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2013, 10(5&6): 1419-1436. doi: 10.3934/mbe.2013.10.1419

Some recent developments on linear determinacy

1. 

Mathematics, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904, Tempe, AZ 85287

2. 

Department of Mathematics, University of Louisville, Louisville, KY 40292

3. 

School of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069, United States

Received  October 2012 Revised  April 2013 Published  August 2013

The process of invasion is fundamental to the study of the dynamics of ecological and epidemiological systems. Quantitatively, a crucial measure of species' invasiveness is given by the rate at which it spreads into new open environments. The so-called ``linear determinacy'' conjecture equates full nonlinear model spread rates with the spread rates computed from linearized systems with the linearization carried out around the leading edge of the invasion. A survey that accounts for recent developments in the identification of conditions under which linear determinacy gives the ``right" answer, particularly in the context of non-compact and non-cooperative systems, is the thrust of this contribution. Novel results that extend some of the research linked to some the contributions covered in this survey are also discussed.
Citation: Carlos Castillo-Chavez, Bingtuan Li, Haiyan Wang. Some recent developments on linear determinacy. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1419-1436. doi: 10.3934/mbe.2013.10.1419
References:
[1]

D. A. Andow, P. M. Kareiva, S. A. Levin and A. Okubo, Spread of invading organisms,, Landscape Ecol., 4 (1990), 177.

[2]

D. A. Andow, P. M. Kareiva, S. A. Levin and A. Okubo, Spread of invading organisms: Patterns of spread,, in, (1993), 219.

[3]

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

[4]

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.

[5]

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

[6]

F. Brauer, and C. Castillo-Chvez, "Mathematical Models in Population Biology and Epidemiology,", Second edition, 40 (2012). doi: 10.1007/978-1-4614-1686-9.

[7]

V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases,, J. Math. Biol., 13 (1981), 173. doi: 10.1007/BF00275212.

[8]

M. M. Crow, Organizing teaching and research to address the grand challenges of sustainable development,, BioScience, 60 (2010), 488. doi: 10.1525/bio.2010.60.7.2.

[9]

O. Diekmann, Thresholds and traveling waves for the geographical spread of an infection,, J. Math. Biol., 6 (1978), 109. doi: 10.1007/BF02450783.

[10]

S. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $R^4$,, Trans. Amer. Math. Soc., 286 (1984), 557. doi: 10.2307/1999810.

[11]

J. M. Epstein and R. Axtell, "Growing Artificial Societies: Social Science from the Bottom Up,", MIT Press, (1996).

[12]

J. M. Epstein, "Generative Social Science: Studies in Agent-Based Computational Modeling,", Princeton University Press, (2007).

[13]

S. Eubank, H. Guclu, V. S. Kumar, M. V. Marathe, A. Srinivasan, Z. Toroczkai and N. Wang, Modelling disease outbreaks in realistic urban social networks,, Nature, 429 (2004), 180. doi: 10.1038/nature02541.

[14]

J. Fang and X. Q. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems,, J. Dynamics and Differential Equations, 21 (2009), 663. doi: 10.1007/s10884-009-9152-7.

[15]

R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[16]

N. M. Ferguson, D. A. Cummings, S. Cauchemez, C. Fraser, S. Riley, A. Meeyai, S. Iamsirithaworn and D. S. Burke, Strategies for containing an emerging influenza pandemic in Southeast Asia,, Nature, 437 (2005), 209. doi: 10.1038/nature04017.

[17]

N. M. Ferguson, D. A. Cummings, C. Fraser, J. C. Cajka, P. C. Cooley and D. S. Burke, Strategies for mitigating an influenza pandemic,, Nature, 442 (2006), 448. doi: 10.1038/nature04795.

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R. Gardner, Existence of traveling wave solutions of predator-prey systems via the connection index,, SIAM J. Appl. Math., 44 (1984), 56. doi: 10.1137/0144006.

[19]

T. C. Germann, K. Kadau, I. M. Longini, Jr. and C. A. Macken, Mitigation strategies for pandemic influenza in the United States,, Proc Natl. Acad. Sci. USA, 103 (2006), 5935. doi: 10.1073/pnas.0601266103.

[20]

K. P Hadeler and F. Rothe, Traveling fronts in nonlinear diffusion equation,, J. Math. Bio., 2 (1975), 251. doi: 10.1007/BF00277154.

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

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K. P Hadeler, Reaction transport systems,, in, 1714 (1999). doi: 10.1007/BFb0092376.

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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. doi: 10.1111/j.1461-0248.2004.00687.x.

