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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. Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[5]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

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[14]

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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. Google Scholar

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

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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. Google Scholar

[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. Google Scholar

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

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

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[32]

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[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. Google Scholar

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[35]

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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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. Google Scholar

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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. Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[5]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[41]

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

[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. Google Scholar

[43]

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

[44]

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

[45]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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