On sufficient conditions for a linearly determinate spreading speed

Previous Article
Optimal control of integrodifference equations with growth-harvesting-dispersal order
DCDS-B Home
This Issue
Next Article
Spreading speeds and traveling waves for non-cooperative integro-difference systems

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

Abstract
Related Papers

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

show all references

References:

[1]	W. C. Allee, "Animal Aggregations. A Study in General Sociology,'' U. of Chicago Press, 1931.
[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.
[3]	K. P. Hadeler and F. Rothe, Traveling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.
[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.
[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.

[1]	Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591
[2]	Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126
[3]	Juliette Bouhours, Grégroie Nadin. A variational approach to reaction-diffusion equations with forced speed in dimension 1. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1843-1872. doi: 10.3934/dcds.2015.35.1843
[4]	Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405
[5]	Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785
[6]	Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515
[7]	Laurent Desvillettes, Klemens Fellner. Entropy methods for reaction-diffusion systems. Conference Publications, 2007, 2007 (Special) : 304-312. doi: 10.3934/proc.2007.2007.304
[8]	A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure and Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65
[9]	Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267
[10]	Hans F. Weinberger, Kohkichi Kawasaki, Nanako Shigesada. Spreading speeds for a partially cooperative 2-species reaction-diffusion model. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1087-1098. doi: 10.3934/dcds.2009.23.1087
[11]	Grégory Faye, Thomas Giletti, Matt Holzer. Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity. Discrete and Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021146
[12]	Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 41-67. doi: 10.3934/dcds.2008.21.41
[13]	Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 19-61. doi: 10.3934/dcds.2009.25.19
[14]	Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707
[15]	Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49
[16]	Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[17]	Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21
[18]	Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815
[19]	C. van der Mee, Stella Vernier Piro. Travelling waves for solid-gas reaction-diffusion systems. Conference Publications, 2003, 2003 (Special) : 872-879. doi: 10.3934/proc.2003.2003.872
[20]	Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks and Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Hans Weinberger
on Google Scholar
Hans Weinberger

[Back to Top]