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January  2019, 18(1): 33-50. doi: 10.3934/cpaa.2019003

An N-barrier maximum principle for autonomous systems of $n$ species and its application to problems arising from population dynamics

Chiun-Chuan Chen 1, , Li-Chang Hung 2,, and  Chen-Chih Lai 3,

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 Mathematics, University of British Columbia, Vancouver, Canada

* Corresponding author

Received  March 2017 Revised  January 2018 Published  August 2018

Fund Project: 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

The main contribution of the N-barrier maximum principle is that it provides rather generic a priori upper and lower bounds for the linear combination of the components of a vector-valued solution. We show that the N-barrier maximum principle (NBMP, C.-C. Chen and L.-C. Hung (2016)) remains true for $n$ $(n>2)$ species. In addition, a stronger lower bound in NBMP is given by employing an improved tangent line method. As an application of NBMP, we establish a nonexistence result for traveling wave solutions to the four species Lotka-Volterra system.

Keywords: Maximum principles, traveling wave solutions, reaction-diffusion equations.
Mathematics Subject Classification: Primary: 35B50; Secondary: 35C07, 35K57.
Citation: Chiun-Chuan Chen, Li-Chang Hung, Chen-Chih Lai. An N-barrier maximum principle for autonomous systems of $n$ species and its application to problems arising from population dynamics. Communications on Pure & Applied Analysis, 2019, 18 (1) : 33-50. doi: 10.3934/cpaa.2019003
References:
[1]

R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170. doi: 10.1086/283553. Google Scholar

[2]

R. S. Cantrell and J. R. Ward, Jr., On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327. doi: 10.1137/S0036139995292367. Google Scholar

[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. Google Scholar

[4]

——, 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. Google Scholar

[5]

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. Google Scholar

[6]

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

[7]

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. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competitiondiffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363. doi: 10.1137/S0036141093244556. Google Scholar

[13]

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

[14]

V. Kozlov and S. Vakulenko, On chaos in Lotka-Volterra systems: an analytical approach, Nonlinearity, 26 (2013), 2299-2314. doi: 10.1088/0951-7715/26/8/2299. Google Scholar

[15]

R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29 (1975), 243-253. Special issue on mathematics and the social and biological sciences. doi: 10.1137/0129022. Google Scholar

[16]

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

[17]

M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition-diffusion system, Ecological Complexity, 21 (2015), 215-232. Google Scholar

[18]

J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer-Verlag, Berlin, second ed., 1993. doi: 10.1007/b98869. Google Scholar

[19]

S. Oancea, I. Grosu and A. Oancea, Biological control based on the synchronization of lotkavolterra systems with four competitive species, Rom. J. Biophys, 21 (2011), 17-26. Google Scholar

[20]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270. Google Scholar

[21]

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. Google Scholar

[22]

L. Roques and M. D. Chekroun, Probing chaos and biodiversity in a simple competition model, Ecological Complexity, 8 (2011), 98-104. Google Scholar

[23]

P. Schuster, K. Sigmund and R. Wolff, On ω-limits for competition between three species, SIAM J. Appl. Math., 37 (1979), 49-54. doi: 10.1137/0137004. Google Scholar

[24]

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

[25]

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

[26]

J. A. Vano, J. C. Wildenberg, M. B. Anderson, J. K. Noel and J. C. Sprott, Chaos in lowdimensional Lotka-Volterra models of competition, Nonlinearity, 19 (2006), 2391-2404. doi: 10.1088/0951-7715/19/10/006. Google Scholar

[27]

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

[28]

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]

R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170. doi: 10.1086/283553. Google Scholar

[2]

R. S. Cantrell and J. R. Ward, Jr., On competition-mediated coexistence, SIAM J. Appl. Math., 57 (1997), 1311-1327. doi: 10.1137/S0036139995292367. Google Scholar

[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. Google Scholar

[4]

——, 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. Google Scholar

[5]

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. Google Scholar

[6]

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

[7]

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. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competitiondiffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363. doi: 10.1137/S0036141093244556. Google Scholar

[13]

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

[14]

V. Kozlov and S. Vakulenko, On chaos in Lotka-Volterra systems: an analytical approach, Nonlinearity, 26 (2013), 2299-2314. doi: 10.1088/0951-7715/26/8/2299. Google Scholar

[15]

R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29 (1975), 243-253. Special issue on mathematics and the social and biological sciences. doi: 10.1137/0129022. Google Scholar

[16]

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

[17]

M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition-diffusion system, Ecological Complexity, 21 (2015), 215-232. Google Scholar

[18]

J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer-Verlag, Berlin, second ed., 1993. doi: 10.1007/b98869. Google Scholar

[19]

S. Oancea, I. Grosu and A. Oancea, Biological control based on the synchronization of lotkavolterra systems with four competitive species, Rom. J. Biophys, 21 (2011), 17-26. Google Scholar

[20]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270. Google Scholar

[21]

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. Google Scholar

[22]

L. Roques and M. D. Chekroun, Probing chaos and biodiversity in a simple competition model, Ecological Complexity, 8 (2011), 98-104. Google Scholar

[23]

P. Schuster, K. Sigmund and R. Wolff, On ω-limits for competition between three species, SIAM J. Appl. Math., 37 (1979), 49-54. doi: 10.1137/0137004. Google Scholar

[24]

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

[25]

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

[26]

J. A. Vano, J. C. Wildenberg, M. B. Anderson, J. K. Noel and J. C. Sprott, Chaos in lowdimensional Lotka-Volterra models of competition, Nonlinearity, 19 (2006), 2391-2404. doi: 10.1088/0951-7715/19/10/006. Google Scholar

[27]

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

[28]

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 2.  u1(x) (red); u2(x) (green); u3(x) (blue); u4(x) (magenta). Figure 2(a) and Figure 2(b) show Type Ⅰ and Type Ⅱ solutions in Theorem 5.1, respectively
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Figure 1.  N-barrier for cases (ⅰ), (ⅱ) and (ⅲ) (from the left to the right)
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