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Exponential trichotomy and $(r, p)$-admissibility for discrete dynamical systems
Monotone traveling waves in a general discrete model for populations
1. | Department of Mathematics, Vietnam National University, 334 Nguyen Trai, Hanoi, Viet Nam |
2. | Department of Mathematics and Statistics, University of Arkansas at Little Rock, 2801 S. University Ave, Little Rock, AR 72204, USA |
In this paper we consider the existence of monotone traveling waves for a class of general integral difference model for populations that allows the dispersal probability to have no continuous density functions but the fecundity functions to generate a monotone dynamical systems. In this setting we deal with the non-compactness of the evolution operator by using the monotone iteration method.
References:
[1] |
D. G. Aronson and H. Weinberger,
Nonlinear diffusion in population genetics, combustion, and nerve propagation, Partial Differential Equations and Related Topics (J. Golstein ed.), Lecture Notes in Mathematics, Springer. Berlin, 466 (1975), 5-49.
|
[2] |
D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
K. Gopalsamy,
Stability and Oscillations in Delay Differential Equations of Population Dynamics
Kluwer Academic Publishers. Dordrecht, 1992.
doi: 10.1007/978-94-015-7920-9. |
[4] |
S.-B. Hsu and X.-Q. Zhao,
Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.
doi: 10.1137/070703016. |
[5] |
M. Kot,
Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436.
doi: 10.1007/BF00173295. |
[6] |
M. Kot, M. A. Lewis and P. van den Driessche,
Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042.
doi: 10.2307/2265698. |
[7] |
M. A. Lewis, B. Li and H. F. Weinberger,
Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.
doi: 10.1007/s002850200144. |
[8] |
B. Li,
Traveling wave solutions in a plant population model with a seed bank, J. Math. Biol., 65 (2012), 855-873.
doi: 10.1007/s00285-011-0481-x. |
[9] |
B. Li, M. A. Lewis and H. F. Weinberger,
Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Bio., 58 (2009), 323-338.
doi: 10.1007/s00285-008-0175-1. |
[10] |
B. Li, M. A. Lewis and H. F. Weinberger,
Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.
doi: 10.1016/j.mbs.2005.03.008. |
[11] |
X. Liang and X.-Q. Zhao,
Asymptotic speeds of spread and traveling waves for monotone semifows with applications, Communications on Pure and Applied Mathematics, 60 (2007), 1-40.
doi: 10.1002/cpa.20154. |
[12] |
X. Liang and X.-Q. Zhao,
Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903.
doi: 10.1016/j.jfa.2010.04.018. |
[13] |
R. Lui,
Existence and stability of traveling wave solutions of a nonlinear integral operator, J. Math. Biol., 16 (1983), 199-220.
doi: 10.1007/BF00276502. |
[14] |
R. Lui,
Biological growth and spread modeled by systems of recursions. Ⅰ Mathematical theory, Math. Biosci., 93 (1989), 269-295.
doi: 10.1016/0025-5564(89)90026-6. |
[15] |
R. Lui,
Biological growth and spread modeled by systems of recursions. Ⅱ Biological theory, Math. Biosci., 93 (1989), 297-312.
doi: 10.1016/0025-5564(89)90027-8. |
[16] |
F. Lutscher,
Density-dependent dispersal in integrodifference equations, J. Math. Biol., 56 (2008), 499-524.
doi: 10.1007/s00285-007-0127-1. |
[17] |
F. Lutscher and N. Van Minh,
Spreading speeds and traveling waves in discrete models of biological populations with sessile stages, Nonlinear Analysis: Real World Applications, 14 (2013), 495-506.
doi: 10.1016/j.nonrwa.2012.07.011. |
[18] |
N. MacDonald and A. R. Watkinson,
Models of an annual plant population with a seed bank, J Theor Biol., 93 (1981), 643-653.
doi: 10.1016/0022-5193(81)90226-5. |
[19] |
M. G. Neubert, M. Kot and M. A. Lewis,
Dispersal and pattern formation in a discrete-time predator-prey model, Theor. Pop. Biol., 48 (1995), 7-43.
doi: 10.1006/tpbi.1995.1020. |
[20] |
M. Neubert and H. Caswell,
Demography and Dispersal: Calculation and sensitivity analysis of invasion speeds for structured populations, Ecology, 81 (2000), 1613-1628.
