $\overline{u}=\overline{v}=1$; $d_1=3$, $a_1=2$, $a_2=3$, $\alpha=1$
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Figure 3.  Red: $u(x) =60\, \big(1-\tanh x\big)^2$; green: $v(x) =8\, \big(1+\tanh x\big)$
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Figure 4.  Red: $u(x) =\displaystyle\frac{1}{10}\big(1-\cos\, (2\, x)\big)$; green: $v(x) =\displaystyle\frac{1}{11}\big(1+\cos\, (2\, x)\big)$; blue: $w(x) =\displaystyle\frac{1}{12}\big(1+\cos\, (2\, x)\big)$
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[3]

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