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R. Hengeveld, "Dynamics of Biological Invasions,", Chapman and Hall, (1989).

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S. I. Higgins, R. Nathan and M. L. Cain, Are long-distance dispersal events in plants usually caused by nonstandard means of dispersal?,, Ecology, 84 (2003), 1945. doi: 10.1890/01-0616.

[28]

Y. Hosono and B. Ilyas, Existence of traveling waves with any positive speed for a diffusive epidemic model,, Nonlinear World, 1 (1994), 277.

[29]

Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model,, Math. Models Methods Appl. Sci., 5 (1995), 935. doi: 10.1142/S0218202595000504.

[30]

S. 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.

[31]

J. Huang, G. Lu and S. Ruan, Existence of traveling wave solutions in a diffusive predator-prey model,, J. Math. Biol., 46 (2003), 132. doi: 10.1007/s00285-002-0171-9.

[32]

W. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model,, J. Diff. Equations, 251 (2011), 1549. doi: 10.1016/j.jde.2011.05.012.

[33]

W. Huang, Traveling wave solutions for a class of predator-prey,, J. Dyn. Diff. Equat., 24 (2012), 633. doi: 10.1007/s10884-012-9255-4.

[34]

Y. Hosono, The minimal speed of traveling fronts for a diffusive Lotka Volterra competition model,, Bull. Math. Biol., 60 (1998), 435. doi: 10.1006/bulm.1997.0008.

[35]

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

[36]

T. K. Theodoropoulos, Y.-H. Qian and I. G. Kevrekidis, Coarse stability and bifurcation analysis using timesteppers: A reaction diffusion example,, Proc. Natl. Acad. Sci., 97 (2000), 9840.

[37]

A. Makeev, D. Maroudas and I. G. Kevrekidis, Coarse stability and bifurcation analysis using stochastic simulators: Kinetic Monte Carlo examples,, J. Chem. Phys., 116 (2002), 10083. doi: 10.1063/1.1476929.

[38]

I. M. Longini, Jr., M. E. Halloran, A. Nizam, Y. Yang, S. Xu, D. S. Burke, D. A. Cummings and J. M. Epstein, Containing a large bioterrorist smallpox attack: A computer simulation approach,, Int. J. Infect. Dis., 11 (2007), 98. doi: 10.1016/j.ijid.2006.03.002.

[39]

D. Mollison, Dependence of epidemic and population velocities on basic parameters,, Mathematical Biosciences, 107 (1991), 255. doi: 10.1016/0025-5564(91)90009-8.

[40]

A. Kolmogorov, I. G. Petrovsky and N. S. Piscounov, Etude de lequation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique,, Bull. Moscow Univ. Math. Mech., 1 (1937), 1.

[41]

M. Kot, Discrete-time traveling waves: Ecological examples,, J. of Math. Biol., 30 (1992), 413. doi: 10.1007/BF00173295.

[42]

M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms,, Ecology, 77 (1996), 2027. doi: 10.2307/2265698.

[43]

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

[44]

S. A. Levin and R. T.Paine, Disturbance, patch formation, and community structure,, Proc. Nat. Acad. Sci. USA, 71 (1974), 2744.

[45]

A. Okubo, "Diffusion and Ecological Problems: Mathematical Models,", Biomathematics, 10 (1980).

[46]

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.

[47]

M. A. Lewis and G. Schmitz, Biological invasion of an organism with separate mobile and stationary states: Modelling and analysis,, Forma, 11 (1996), 1.

[48]

B. Li, Traveling wave solutions in partially degenerate cooperative reaction-diffusion systems,, Journal of Differential Equations, 252 (2012), 4842. doi: 10.1016/j.jde.2012.01.018.

[49]

B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems,, Nonlinearity, 24 (2011), 1759. doi: 10.1088/0951-7715/24/6/004.

[50]

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.

[51]

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.

[52]

W. Li and S. Wu, Traveling waves in a diffusive predator-prey model with holling type-III functional response,, Chaos Solitons Fractals, 3 (2008), 476. doi: 10.1016/j.chaos.2006.09.039.

[53]

X. Lin, P. Weng and C. Wu, Traveling wave solutions for a predator-prey system with sigmoidal response function,, J. Dynam. Di . Equ., 23 (2011), 903. doi: 10.1007/s10884-011-9220-7.

[54]

X. Liang and X. Zhao, Asymptotic speed of pread and traveling waves for monotone semiflows with applications,, Commun. Pure Appl. Math., 60 (2007), 1. doi: 10.1002/cpa.20154.

[55]

M. Owen and M. Lewis, How predation can slow, stop or reverse a prey invasion,, Bull. Math. Biol., 63 (2001), 655. doi: 10.1006/bulm.2001.0239.

