\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author

    * 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

Abstract Full Text(HTML) Figure(4) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 35B50; Secondary: 35C07, 35K57.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • 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): $\displaystyle\frac{u}{\sqrt{\frac{\lambda_2}{\alpha\, d_1}}}+\displaystyle\frac{v}{\sqrt{\frac{\lambda_2}{\beta\, d_2}}}=1$

    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); $\overline{u}=\overline{v}=1$; $d_1=3$, $a_1=2$, $a_2=3$, $\alpha=1$

    Figure 3.  Red: $u(x) =60\, \big(1-\tanh x\big)^2$; green: $v(x) =8\, \big(1+\tanh x\big)$

    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: $w(x) =\displaystyle\frac{1}{12}\big(1+\cos\, (2\, x)\big)$

  • [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.
    [2] R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170.  doi: 10.1086/283553.
    [3] A. J. BaczkowskiD. 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.
    [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.
    [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.
    [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.
    [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.
    [8] C.-C. ChenL.-C. HungM. 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.
    [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.
    [10] S.-I. EiR. Ikota and M. Mimura, Segregating partition problem in competition-diffusion systems, Interfaces Free Bound, 1 (1999), 57-80.  doi: 10.4171/IFB/4.
    [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.
    [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.
    [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.
    [14] M. GueddaR. KersnerM. 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.
    [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.
    [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.
    [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.
    [18] T. G. HallamL. 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.
    [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.
    [20] 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.
    [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.
    [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.
    [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.
    [24] W. KoK. 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.
    [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.
    [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.
    [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.
    [28] J. D. Murray, Mathematical Biology Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.
    [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.
    [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.
    [31] S. PetrovskiiK. KawasakiF. 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.
    [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.
    [33] M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270. 
    [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.
    [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.
    [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.
    [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.
    [38] N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.
    [39] E. H. Simpson, Measurement of diversity Nature, 163 (1949), p688. doi: 10.1038/163688a0.
    [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.
    [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.
    [42] M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.  doi: 10.1080/02681119308806158.
    [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.
    [44] Y. B. Zeldovich, Theory of Flame Propagation 1951.
  • 加载中

Figures(4)

SHARE

Article Metrics

HTML views(573) PDF downloads(175) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return