|
[21] |
J. A. Powell, I. Slapničar and W. van der Werf,
Epidemic spread of a lesion-forming plant pathogen - analysis of a mechanistic model with infinite age structure, J. Lin. Alg. Appl., 398 (2005), 117-140.
doi: 10.1016/j.laa.2004.10.020. |
[22] |
M. Rees and M. J. Long,
The analysis and interpretation of seedling recruitment curves, Am. Nat., 141 (1993), 233-262.
doi: 10.1086/285471. |
[23] |
A. R. Templeton and D. A. Levin,
Evolutionary consequences of seed pools, Am. Nat., 114 (1979), 232-249.
doi: 10.1086/283471. |
[24] |
H. Thieme,
Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187.
doi: 10.1007/BF00279720. |
[25] |
H. Thieme,
Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. Reine Angew. Math., 306 (1979), 94-121.
doi: 10.1515/crll.1979.306.94. |
[26] |
L. M. Thuc, F. Lutscher and N. Van Minh,
Traveling wave dispersal in partially sedentary age-structured populations, Acta Mathematica Vietnamica, 36 (2011), 319-330.
|
[27] |
A. Valleriani and K. Tielbörger,
Effect of age on germination of dormant seeds, Theor. Pop. Biol., 70 (2006), 1-9.
doi: 10.1016/j.tpb.2006.02.003. |
[28] |
R. R. Veit and M. A. Lewis,
Dispersal, population growth, and the Allee effect: Dynamics of the House Finch invasion of eastern North America, Am. Nat., 148 (1996), 255-274.
doi: 10.1086/285924. |
[29] |
D. Volkov and R. Lui,
Spreading speed and travelling wave solutions of a partially sedentary population, IMA Journal of Applied Mathematics, 72 (2007), 801-816.
doi: 10.1093/imamat/hxm025. |
[30] |
H. Weinberger,
Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[31] |
H. Weinberger,
Asymptotic behavior of a model in population genetics, Partial Differential Equations and Applications (J. Chadam ed.), Lecture Notes in Mathematics, Springer, New York, 648 (1978), 47-98.
|
[32] |
H. F. Weinberger, M. A. Lewis and B. Li,
Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222.
doi: 10.1007/s00285-007-0078-6. |
[33] |
X.-Q. Zhao,
Spatial dynamics of some evolution systems in biology, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Sci. Publ., Hackensack, NJ, (2009), 332-363.
doi: 10.1142/9789812834744_0015. |
show all references
References:
[1] |
D. G. Aronson and H. Weinberger,
Nonlinear diffusion in population genetics, combustion, and nerve propagation, Partial Differential Equations and Related Topics (J. Golstein ed.), Lecture Notes in Mathematics, Springer. Berlin, 466 (1975), 5-49.
|
[2] |
D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
K. Gopalsamy,
Stability and Oscillations in Delay Differential Equations of Population Dynamics
Kluwer Academic Publishers. Dordrecht, 1992.
doi: 10.1007/978-94-015-7920-9. |
[4] |
S.-B. Hsu and X.-Q. Zhao,
Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.
doi: 10.1137/070703016. |
[5] |
M. Kot,
Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436.
doi: 10.1007/BF00173295. |
[6] |
M. Kot, M. A. Lewis and P. van den Driessche,
Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042.
doi: 10.2307/2265698. |
[7] |
M. A. Lewis, B. Li and H. F. Weinberger,
Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.
doi: 10.1007/s002850200144. |
[8] |
B. Li,
Traveling wave solutions in a plant population model with a seed bank, J. Math. Biol., 65 (2012), 855-873.
doi: 10.1007/s00285-011-0481-x. |
[9] |
B. Li, M. A. Lewis and H. F. Weinberger,
Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Bio., 58 (2009), 323-338.
doi: 10.1007/s00285-008-0175-1. |
[10] |
B. Li, M. A. Lewis and H. F. Weinberger,
Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.
doi: 10.1016/j.mbs.2005.03.008. |
[11] |
X. Liang and X.-Q. Zhao,
Asymptotic speeds of spread and traveling waves for monotone semifows with applications, Communications on Pure and Applied Mathematics, 60 (2007), 1-40.
doi: 10.1002/cpa.20154. |
[12] |
X. Liang and X.-Q. Zhao,
Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903.
