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June  2018, 23(4): 1503-1521. doi: 10.3934/dcdsb.2018054

An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems

Chiun-Chuan Chen 1, and  Li-Chang Hung 2,,

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

* Corresponding author.

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.

Received  April 2017 Published  February 2018

By employing the N-barrier method developed in C.-C. Chen and L.-C. Hung, 2016 ([6]), we establish a new N-barrier maximum principle for diffusive Lotka-Volterra systems of two competing species. To this end, this gives rise to the N-barrier maximum principle for a second-order elliptic equation involving two distinct unknown functions and a quadratic nonlinearity. An immediate consequence of the N-barrier maximum principle is an a priori estimate for the total populations of the two species. As an application of this maximum principle, we show under certain conditions the existence and nonexistence of traveling waves solutions for systems of three competing species. In addition, new $(1, 0, 0)$-$(u^{*}, v^{*}, 0)$ waves are given in terms of the tanh function, provided that the system's parameters satisfy certain conditions.

Keywords: Maximum principles, traveling wave solutions, reaction-diffusion equations.
Mathematics Subject Classification: Primary: 35B50; Secondary: 35C07, 35K57.
Citation: Chiun-Chuan Chen, Li-Chang Hung. An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1503-1521. doi: 10.3934/dcdsb.2018054
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.   Google Scholar

[2]

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

[4]

H. Berestycki, O. Diekmann, C. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2009), 399-429.   Google Scholar

[5]

Cantrell, Ward and Jr., On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.   Google Scholar

[6]

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.   Google Scholar

[7]

CC. Chen and LC. Hung, Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive lotka-volterra systems of three competing species., Communications on Pure & Applied Analysis, 15 (2016), 1451-1469.   Google Scholar

[8]

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

[9]

C.-C.Chen, L.-C.Hung and H.-F.Liu, N-barrier maximum principle for degenerate elliptic systems and its application, Discrete Contin.Dyn.Syst., to appear. Google Scholar

[10]

C.-C. Chen, L.-C. Hung, M. Mimura, M. Tohma and D. Ueyama, Semi-exact equilibrium solutions for three-species competition-diffusion systems, Hiroshima Math J., 43 (2013), 176-206.   Google Scholar

[11]

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.   Google Scholar

[12]

P.de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Institute of Math., Polish Academy Sci., zam, 11 (1979), p.190. Google Scholar

[13]

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

[14]

I. J. Good, The population frequencies of species and the estimation of population parameters, Biometrika, 40 (1953), 237-264.   Google Scholar

[15]

S. Grossberg, Decisions, patterns, and oscillations in nonlinear competitve systems with applications to Volterra-Lotka systems, J. Theoret. Biol., 73 (1978), 101-130.   Google Scholar

[16]

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.   Google Scholar

[17]

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.   Google Scholar

[18]

M. W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows, Contemp. Math., 17 (1983), 267-285.   Google Scholar

[19]

X. Hou and A. W. Leung, Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics, Nonlinear Anal. Real World Appl., 9 (2008), 2196-2213.   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

J. I. Kanel, On the wave front solution of a competition-diffusion system in population dynamics, Nonlinear Anal., 65 (2006), 301-320.  doi: 10.1016/j.na.2005.05.014.  Google Scholar

[25]

J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlinear Anal., 27 (1996), 579-587.  doi: 10.1016/0362-546X(95)00221-G.  Google Scholar

[26]

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

[27]

K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21.  doi: 10.1016/0022-0396(85)90020-8.  Google Scholar

[28]

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

[29]

P. Koch Medina and G. Schätti, Long-time behaviour for reaction-diffusion equations on $\mathbf R^N$, Nonlinear Anal., 25 (1995), 831-870.  doi: 10.1016/0362-546X(94)00174-G.  Google Scholar

[30]

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.  Google Scholar

[31]

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

[32]

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

[33]

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

[34]

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

[35]

L. Sanchez, A note on a nonautonomous O.D.E. related to the Fisher equation, J. Comput. Appl. Math., 113 (2000), 201-209.  doi: 10.1016/S0377-0427(99)00254-X.  Google Scholar

[36]

