October  2017, 22(8): 3221-3234. doi: 10.3934/dcdsb.2017171

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

Received  July 2016 Revised  September 2016 Published  June 2017

Fund Project: We thank the anonymous referee for carefully reading the manuscript and suggestions that help us improve the justifications of some conditions.

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.

Citation: Thuc Manh Le, Nguyen Van Minh. Monotone traveling waves in a general discrete model for populations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3221-3234. doi: 10.3934/dcdsb.2017171
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. KotM. 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. LewisB. 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. LiM. 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. LiM. 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. NeubertM. 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. PowellI. 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. ThucF. 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. WeinbergerM. 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. KotM. 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. LewisB. 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. LiM. 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. LiM. 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. NeubertM. 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. PowellI. 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. ThucF. 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. WeinbergerM. 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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