doi: 10.1016/j.jfa.2010.04.018. |
[13] |
R. Lui,
Existence and stability of traveling wave solutions of a nonlinear integral operator, J. Math. Biol., 16 (1983), 199-220.
doi: 10.1007/BF00276502. |
[14] |
R. Lui,
Biological growth and spread modeled by systems of recursions. Ⅰ Mathematical theory, Math. Biosci., 93 (1989), 269-295.
doi: 10.1016/0025-5564(89)90026-6. |
[15] |
R. Lui,
Biological growth and spread modeled by systems of recursions. Ⅱ Biological theory, Math. Biosci., 93 (1989), 297-312.
doi: 10.1016/0025-5564(89)90027-8. |
[16] |
F. Lutscher,
Density-dependent dispersal in integrodifference equations, J. Math. Biol., 56 (2008), 499-524.
doi: 10.1007/s00285-007-0127-1. |
[17] |
F. Lutscher and N. Van Minh,
Spreading speeds and traveling waves in discrete models of biological populations with sessile stages, Nonlinear Analysis: Real World Applications, 14 (2013), 495-506.
doi: 10.1016/j.nonrwa.2012.07.011. |
[18] |
N. MacDonald and A. R. Watkinson,
Models of an annual plant population with a seed bank, J Theor Biol., 93 (1981), 643-653.
doi: 10.1016/0022-5193(81)90226-5. |
[19] |
M. G. Neubert, M. Kot and M. A. Lewis,
Dispersal and pattern formation in a discrete-time predator-prey model, Theor. Pop. Biol., 48 (1995), 7-43.
doi: 10.1006/tpbi.1995.1020. |
[20] |
M. Neubert and H. Caswell,
Demography and Dispersal: Calculation and sensitivity analysis of invasion speeds for structured populations, Ecology, 81 (2000), 1613-1628.
|
[21] |
J. A. Powell, I. Slapničar and W. van der Werf,
Epidemic spread of a lesion-forming plant pathogen - analysis of a mechanistic model with infinite age structure, J. Lin. Alg. Appl., 398 (2005), 117-140.
doi: 10.1016/j.laa.2004.10.020. |
[22] |
M. Rees and M. J. Long,
The analysis and interpretation of seedling recruitment curves, Am. Nat., 141 (1993), 233-262.
doi: 10.1086/285471. |
[23] |
A. R. Templeton and D. A. Levin,
Evolutionary consequences of seed pools, Am. Nat., 114 (1979), 232-249.
doi: 10.1086/283471. |
[24] |
H. Thieme,
Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187.
doi: 10.1007/BF00279720. |
[25] |
H. Thieme,
Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. Reine Angew. Math., 306 (1979), 94-121.
doi: 10.1515/crll.1979.306.94. |
[26] |
L. M. Thuc, F. Lutscher and N. Van Minh,
Traveling wave dispersal in partially sedentary age-structured populations, Acta Mathematica Vietnamica, 36 (2011), 319-330.
|
[27] |
A. Valleriani and K. Tielbörger,
Effect of age on germination of dormant seeds, Theor. Pop. Biol., 70 (2006), 1-9.
doi: 10.1016/j.tpb.2006.02.003. |
[28] |
R. R. Veit and M. A. Lewis,
Dispersal, population growth, and the Allee effect: Dynamics of the House Finch invasion of eastern North America, Am. Nat., 148 (1996), 255-274.
doi: 10.1086/285924. |
[29] |
D. Volkov and R. Lui,
Spreading speed and travelling wave solutions of a partially sedentary population, IMA Journal of Applied Mathematics, 72 (2007), 801-816.
doi: 10.1093/imamat/hxm025. |
[30] |
H. Weinberger,
Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[31] |
H. Weinberger,
Asymptotic behavior of a model in population genetics, Partial Differential Equations and Applications (J. Chadam ed.), Lecture Notes in Mathematics, Springer, New York, 648 (1978), 47-98.
|
[32] |
H. F. Weinberger, M. A. Lewis and B. Li,
Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222.
doi: 10.1007/s00285-007-0078-6. |
[33] |
X.-Q. Zhao,
Spatial dynamics of some evolution systems in biology, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Sci. Publ., Hackensack, NJ, (2009), 332-363.
doi: 10.1142/9789812834744_0015. |
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