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

[37]

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

[38]

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

[39]

A.I.Volpert, V.A.Volpert and V.A.Volpert, Traveling Wave Solutions of Parabolic Systems, vol.140 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1994.Translated from the Russian manuscript by James F.Heyda.  Google Scholar

[40]

V. A. Volpert and Y. M. Suhov, Stationary solutions of non-autonomous Kolmogorov-Petrovsky-Piskunov equations, Ergodic Theory Dynam. Systems, 19 (1999), 809-835.  doi: 10.1017/S0143385799138823.  Google Scholar

[41]

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

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.   Google Scholar

[2]

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

[4]

H. Berestycki, O. Diekmann, C. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2009), 399-429.   Google Scholar

[5]

Cantrell, Ward and Jr., On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.   Google Scholar

[6]

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.   Google Scholar

[7]

CC. Chen and LC. Hung, Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive lotka-volterra systems of three competing species., Communications on Pure & Applied Analysis, 15 (2016), 1451-1469.   Google Scholar

[8]

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

[9]

C.-C.Chen, L.-C.Hung and H.-F.Liu, N-barrier maximum principle for degenerate elliptic systems and its application, Discrete Contin.Dyn.Syst., to appear. Google Scholar

[10]

C.-C. Chen, L.-C. Hung, M. Mimura, M. Tohma and D. Ueyama, Semi-exact equilibrium solutions for three-species competition-diffusion systems, Hiroshima Math J., 43 (2013), 176-206.   Google Scholar

[11]

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.   Google Scholar

[12]

P.de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Institute of Math., Polish Academy Sci., zam, 11 (1979), p.190. Google Scholar

[13]

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

[14]

I. J. Good, The population frequencies of species and the estimation of population parameters, Biometrika, 40 (1953), 237-264.   Google Scholar

[15]

S. Grossberg, Decisions, patterns, and oscillations in nonlinear competitve systems with applications to Volterra-Lotka systems, J. Theoret. Biol., 73 (1978), 101-130.   Google Scholar

[16]

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.   Google Scholar

[17]

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.   Google Scholar

[18]

M. W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows, Contemp. Math., 17 (1983), 267-285.   Google Scholar

[19]

X. Hou and A. W. Leung, Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics, Nonlinear Anal. Real World Appl., 9 (2008), 2196-2213.   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

J. I. Kanel, On the wave front solution of a competition-diffusion system in population dynamics, Nonlinear Anal., 65 (2006), 301-320.  doi: 10.1016/j.na.2005.05.014.  Google Scholar

[25]

J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlinear Anal., 27 (1996), 579-587.  doi: 10.1016/0362-546X(95)00221-G.  Google Scholar

[26]

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

[27]

K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21.  doi: 10.1016/0022-0396(85)90020-8.  Google Scholar

[28]

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

[29]

P. Koch Medina and G. Schätti, Long-time behaviour for reaction-diffusion equations on $\mathbf R^N$, Nonlinear Anal., 25 (1995), 831-870.  doi: 10.1016/0362-546X(94)00174-G.  Google Scholar

[30]

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.  Google Scholar

[31]

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

[32]

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

[33]

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

[34]

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

[35]

L. Sanchez, A note on a nonautonomous O.D.E. related to the Fisher equation, J. Comput. Appl. Math., 113 (2000), 201-209.  doi: 10.1016/S0377-0427(99)00254-X.  Google Scholar

[36]

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

[37]

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

[38]

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

[39]

A.I.Volpert, V.A.Volpert and V.A.Volpert, Traveling Wave Solutions of Parabolic Systems, vol.140 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1994.Translated from the Russian manuscript by James F.Heyda.  Google Scholar

[40]

V. A. Volpert and Y. M. Suhov, Stationary solutions of non-autonomous Kolmogorov-Petrovsky-Piskunov equations, Ergodic Theory Dynam. Systems, 19 (1999), 809-835.  doi: 10.1017/S0143385799138823.  Google Scholar

[41]