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A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives,", Springer-Verlag, (2002).

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[65]

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[66]

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

[67]

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[69]

M.-H. Wang, M. Kot and M. Neubert, Integrodifference equations, Allee effects, and invasions,, J. Math. Biol., 44 (2002), 150. doi: 10.1007/s002850100116.

[70]

X.-S. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation,, Discrete and Continuous Dynamical Systems A, 32 (2012), 3303. doi: 10.3934/dcds.2012.32.3303.

[71]

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.

[72]

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.

[73]

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

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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.

[77]

H. F. Weinberger and X.-Q. Zhao, An extension of the formula for spreading speeds,, Math. Biosci. Eng., 7 (2010), 187. doi: 10.3934/mbe.2010.7.187.

[78]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dyn. Diff. Eqs., 13 (2001), 651. doi: 10.1023/A:1016690424892.

show all references

References:
[1]

D. A. Andow, P. M. Kareiva, S. A. Levin and A. Okubo, Spread of invading organisms,, Landscape Ecol., 4 (1990), 177.

[2]

D. A. Andow, P. M. Kareiva, S. A. Levin and A. Okubo, Spread of invading organisms: Patterns of spread,, in, (1993), 219.

[3]

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

[4]

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.

[5]

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

[6]

F. Brauer, and C. Castillo-Chvez, "Mathematical Models in Population Biology and Epidemiology,", Second edition, 40 (2012). doi: 10.1007/978-1-4614-1686-9.

[7]

V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases,, J. Math. Biol., 13 (1981), 173. doi: 10.1007/BF00275212.

[8]

M. M. Crow, Organizing teaching and research to address the grand challenges of sustainable development,, BioScience, 60 (2010), 488. doi: 10.1525/bio.2010.60.7.2.

[9]

O. Diekmann, Thresholds and traveling waves for the geographical spread of an infection,, J. Math. Biol., 6 (1978), 109. doi: 10.1007/BF02450783.

[10]

S. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $R^4$,, Trans. Amer. Math. Soc., 286 (1984), 557. doi: 10.2307/1999810.

[11]

J. M. Epstein and R. Axtell, "Growing Artificial Societies: Social Science from the Bottom Up,", MIT Press, (1996).

[12]

J. M. Epstein, "Generative Social Science: Studies in Agent-Based Computational Modeling,", Princeton University Press, (2007).

[13]

S. Eubank, H. Guclu, V. S. Kumar, M. V. Marathe, A. Srinivasan, Z. Toroczkai and N. Wang, Modelling disease outbreaks in realistic urban social networks,, Nature, 429 (2004), 180. doi: 10.1038/nature02541.

[14]

J. Fang and X. Q. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems,, J. Dynamics and Differential Equations, 21 (2009), 663. doi: 10.1007/s10884-009-9152-7.

[15]

R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[16]

N. M. Ferguson, D. A. Cummings, S. Cauchemez, C. Fraser, S. Riley, A. Meeyai, S. Iamsirithaworn and D. S. Burke, Strategies for containing an emerging influenza pandemic in Southeast Asia,, Nature, 437 (2005), 209. doi: 10.1038/nature04017.

[17]

N. M. Ferguson, D. A. Cummings, C. Fraser, J. C. Cajka, P. C. Cooley and D. S. Burke, Strategies for mitigating an influenza pandemic,, Nature, 442 (2006), 448. doi: 10.1038/nature04795.

[18]

R. Gardner, Existence of traveling wave solutions of predator-prey systems via the connection index,, SIAM J. Appl. Math., 44 (1984), 56. doi: 10.1137/0144006.

[19]

T. C. Germann, K. Kadau, I. M. Longini, Jr. and C. A. Macken, Mitigation strategies for pandemic influenza in the United States,, Proc Natl. Acad. Sci. USA, 103 (2006), 5935. doi: 10.1073/pnas.0601266103.

[20]

K. P Hadeler and F. Rothe, Traveling fronts in nonlinear diffusion equation,, J. Math. Bio., 2 (1975), 251. doi: 10.1007/BF00277154.

[21]

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

[22]

K. P Hadeler, Reaction transport systems,, in, 1714 (1999). doi: 10.1007/BFb0092376.

[23]

K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment,, Can. Appl. Math. Q., 10 (2002), 473.

[24]

M. Hassell and H. Comins, Discrete time models for two-species competition,, Theoretical Population Biology, 9 (1976), 202. doi: 10.1016/0040-5809(76)90045-9.

[25]

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. doi: 10.1111/j.1461-0248.2004.00687.x.

