American Institute of Mathematical Sciences

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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

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 & Continuous Dynamical Systems - B, 2012, 17 (6) : 2243-2266. doi: 10.3934/dcdsb.2012.17.2243
References:
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[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics,, Adv. Math., 30 (1978), 33.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

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K. P. Hadeler and F. Rothe, Traveling fronts in nonlinear diffusion equation,, J. Math. Bio., 2 (1975), 251.   Google Scholar

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K. P. Hadeler, Reaction transport systems,, in V.Capasso, 1714 (1999), 95.   Google Scholar

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M. Hassell and H. Comins, Discrete time models for two-species competition,, Theoretical Population Biology, 9 (1976), 202.   Google Scholar

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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.   Google Scholar

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S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776.  doi: 10.1137/070703016.  Google Scholar

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H. Kierstad and L. B. Slobodkin, The size of water masses containing plankton blooms,, J. Mar. Res., 12 (1953), 141.   Google Scholar

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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.   Google Scholar

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S. A. Levin and R. T. Paine, Disturbance, patch formation, and community structure,, Proc. Nat. Acad. Sci. USA, 71 (1974), 2744.  doi: 10.1073/pnas.71.7.2744.  Google Scholar

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A. Okubo, "Diffusion and Ecological Problems: Mathematical Models," An extended version of the Japanese edition, Ecology and Diffusion,, Biomathematics, 10 (1980).   Google Scholar

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M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models,, Journal of Mathematical Biology, 45 (2002), 219.  doi: 10.1007/s002850200144.  Google Scholar

[22]

B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosciences, 196 (2005), 82.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[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.  doi: 10.1007/s00285-008-0175-1.  Google Scholar

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R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory,, Math. Biosciences, 93 (1989), 269.  doi: 10.1016/0025-5564(89)90027-8.  Google Scholar

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L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics,", Mathematical Surveys and Monographs, 102 (2003).   Google Scholar

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S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equation,, Journal of Differential Equations, 237 (2007), 259.  doi: 10.1016/j.jde.2007.03.014.  Google Scholar

[28]

H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread,, J. of Math. Biol., 8 (1979), 173.   Google Scholar

[29]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196.  doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[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, 410 (2006), 297.   Google Scholar

[31]

H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations,, Journal of Differential Equations, 247 (2009), 887.  doi: 10.1016/j.jde.2009.04.002.  Google Scholar

[32]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems,, Journal of Nonlinear Science, 21 (2011), 747.  doi: 10.1007/s00332-011-9099-9.  Google Scholar

[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.  doi: 10.1007/s002850200145.  Google Scholar

[34]

H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems,, J. Math. Biol., 55 (2007), 207.  doi: 10.1007/s00285-007-0078-6.  Google Scholar

[35]

H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353.  doi: 10.1137/0513028.  Google Scholar

[36]

H. F. Weinberger, Asymptotic behavior of a model in population genetics,, in, (1978), 1976.   Google Scholar

[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.  doi: 10.3934/dcds.2009.23.1087.  Google Scholar

[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.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, in, 446 (1975), 5.   Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics,, Adv. Math., 30 (1978), 33.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology,", Texts in Applied Mathematics, 40 (2001).   Google Scholar

[4]

K. Brown and J. Carr, Deterministic epidemic waves of critical velocity,, Math. Proc. Cambridge Philos. Soc., 81 (1977), 431.  doi: 10.1017/S0305004100053494.  Google Scholar

[5]

M. M. Crow, "Organizing Teaching and Research to Address the Grand Challenges of Sustainable Development,", BioScience, (2010), 488.   Google Scholar

[6]

O. Diekmann, Thresholds and traveling waves for the geographical spread of an infection,, J. Math. Biol., 6 (1978), 109.   Google Scholar

[7]

R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355.   Google Scholar

[8]

K. P. Hadeler and F. Rothe, Traveling fronts in nonlinear diffusion equation,, J. Math. Bio., 2 (1975), 251.   Google Scholar

[9]

K. P. Hadeler, Hyperbolic travelling fronts,, Proc. Edinb. Math. Soc. (2), 31 (1988), 89.  doi: 10.1017/S001309150000660X.  Google Scholar

[10]

K. P. Hadeler, Reaction transport systems,, in V.Capasso, 1714 (1999), 95.   Google Scholar

[11]

M. Hassell and H. Comins, Discrete time models for two-species competition,, Theoretical Population Biology, 9 (1976), 202.   Google Scholar

[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.   Google Scholar

[13]

R. Horn, C. Johnson and R. Charles, "Matrix Analysis,", Cambridge University Press, (1985).   Google Scholar

[14]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776.  doi: 10.1137/070703016.  Google Scholar

[15]

H. Kierstad and L. B. Slobodkin, The size of water masses containing plankton blooms,, J. Mar. Res., 12 (1953), 141.   Google Scholar

[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.   Google Scholar

[17]

M. Kot, Discrete-time traveling waves: Ecological examples,, J. of Math. Biol., 30 (1992), 413.   Google Scholar

[18]

S. A. Levin, Toward a science of sustainability: Executive summary,, in, (2009), 4.   Google Scholar

[19]

S. A. Levin and R. T. Paine, Disturbance, patch formation, and community structure,, Proc. Nat. Acad. Sci. USA, 71 (1974), 2744.  doi: 10.1073/pnas.71.7.2744.  Google Scholar

[20]

A. Okubo, "Diffusion and Ecological Problems: Mathematical Models," An extended version of the Japanese edition, Ecology and Diffusion,, Biomathematics, 10 (1980).   Google Scholar

[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.  doi: 10.1007/s002850200144.  Google Scholar

[22]

B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosciences, 196 (2005), 82.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[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.  doi: 10.1007/s00285-008-0175-1.  Google Scholar

[24]

, B. Li,, Personal communication., ().   Google Scholar

[25]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory,, Math. Biosciences, 93 (1989), 269.  doi: 10.1016/0025-5564(89)90027-8.  Google Scholar

[26]

L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics,", Mathematical Surveys and Monographs, 102 (2003).   Google Scholar

[27]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equation,, Journal of Differential Equations, 237 (2007), 259.  doi: 10.1016/j.jde.2007.03.014.  Google Scholar

[28]

H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread,, J. of Math. Biol., 8 (1979), 173.   Google Scholar

[29]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196.  doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[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, 410 (2006), 297.   Google Scholar

[31]

H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations,, Journal of Differential Equations, 247 (2009), 887.  doi: 10.1016/j.jde.2009.04.002.  Google Scholar

[32]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems,, Journal of Nonlinear Science, 21 (2011), 747.  doi: 10.1007/s00332-011-9099-9.  Google Scholar

[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.  doi: 10.1007/s002850200145.  Google Scholar

[34]

H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems,, J. Math. Biol., 55 (2007), 207.  doi: 10.1007/s00285-007-0078-6.  Google Scholar

[35]

H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353.  doi: 10.1137/0513028.  Google Scholar

[36]

H. F. Weinberger, Asymptotic behavior of a model in population genetics,, in, (1978), 1976.   Google Scholar

[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.  doi: 10.3934/dcds.2009.23.1087.  Google Scholar

[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.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar

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