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

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

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

[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.

[3]

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

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

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

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

[7]

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

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

[9]

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

[10]

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

[11]

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

[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.

[13]

R. Horn, C. Johnson and R. Charles, "Matrix Analysis,", Cambridge University Press, (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. 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.

[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.

[17]

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

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

[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.

[20]

A. Okubo, "Diffusion and Ecological Problems: Mathematical Models," An extended version of the Japanese edition, Ecology and Diffusion,, Biomathematics, 10 (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. 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. 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. 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. doi: 10.1016/0025-5564(89)90027-8.

[26]

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

[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.

[28]

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

[29]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196. 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, 410 (2006), 297.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[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.

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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[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.

[13]

R. Horn, C. Johnson and R. Charles, "Matrix Analysis,", Cambridge University Press, (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. 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.

[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.

[17]

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

[18]

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

[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.

[20]

A. Okubo, "Diffusion and Ecological Problems: Mathematical Models," An extended version of the Japanese edition, Ecology and Diffusion,, Biomathematics, 10 (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. 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. 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. 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. doi: 10.1016/0025-5564(89)90027-8.

[26]

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

[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.

[28]

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

[29]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196. 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, 410 (2006), 297.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[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.

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