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Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity
An N-barrier maximum principle for autonomous systems of $n$ species and its application to problems arising from population dynamics
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 |
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.
References:
[1] |
R. A. Armstrong and R. McGehee,
Competitive exclusion, Amer. Natur., 115 (1980), 151-170.
doi: 10.1086/283553. |
[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. |
[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] |
——, 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, 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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
J. Kastendiek,
Competitor-mediated coexistence: interactions among three species of benthic
macroalgae, Journal of Experimental Marine Biology and Ecology, 62 (1982), 201-210.
|
[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. |
[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. |
[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. |
[17] |
M. Mimura and M. Tohma,
Dynamic coexistence in a three-species competition-diffusion
system, Ecological Complexity, 21 (2015), 215-232.
|
[18] |
J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer-Verlag, Berlin, second ed., 1993.
doi: 10.1007/b98869. |
[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.
|
[20] |
M. Rodrigo and M. Mimura,
Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270.
|
[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. |
[22] |
L. Roques and M. D. Chekroun,
Probing chaos and biodiversity in a simple competition
model, Ecological Complexity, 8 (2011), 98-104.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[28] |
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] |
R. A. Armstrong and R. McGehee,
Competitive exclusion, Amer. Natur., 115 (1980), 151-170.
doi: 10.1086/283553. |
[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. |
[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] |
——, 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, 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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
J. Kastendiek,
Competitor-mediated coexistence: interactions among three species of benthic
macroalgae, Journal of Experimental Marine Biology and Ecology, 62 (1982), 201-210.
|
[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. |
[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. |
[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. |
[17] |
M. Mimura and M. Tohma,
Dynamic coexistence in a three-species competition-diffusion
system, Ecological Complexity, 21 (2015), 215-232.
|
[18] |
J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer-Verlag, Berlin, second ed., 1993.
doi: 10.1007/b98869. |
[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.
|
[20] |
M. Rodrigo and M. Mimura,
Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270.
|
[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. |
[22] |
L. Roques and M. D. Chekroun,
Probing chaos and biodiversity in a simple competition
model, Ecological Complexity, 8 (2011), 98-104.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[28] |
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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