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

Figure 1.  Red line: $\sigma_1-c_{11}\,u-c_{12}\,v = 0$; blue line: $\sigma_2-c_{21}\,u-c_{22}\,v = 0$; green curve: $\alpha\,u\,(\sigma_1-c_{11}\,u-c_{12}\,v)+\beta\,v\,(\sigma_2-c_{21}\,u-c_{22}\,v) = 0$. $\sigma_1 = \sigma_2 = c_{11} = c_{22} = 1,c_{12} = \frac{1}{2},c_{21} = \frac{2}{3}$. (a) $\alpha = \frac{1}{2}$, $\beta = 4$ (hyperbola). (b) $\alpha = 2$, $\beta = \frac{3}{20}$ (hyperbola). (c) $\alpha = 2$, $\beta = \frac{15}{2}+3 \sqrt{6}\approx14.8485$ (parabola). (d) $\alpha = 2$, $\beta = \frac{15}{2}-3 \sqrt{6}\approx0.1515$ (parabola). (e) $\alpha = 2$, $\beta = 3$ (ellipse). (f) zooming out of (a).
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Figure 2.  Red line: $\sigma_1-c_{11}\,u-c_{12}\,v = 0$; blue line: $\sigma_2-c_{21}\,u-c_{22}\,v = 0$; green curve: $\alpha\,u\,(\sigma_1-c_{11}\,u-c_{12}\,v)+\beta\,v\,(\sigma_2-c_{21}\,u-c_{22}\,v) = 0$; magenta line (above): $\alpha\,d_1\,u+\beta\,d_2\,v = \lambda_2$; magenta line (below): $\alpha\,d_1\,u+\beta\,d_2\,v = \lambda_1$; yellow line: $\alpha\,u+\beta\,v = \eta$; dashed curve: $(u(x),v(x))$. $d_1 = \sigma_1 = \sigma_2 = c_{11} = c_{22} = 1$. (a) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 18$, and $d_2 = 2$ give $\lambda_1 = \frac{17}{6}$, $\lambda_2 = \frac{17}{3}$, and $\eta = \frac{17}{6}$. (b) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 5$, and $d_2 = 2$ give $\lambda_1 = \frac{5}{2}$, $\lambda_2 = 5$, and $\eta = \frac{5}{2}$. (c) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 18$, and $d_2 = \frac{2}{3}$ give $\lambda_1 = \frac{34}{9}$, $\lambda_2 = \frac{17}{3}$, and $\eta = \frac{17}{3}$. (d) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 18$, and $d_2 = \frac{1}{2}$ give $\lambda_1 = \frac{9}{4}$, $\lambda_2 = \frac{9}{2}$, and $\eta = \frac{9}{2}$.
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Figure 3.  Red line: $\sigma_1-c_{11}\,u-c_{12}\,v = 0$; blue line: $\sigma_2-c_{21}\,u-c_{22}\,v = 0$; green curve: $\alpha\,u\,(\sigma_1-c_{11}\,u-c_{12}\,v)+\beta\,v\,(\sigma_2-c_{21}\,u-c_{22}\,v) = 0$; magenta line (below): $\alpha\,d_1\,u+\beta\,d_2\,v = \lambda_2$; magenta line (above): $\alpha\,d_1\,u+\beta\,d_2\,v = \lambda_1$; yellow line: $\alpha\,u+\beta\,v = \eta$; dashed curve: $(u(x),v(x))$. $d_1 = \sigma_1 = \sigma_2 = c_{11} = c_{22} = 1$. (a) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 18$, and $d_2 = 2$ give $\lambda_1 = 72$, $\lambda_2 = 36$, and $\eta = 36$. (b) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 5$, and $d_2 = 2$ give $\lambda_1 = 34$, $\lambda_{2} = 17$, and $\eta = 17$. (c) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 33$, and $d_2 = \frac{2}{3}$ give $\lambda_1 = 33$, $\lambda_2 = 22$, and $\eta = 33$. (d) $c_{12} = 2$, $c_{21} = 3$, $\alpha = 17$, $\beta = 18$, and $d_2 = \frac{1}{2}$ give $\lambda_1 = 34$, $\lambda_2 = 17$, and $\eta = 34$.
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Figure 4.  Profiles of the solution $(u(x),v(x),w(x))$.
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