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Discrete N-barrier maximum principle for a lattice dynamical system arising in competition models
1. | Department of Mathematics, National Taiwan University, National Center for Theoretical Sciences, Taipei, Taiwan |
2. | Department of Mathematics, National Taiwan University, Taipei, Taiwan |
In the present paper, we show that an analogous N-barrier maximum principle (see [
References:
[1] |
R. A. Armstrong and R. McGehee,
Competitive exclusion, Amer. Natur., 115 (1980), 151-170.
doi: 10.1086/283553. |
[2] |
Cantrell, Ward and Jr.,
On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.
doi: 10.1137/S0036139995292367. |
[3] |
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. |
[4] |
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. |
[5] |
C.-C. Chen and L.-C. Hung,
An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems, Discrete Contin. Dyn. Syst. B, 23 (2018), 1503-1521.
doi: 10.3934/dcdsb.2018054. |
[6] |
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, Commun. Pure Appl. Anal., 18 (2019), 33-50.
doi: 10.3934/cpaa.2019003. |
[7] |
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. A, 38 (2018), 791-821.
doi: 10.3934/dcds.2018034. |
[8] |
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), 179-206.
doi: 10.32917/hmj/1372180511. |
[9] |
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. |
[10] |
P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Institute of Math., Polish Academy Sci., 11 (1979), p190. Google Scholar |
[11] |
J.-S. Guo and C.-C. Wu,
The existence of traveling wave solutions for a bistable three-component lattice dynamical system, J. Differential Equations, 260 (2016), 1445-1455.
doi: 10.1016/j.jde.2015.09.036. |
[12] |
J.-S. Guo and C.-H. Wu,
Wave propagation for a two-component lattice dynamical system arising in strong competition models, Journal of Differential Equations, 250 (2011), 3504-3533.
doi: 10.1016/j.jde.2010.12.004. |
[13] |
J.-S. Guo and C.-H. Wu,
Traveling wave front for a two-component lattice dynamical system arising in competition models, Journal of Differential Equations, 252 (2012), 4357-4391.
doi: 10.1016/j.jde.2012.01.009. |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
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. |
[18] |
Y. Kan-on,
Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.
doi: 10.1137/S0036141093244556. |
[19] |
J. Kastendiek, Competitor-mediated coexistence: Interactions among three species of benthic macroalgae, Journal of Experimental Marine Biology and Ecology, 62 (1982), 201-210. Google Scholar |
[20] |
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. |
[21] |
M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition–diffusion system, Ecological Complexity, 21 (2015), 215-232. Google Scholar |
[22] |
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. |
show all references
References:
[1] |
R. A. Armstrong and R. McGehee,
Competitive exclusion, Amer. Natur., 115 (1980), 151-170.
doi: 10.1086/283553. |
[2] |
Cantrell, Ward and Jr.,
On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327.
doi: 10.1137/S0036139995292367. |
[3] |
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. |
[4] |
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. |
[5] |
C.-C. Chen and L.-C. Hung,
An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems, Discrete Contin. Dyn. Syst. B, 23 (2018), 1503-1521.
doi: 10.3934/dcdsb.2018054. |
[6] |
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, Commun. Pure Appl. Anal., 18 (2019), 33-50.
doi: 10.3934/cpaa.2019003. |
[7] |
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. A, 38 (2018), 791-821.
doi: 10.3934/dcds.2018034. |
[8] |
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), 179-206.
doi: 10.32917/hmj/1372180511. |
[9] |
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. |
[10] |
P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Institute of Math., Polish Academy Sci., 11 (1979), p190. Google Scholar |
[11] |
J.-S. Guo and C.-C. Wu,
The existence of traveling wave solutions for a bistable three-component lattice dynamical system, J. Differential Equations, 260 (2016), 1445-1455.
doi: 10.1016/j.jde.2015.09.036. |
[12] |
J.-S. Guo and C.-H. Wu,
Wave propagation for a two-component lattice dynamical system arising in strong competition models, Journal of Differential Equations, 250 (2011), 3504-3533.
doi: 10.1016/j.jde.2010.12.004. |
[13] |
J.-S. Guo and C.-H. Wu,
Traveling wave front for a two-component lattice dynamical system arising in competition models, Journal of Differential Equations, 252 (2012), 4357-4391.
doi: 10.1016/j.jde.2012.01.009. |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
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. |
[18] |
Y. Kan-on,
Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.
doi: 10.1137/S0036141093244556. |
[19] |
J. Kastendiek, Competitor-mediated coexistence: Interactions among three species of benthic macroalgae, Journal of Experimental Marine Biology and Ecology, 62 (1982), 201-210. Google Scholar |
[20] |
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. |
[21] |
M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition–diffusion system, Ecological Complexity, 21 (2015), 215-232. Google Scholar |
[22] |
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. |





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