Spreading speeds and traveling waves for non-cooperative integro-difference systems

Previous Article
On sufficient conditions for a linearly determinate spreading speed
DCDS-B Home
This Issue
Next Article
Convex spacelike hypersurfaces of constant curvature in de Sitter space

September 2012, 17(6): 2243-2266. doi: 10.3934/dcdsb.2012.17.2243

Spreading speeds and traveling waves for non-cooperative integro-difference systems

Haiyan Wang ^1, and Carlos Castillo-Chavez ^2,

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

Received July 2011 Revised March 2012 Published May 2012

Abstract
Related Papers
Under a Creative Commons license

The study of spatially explicit integro-difference systems when the local population dynamics are given in terms of discrete-time generations models has gained considerable attention over the past two decades. These nonlinear systems arise naturally in the study of the spatial dispersal of organisms. The brunt of the mathematical research on these systems, particularly, when dealing with cooperative systems, has focused on the study of the existence of traveling wave solutions and the characterization of their spreading speed. Here, we characterize the minimum propagation (spreading) speed, via the convergence of initial data to wave solutions, for a large class of non cooperative nonlinear systems of integro-difference equations. The spreading speed turns out to be the slowest speed from a family of non-constant traveling wave solutions. The applicability of these theoretical results is illustrated through the explicit study of an integro-difference system with local population dynamics governed by Hassell and Comins' non-cooperative competition model (1976). The corresponding integro-difference nonlinear systems that results from the redistribution of individuals via a dispersal kernel is shown to satisfy conditions that guarantee the existence of minimum speeds and traveling waves. This paper is dedicated to Avner Friedman as we celebrate his immense contributions to the fields of partial differential equations, integral equations, mathematical biology, industrial mathematics and applied mathematics in general. His leadership in the mathematical sciences and his mentorship of students and friends over several decades has made a huge difference in the personal and professional lives of many, including both of us.

Keywords: biological invasion, traveling wave, Minimum speed, dispersal, integro-difference systems..

Mathematics Subject Classification: 39A11, 92D25, 92D4.

Citation: Haiyan Wang, Carlos Castillo-Chavez. Spreading speeds and traveling waves for non-cooperative integro-difference systems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2243-2266. doi: 10.3934/dcdsb.2012.17.2243

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]	, B. Li,, Personal communication., ().
[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]	, B. Li,, Personal communication., ().
[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.

[1]	Weiwei Ding, Xing Liang, Bin Xu. Spreading speeds of $N$-season spatially periodic integro-difference models. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3443-3472. doi: 10.3934/dcds.2013.33.3443
[2]	Hui Xue, Jianhua Huang, Zhixian Yu. Invasion traveling wave solutions in temporally discrete random-diffusion systems with delays. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1107-1131. doi: 10.3934/dcdss.2017060
[3]	Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047
[4]	Caleb Mayer, Eric Stachura. Traveling wave solutions for a cancer stem cell invasion model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5067-5093. doi: 10.3934/dcdsb.2020333
[5]	Aiyong Chen, Chi Zhang, Wentao Huang. Limit speed of traveling wave solutions for the perturbed generalized KdV equation. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022048
[6]	Liang Zhang, Bingtuan Li. Traveling wave solutions in an integro-differential competition model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 417-428. doi: 10.3934/dcdsb.2012.17.417
[7]	Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405
[8]	Mohammad El Smaily, François Hamel, Lionel Roques. Homogenization and influence of fragmentation in a biological invasion model. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 321-342. doi: 10.3934/dcds.2009.25.321
[9]	Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067
[10]	Wan-Tong Li, Li Zhang, Guo-Bao Zhang. Invasion entire solutions in a competition system with nonlocal dispersal. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531
[11]	Aijun Zhang. Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations. Conference Publications, 2013, 2013 (special) : 815-824. doi: 10.3934/proc.2013.2013.815
[12]	Kun Li, Jianhua Huang, Xiong Li. Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system. Communications on Pure and Applied Analysis, 2017, 16 (1) : 131-150. doi: 10.3934/cpaa.2017006
[13]	Guo Lin, Shuxia Pan. Periodic traveling wave solutions of periodic integrodifference systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3005-3031. doi: 10.3934/dcdsb.2020049
[14]	Xiongxiong Bao, Wenxian Shen, Zhongwei Shen. Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems. Communications on Pure and Applied Analysis, 2019, 18 (1) : 361-396. doi: 10.3934/cpaa.2019019
[15]	Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[16]	Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735
[17]	S. Rüdiger, J. Casademunt, L. Kramer. Kinks in stripe forming systems under traveling wave forcing. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 1027-1042. doi: 10.3934/dcdsb.2005.5.1027
[18]	Changbing Hu, Yang Kuang, Bingtuan Li, Hao Liu. Spreading speeds and traveling wave solutions in cooperative integral-differential systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1663-1684. doi: 10.3934/dcdsb.2015.20.1663
[19]	Franziska Hinkelmann, Reinhard Laubenbacher. Boolean models of bistable biological systems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1443-1456. doi: 10.3934/dcdss.2011.4.1443
[20]	P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Haiyan Wang Carlos Castillo-Chavez
on Google Scholar
Haiyan Wang Carlos Castillo-Chavez

[Back to Top]