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Spreading speeds and traveling waves for non-cooperative integro-difference systems
1. | Division of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069-7100 |
2. | Department of Mathematics & Statistics, Arizona State University, Tempe, AZ 85287-1804 |
References:
[1] |
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" (ed. J. A. Goldstein) (Program, Tulane Univ., New Orleans, La., 1974), Lecture Notes in Mathematics, 446, Springer, Berlin, (1975), 5-49. |
[2] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001. |
[4] |
K. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Cambridge Philos. Soc., 81 (1977), 431-433.
doi: 10.1017/S0305004100053494. |
[5] |
M. M. Crow, "Organizing Teaching and Research to Address the Grand Challenges of Sustainable Development," BioScience, Vol. 60, University of California Press, (2010), 488-489. |
[6] |
O. Diekmann, Thresholds and traveling waves for the geographical spread of an infection, J. Math. Biol., 6 (1978), 109-130. |
[7] |
R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. |
[8] |
K. P. Hadeler and F. Rothe, Traveling fronts in nonlinear diffusion equation, J. Math. Bio., 2 (1975), 251-263. |
[9] |
K. P. Hadeler, Hyperbolic travelling fronts, Proc. Edinb. Math. Soc. (2), 31 (1988), 89-97.
doi: 10.1017/S001309150000660X. |
[10] |
K. P. Hadeler, Reaction transport systems, in V.Capasso, O.Diekmann, "Mathematics Inspired by Biology" (eds. V. Capasso and O. Diekmann) (Martina Franca, 1997), Lecture Notes in Mathematics, 1714, Springer, Berlin, (1999), 95-150. |
[11] |
M. Hassell and H. Comins, Discrete time models for two-species competition, Theoretical Population Biology, 9 (1976), 202-221. |
[12] |
A. Hastings, K. Cuddington, K. Davies, C. Dugaw, S. Elmendorf, A. Freestone, S. Harrison, M. Holland, J. Lambrinos, U. Malvadkar, B. Melbourne, K. Moore, C. Taylor and D. Thomson, The spatial spread of invasions: New developments in theory and evidence, Ecology Letters, 8 (2005), 91-101. |
[13] |
R. Horn, C. Johnson and R. Charles, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. |
[14] |
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. |
[15] |
H. Kierstad and L. B. Slobodkin, The size of water masses containing plankton blooms, J. Mar. Res., 12 (1953), 141-147. |
[16] |
A. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matiére et son application á un probléme biologique, Bull. Univ. Moscov. Ser. Internat. Sect., 1 (1937), 1-26. |
[17] |
M. Kot, Discrete-time traveling waves: Ecological examples, J. of Math. Biol., 30 (1992), 413-436. |
[18] |
S. A. Levin, Toward a science of sustainability: Executive summary, in "Report from Toward a Science of Sustainability Conference Airlie Center," March, National Science Foundation, (2009), 4-10. |
[19] |
S. A. Levin and R. T. Paine, Disturbance, patch formation, and community structure, Proc. Nat. Acad. Sci. USA, 71 (1974), 2744-2747.
doi: 10.1073/pnas.71.7.2744. |
[20] |
A. Okubo, "Diffusion and Ecological Problems: Mathematical Models," An extended version of the Japanese edition, Ecology and Diffusion, Biomathematics, 10, Springer-Verlag, Berlin-New York, 1980. |
[21] |
M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models, Journal of Mathematical Biology, 45 (2002), 219-233.
doi: 10.1007/s002850200144. |
[22] |
B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosciences, 196 (2005), 82-98.
doi: 10.1016/j.mbs.2005.03.008. |
[23] |
B. Li, M. Lewis and H. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, Journal of Mathematical Biology, 58 (2009), 323-338.
doi: 10.1007/s00285-008-0175-1. |
[24] | |
[25] |
R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosciences, 93 (1989), 269-295.
doi: 10.1016/0025-5564(89)90027-8. |
[26] |
L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics," Mathematical Surveys and Monographs, 102, American Mathematical Society, Providence, RI, 2003. |
[27] |
S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equation, Journal of Differential Equations, 237 (2007), 259-277.
doi: 10.1016/j.jde.2007.03.014. |
[28] |
H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. of Math. Biol., 8 (1979), 173-187. |
[29] |
J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.
doi: 10.1093/biomet/38.1-2.196. |
[30] |
K. R. Ríos-Soto, C. Castillo-Chavez, M. Neubert, E. S. Titi and A.-A. Yakubu, Epidemic spread in populations at demographic equilibrium, in "Mathematical Studies on Human Disease Dynamics" (eds. A. Gumel, C. Castillo-Chavez, D. P. Clemence and R. E. Mickens), Contemp. Math., 410, American Mathematical Society, Providence, RI, (2006), 297-309. |
[31] |
H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations, Journal of Differential Equations, 247 (2009), 887-905.
doi: 10.1016/j.jde.2009.04.002. |
[32] |
H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, Journal of Nonlinear Science, 21 (2011), 747-783.
doi: 10.1007/s00332-011-9099-9. |
[33] |
H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.
doi: 10.1007/s002850200145. |
[34] |
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. |
[35] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[36] |
H. F. Weinberger, Asymptotic behavior of a model in population genetics, in "Nonlinear Partial Differential Equations and Applications" (ed. J. M. Chadam) (Proc. Special Sem., Indiana Univ., Bloomington, Ind., 1976-1977), Lecture Notes in Mathematics, Vol. 648, Springer, Berlin, (1978), 47-96. |
[37] |
H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds for a partially cooperative 2-species reaction-diffusion model, Discrete and Continuous Dynamical Systems, 23 (2009), 1087-1098.
