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Dispersive effects of weakly compressible and fast rotating inviscid fluids
N-barrier maximum principle for degenerate elliptic systems and its application
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 |
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.
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. |
[2] |
R. A. Armstrong and R. McGehee,
Competitive exclusion, Amer. Natur., 115 (1980), 151-170.
doi: 10.1086/283553. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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] |
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. |
[2] |
R. A. Armstrong and R. McGehee,
Competitive exclusion, Amer. Natur., 115 (1980), 151-170.
doi: 10.1086/283553. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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] |



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