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September  2012, 17(6): 2267-2280. doi: 10.3934/dcdsb.2012.17.2267

On sufficient conditions for a linearly determinate spreading speed

Hans Weinberger 1,

1. 

School of Mathematics, University of Minnesota, 206 Church Street, Minneapolis, MN 55455, United States

Received  June 2011 Revised  December 2011 Published  May 2012

It is shown how to construct criteria of the form $f(u)\le f'(0)K(u)$ which guarantee that the spreading speed $c^*$ of a reaction-diffusion equation with the reaction term $f(u)$ is linearly determinate in the sense that $c^*=2\sqrt{f'(0)}$. Some of these criteria improve the classical condition $f(u)\le f'(0)u$, and permit the presence of sharp Allee effects. Inequalities which guarantee the failure of linear determinacy are also presented.
Keywords: spreading speed., Reaction-diffusion systems.
Mathematics Subject Classification: Primary: 35K60, 35B40; Secondary: 92A1.
Citation: Hans Weinberger. On sufficient conditions for a linearly determinate spreading speed. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2267-2280. doi: 10.3934/dcdsb.2012.17.2267
References:
[1]

W. C. Allee, "Animal Aggregations. A Study in General Sociology,'' U. of Chicago Press, 1931. Google Scholar

[2]

B. H. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction," Progress in Nonlinear Differential Equations and their Applications, 60, Birkhäuser Verlag, Basel, 2004.  Google Scholar

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K. P. Hadeler and F. Rothe, Traveling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.  Google Scholar

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A. N. Kolmogorov, I. G. Petrovski and N. S. Piscounov, Étude de l'équation de la diffusion avec croissance de la quantité de matiére et son application á un probléme biologique, Bull. Univ. d'État á Moscou Ser. Intern., 1 (1937), 1-26. Google Scholar

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M.-H. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Mathematical Biosciences, 171 (2001), 83-97. doi: 10.1016/S0025-5564(01)00048-7.  Google Scholar

show all references

References:
[1]

W. C. Allee, "Animal Aggregations. A Study in General Sociology,'' U. of Chicago Press, 1931. Google Scholar

[2]

B. H. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction," Progress in Nonlinear Differential Equations and their Applications, 60, Birkhäuser Verlag, Basel, 2004.  Google Scholar

[3]

K. P. Hadeler and F. Rothe, Traveling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.  Google Scholar

[4]

A. N. Kolmogorov, I. G. Petrovski and N. S. Piscounov, Étude de l'équation de la diffusion avec croissance de la quantité de matiére et son application á un probléme biologique, Bull. Univ. d'État á Moscou Ser. Intern., 1 (1937), 1-26. Google Scholar

[5]

M.-H. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Mathematical Biosciences, 171 (2001), 83-97. doi: 10.1016/S0025-5564(01)00048-7.  Google Scholar

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