doi: 10.3934/dcds.2009.23.1087. |
[38] |
P. Weng, X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, Journal of Differential Equations, 229 (2006), 270-296.
doi: 10.1016/j.jde.2006.01.020. |
show all references
References:
[1] |
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" (ed. J. A. Goldstein) (Program, Tulane Univ., New Orleans, La., 1974), Lecture Notes in Mathematics, 446, Springer, Berlin, (1975), 5-49. |
[2] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001. |
[4] |
K. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Cambridge Philos. Soc., 81 (1977), 431-433.
doi: 10.1017/S0305004100053494. |
[5] |
M. M. Crow, "Organizing Teaching and Research to Address the Grand Challenges of Sustainable Development," BioScience, Vol. 60, University of California Press, (2010), 488-489. |
[6] |
O. Diekmann, Thresholds and traveling waves for the geographical spread of an infection, J. Math. Biol., 6 (1978), 109-130. |
[7] |
R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. |
[8] |
K. P. Hadeler and F. Rothe, Traveling fronts in nonlinear diffusion equation, J. Math. Bio., 2 (1975), 251-263. |
[9] |
K. P. Hadeler, Hyperbolic travelling fronts, Proc. Edinb. Math. Soc. (2), 31 (1988), 89-97.
doi: 10.1017/S001309150000660X. |
[10] |
K. P. Hadeler, Reaction transport systems, in V.Capasso, O.Diekmann, "Mathematics Inspired by Biology" (eds. V. Capasso and O. Diekmann) (Martina Franca, 1997), Lecture Notes in Mathematics, 1714, Springer, Berlin, (1999), 95-150. |
[11] |
M. Hassell and H. Comins, Discrete time models for two-species competition, Theoretical Population Biology, 9 (1976), 202-221. |
[12] |
A. Hastings, K. Cuddington, K. Davies, C. Dugaw, S. Elmendorf, A. Freestone, S. Harrison, M. Holland, J. Lambrinos, U. Malvadkar, B. Melbourne, K. Moore, C. Taylor and D. Thomson, The spatial spread of invasions: New developments in theory and evidence, Ecology Letters, 8 (2005), 91-101. |
[13] |
R. Horn, C. Johnson and R. Charles, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. |
[14] |
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. |
[15] |
H. Kierstad and L. B. Slobodkin, The size of water masses containing plankton blooms, J. Mar. Res., 12 (1953), 141-147. |
[16] |
A. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matiére et son application á un probléme biologique, Bull. Univ. Moscov. Ser. Internat. Sect., 1 (1937), 1-26. |
[17] |
M. Kot, Discrete-time traveling waves: Ecological examples, J. of Math. Biol., 30 (1992), 413-436. |
[18] |
S. A. Levin, Toward a science of sustainability: Executive summary, in "Report from Toward a Science of Sustainability Conference Airlie Center," March, National Science Foundation, (2009), 4-10. |
[19] |
S. A. Levin and R. T. Paine, Disturbance, patch formation, and community structure, Proc. Nat. Acad. Sci. USA, 71 (1974), 2744-2747.
doi: 10.1073/pnas.71.7.2744. |
[20] |
A. Okubo, "Diffusion and Ecological Problems: Mathematical Models," An extended version of the Japanese edition, Ecology and Diffusion, Biomathematics, 10, Springer-Verlag, Berlin-New York, 1980. |
[21] |
M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models, Journal of Mathematical Biology, 45 (2002), 219-233.
doi: 10.1007/s002850200144. |
[22] |
B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosciences, 196 (2005), 82-98.
doi: 10.1016/j.mbs.2005.03.008. |
[23] |
B. Li, M. Lewis and H. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, Journal of Mathematical Biology, 58 (2009), 323-338.
doi: 10.1007/s00285-008-0175-1. |
[24] | |
[25] |
R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosciences, 93 (1989), 269-295.
doi: 10.1016/0025-5564(89)90027-8. |
[26] |
L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics," Mathematical Surveys and Monographs, 102, American Mathematical Society, Providence, RI, 2003. |
[27] |
S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equation, Journal of Differential Equations, 237 (2007), 259-277.
doi: 10.1016/j.jde.2007.03.014. |
[28] |
H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. of Math. Biol., 8 (1979), 173-187. |
[29] |
J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.
doi: 10.1093/biomet/38.1-2.196. |
[30] |
K. R. Ríos-Soto, C. Castillo-Chavez, M. Neubert, E. S. Titi and A.-A. Yakubu, Epidemic spread in populations at demographic equilibrium, in "Mathematical Studies on Human Disease Dynamics" (eds. A. Gumel, C. Castillo-Chavez, D. P. Clemence and R. E. Mickens), Contemp. Math., 410, American Mathematical Society, Providence, RI, (2006), 297-309. |
[31] |
H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations, Journal of Differential Equations, 247 (2009), 887-905.
doi: 10.1016/j.jde.2009.04.002. |
[32] |
H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, Journal of Nonlinear Science, 21 (2011), 747-783.
doi: 10.1007/s00332-011-9099-9. |
[33] |
H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.
doi: 10.1007/s002850200145. |
[34] |
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. |
[35] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[36] |
H. F. Weinberger, Asymptotic behavior of a model in population genetics, in "Nonlinear Partial Differential Equations and Applications" (ed. J. M. Chadam) (Proc. Special Sem., Indiana Univ., Bloomington, Ind., 1976-1977), Lecture Notes in Mathematics, Vol. 648, Springer, Berlin, (1978), 47-96. |
[37] |
H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds for a partially cooperative 2-species reaction-diffusion model, Discrete and Continuous Dynamical Systems, 23 (2009), 1087-1098.
doi: 10.3934/dcds.2009.23.1087. |
[38] |
P. Weng, X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, Journal of Differential Equations, 229 (2006), 270-296.
doi: 10.1016/j.jde.2006.01.020. |
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