Existence of Waves for a Nonlocal Reaction-Diffusion Equation
Mathematical Modelling of Natural Phenomena (2010)
- Volume: 5, Issue: 5, page 80-101
- ISSN: 0973-5348
Access Full Article
top
Abstract
topHow to cite
topDemin, I., and Volpert, V.. "Existence of Waves for a Nonlocal Reaction-Diffusion Equation." Mathematical Modelling of Natural Phenomena 5.5 (2010): 80-101. <http://eudml.org/doc/197617>.
@article{Demin2010,
abstract = {In this work we study a nonlocal reaction-diffusion equation arising in population
dynamics. The integral term in the nonlinearity describes nonlocal stimulation of
reproduction. We prove existence of travelling wave solutions by the Leray-Schauder method
using topological degree for Fredholm and proper operators and special a priori estimates
of solutions in weighted Hölder spaces.},
author = {Demin, I., Volpert, V.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {integro-differential equation; travelling waves; Leray-Schauder method; topological degree; nonlocal stimulation of reproduction},
language = {eng},
month = {7},
number = {5},
pages = {80-101},
publisher = {EDP Sciences},
title = {Existence of Waves for a Nonlocal Reaction-Diffusion Equation},
url = {http://eudml.org/doc/197617},
volume = {5},
year = {2010},
}
TY - JOUR
AU - Demin, I.
AU - Volpert, V.
TI - Existence of Waves for a Nonlocal Reaction-Diffusion Equation
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/7//
PB - EDP Sciences
VL - 5
IS - 5
SP - 80
EP - 101
AB - In this work we study a nonlocal reaction-diffusion equation arising in population
dynamics. The integral term in the nonlinearity describes nonlocal stimulation of
reproduction. We prove existence of travelling wave solutions by the Leray-Schauder method
using topological degree for Fredholm and proper operators and special a priori estimates
of solutions in weighted Hölder spaces.
LA - eng
KW - integro-differential equation; travelling waves; Leray-Schauder method; topological degree; nonlocal stimulation of reproduction
UR - http://eudml.org/doc/197617
ER -
References
top- S. Ai. Traveling wave fronts for generalized Fisher equations with spatio-temporal delays. J. Differential Equations, 232 (2007), 104–133.
- A. Apreutesei, A. Ducrot, V. Volpert. Competition of species with intra-specific competition. Math. Model. Nat. Phenom., 3 (2008), 1–27.
- N. Apreutesei, A. Ducrot, V. Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete Cont. Dyn. Syst. Ser. B, 11 (2009), 541–561.
- N.F. Britton. Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model. SIAM J. Appl. Math., 6 (1990), 1663–1688.
- A. Ducrot, Travelling wave solutions for a scalar age-structured equation, Discrete Contin. Dyn. Syst. Ser. B , 7 (2007), 251–273.
- P. C. Fife, J. B. McLeod. The approach of solutions of nonlinear diffusion equations to travelling wave solutions. Bull. Amer. Math. Soc., 81 (1975), 1076–1078.
- A. Friedman. Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, 1964.
- S. Génieys, V. Volpert, P. Auger. Pattern and waves for a model in population dynamics with nonlocal consumption of resources. Math. Model. Nat. Phenom., 1 (2006), 63–80.
- S. A. Gourley. Travelling front solutions of a nonlocal Fisher equation. J. Math. Biol.41 (2000), 272–284.
- Ya. I. Kanel. The behavior of solutions of the Cauchy problem when the time tends to infinity, in the case of quasilinear equations arising in the theory of combustion. Soviet Math. Dokl.1 (1960), 533–536.
- R. Lefever, O. Lejeune. On the origin of tiger bush. Bul. Math. Biol., 59 (1997), No. 2, 263–294.
- A.I. Volpert, V.A. Volpert. Applications of the rotation theory of vector fields to the study of wave solutions of parabolic equations. Trans. Moscow Math. Soc., 52 (1990), 59–108.
- A. Volpert, Vl. Volpert, Vit. Volpert. Travelling wave solutions of parabolic systems. 1994, AMS, Providence.
- V. Volpert, A. Volpert, J.F. Collet. Topological degree for elliptic operators in unbounded cylinders. Adv. Diff. Eq., 4 (1999), 777–812.
- V. Volpert, A. Volpert. Properness and topological degree for general elliptic operators. Abstract and Applied Analysis, 2003 (2003), 129–181.
- A. Volpert, V. Volpert. Normal solvability of general linear elliptic problems. Abstract and Applied Analysis, 7 (2005), 733–756.
- Y. Wang, J. Yin, Travelling waves for a biological reaction diffusion model with spatio- temporal delay, J. Math. Anal. Appl., 325 (2007), 1400–1409.
- Z.C. Wang, W.T. Li, S. Ruan, Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Diff. Equations, 222 (2006), 185–232.
NotesEmbed ?
topYou must be logged in to post comments.
To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.