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Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise
An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems
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 |
By employing the N-barrier method developed in C.-C. Chen and L.-C. Hung, 2016 ([
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.
|
[2] |
R. A. Armstrong and R. McGehee,
Competitive exclusion, Amer. Natur., 115 (1980), 151-170.
|
[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.
|
[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.
|
[5] |
Cantrell, Ward and Jr.,
On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.
|
[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.
|
[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.
|
[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. |
[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. |
[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.
|
[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.
|
[12] |
P.de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Institute of Math., Polish Academy Sci., zam, 11 (1979), p.190. |
[13] |
S.-I. Ei, R. Ikota and M. Mimura,
Segregating partition problem in competition-diffusion systems, Interfaces Free Bound., 1 (1999), 57-80.
|
[14] |
I. J. Good,
The population frequencies of species and the estimation of population parameters, Biometrika, 40 (1953), 237-264.
|
[15] |
S. Grossberg,
Decisions, patterns, and oscillations in nonlinear competitve systems with applications to Volterra-Lotka systems, J. Theoret. Biol., 73 (1978), 101-130.
|
[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.
|
[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.
|
[18] |
M. W. Hirsch,
Differential equations and convergence almost everywhere in strongly monotone semiflows, Contemp. Math., 17 (1983), 267-285.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
E.H.Simpson, Measurement of diversity Nature, 163 (1949), p688.
doi: 10.1038/163688a0. |
[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. |
[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. |
[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. |
[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. |
[41] |
M. L. Zeeman,
Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.
doi: 10.1080/02681119308806158. |
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.
|
[2] |
R. A. Armstrong and R. McGehee,
Competitive exclusion, Amer. Natur., 115 (1980), 151-170.
|
[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.
|
[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.
|
[5] |
Cantrell, Ward and Jr.,
On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.
|
[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.
|
[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.
|
[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. |
[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. |
[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.
|
[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.
|
[12] |
P.de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Institute of Math., Polish Academy Sci., zam, 11 (1979), p.190. |
[13] |
S.-I. Ei, R. Ikota and M. Mimura,
Segregating partition problem in competition-diffusion systems, Interfaces Free Bound., 1 (1999), 57-80.
|
[14] |
I. J. Good,
The population frequencies of species and the estimation of population parameters, Biometrika, 40 (1953), 237-264.
|
[15] |
S. Grossberg,
Decisions, patterns, and oscillations in nonlinear competitve systems with applications to Volterra-Lotka systems, J. Theoret. Biol., 73 (1978), 101-130.
|
[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.
|
[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.
|
[18] |
M. W. Hirsch,
Differential equations and convergence almost everywhere in strongly monotone semiflows, Contemp. Math., 17 (1983), 267-285.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
E.H.Simpson, Measurement of diversity Nature, 163 (1949), p688.
doi: 10.1038/163688a0. |
[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. |
[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. |
[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. |
[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. |
[41] |
M. L. Zeeman,
Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.
doi: 10.1080/02681119308806158. |




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