[26]

R. Hengeveld, "Dynamics of Biological Invasions,", Chapman and Hall, (1989).

[27]

S. I. Higgins, R. Nathan and M. L. Cain, Are long-distance dispersal events in plants usually caused by nonstandard means of dispersal?,, Ecology, 84 (2003), 1945. doi: 10.1890/01-0616.

[28]

Y. Hosono and B. Ilyas, Existence of traveling waves with any positive speed for a diffusive epidemic model,, Nonlinear World, 1 (1994), 277.

[29]

Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model,, Math. Models Methods Appl. Sci., 5 (1995), 935. doi: 10.1142/S0218202595000504.

[30]

S. 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.

[31]

J. Huang, G. Lu and S. Ruan, Existence of traveling wave solutions in a diffusive predator-prey model,, J. Math. Biol., 46 (2003), 132. doi: 10.1007/s00285-002-0171-9.

[32]

W. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model,, J. Diff. Equations, 251 (2011), 1549. doi: 10.1016/j.jde.2011.05.012.

[33]

W. Huang, Traveling wave solutions for a class of predator-prey,, J. Dyn. Diff. Equat., 24 (2012), 633. doi: 10.1007/s10884-012-9255-4.

[34]

Y. Hosono, The minimal speed of traveling fronts for a diffusive Lotka Volterra competition model,, Bull. Math. Biol., 60 (1998), 435. doi: 10.1006/bulm.1997.0008.

[35]

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

[36]

T. K. Theodoropoulos, Y.-H. Qian and I. G. Kevrekidis, Coarse stability and bifurcation analysis using timesteppers: A reaction diffusion example,, Proc. Natl. Acad. Sci., 97 (2000), 9840.

[37]

A. Makeev, D. Maroudas and I. G. Kevrekidis, Coarse stability and bifurcation analysis using stochastic simulators: Kinetic Monte Carlo examples,, J. Chem. Phys., 116 (2002), 10083. doi: 10.1063/1.1476929.

[38]

I. M. Longini, Jr., M. E. Halloran, A. Nizam, Y. Yang, S. Xu, D. S. Burke, D. A. Cummings and J. M. Epstein, Containing a large bioterrorist smallpox attack: A computer simulation approach,, Int. J. Infect. Dis., 11 (2007), 98. doi: 10.1016/j.ijid.2006.03.002.

[39]

D. Mollison, Dependence of epidemic and population velocities on basic parameters,, Mathematical Biosciences, 107 (1991), 255. doi: 10.1016/0025-5564(91)90009-8.

[40]

A. Kolmogorov, I. G. Petrovsky and N. S. Piscounov, Etude de lequation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique,, Bull. Moscow Univ. Math. Mech., 1 (1937), 1.

[41]

M. Kot, Discrete-time traveling waves: Ecological examples,, J. of Math. Biol., 30 (1992), 413. doi: 10.1007/BF00173295.

[42]

M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms,, Ecology, 77 (1996), 2027. doi: 10.2307/2265698.

[43]

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

[44]

S. A. Levin and R. T.Paine, Disturbance, patch formation, and community structure,, Proc. Nat. Acad. Sci. USA, 71 (1974), 2744.

[45]

A. Okubo, "Diffusion and Ecological Problems: Mathematical Models,", Biomathematics, 10 (1980).

[46]

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.

[47]

M. A. Lewis and G. Schmitz, Biological invasion of an organism with separate mobile and stationary states: Modelling and analysis,, Forma, 11 (1996), 1.

[48]

B. Li, Traveling wave solutions in partially degenerate cooperative reaction-diffusion systems,, Journal of Differential Equations, 252 (2012), 4842. doi: 10.1016/j.jde.2012.01.018.

[49]

B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems,, Nonlinearity, 24 (2011), 1759. doi: 10.1088/0951-7715/24/6/004.

[50]

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.

[51]

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.

[52]

W. Li and S. Wu, Traveling waves in a diffusive predator-prey model with holling type-III functional response,, Chaos Solitons Fractals, 3 (2008), 476. doi: 10.1016/j.chaos.2006.09.039.

[53]

X. Lin, P. Weng and C. Wu, Traveling wave solutions for a predator-prey system with sigmoidal response function,, J. Dynam. Di . Equ., 23 (2011), 903. doi: 10.1007/s10884-011-9220-7.

[54]

X. Liang and X. Zhao, Asymptotic speed of pread and traveling waves for monotone semiflows with applications,, Commun. Pure Appl. Math., 60 (2007), 1. doi: 10.1002/cpa.20154.

[55]

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