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Existence of Waves for a Nonlocal Reaction-Diffusion Equation

Published online by Cambridge University Press:  27 July 2010

I. Demin  and
V. Volpert
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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.

Keywords

integro-differential equationtravelling wavesLeray-Schauder method
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 5: Reaction-diffusion waves , 2010 , pp. 80 - 101
Copyright
© EDP Sciences, 2010

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References

Ai, S.. Traveling wave fronts for generalized Fisher equations with spatio-temporal delays. J. Differential Equations, 232 (2007), 104–133.CrossRefGoogle Scholar
Apreutesei, A., Ducrot, A., Volpert, V.. Competition of species with intra-specific competition. Math. Model. Nat. Phenom., 3 (2008), 1–27.CrossRefGoogle Scholar
Apreutesei, N., Ducrot, A., Volpert, V.. Travelling waves for integro-differential equations in population dynamics. Discrete Cont. Dyn. Syst. Ser. B, 11 (2009), 541–561.CrossRefGoogle Scholar
Britton, N.F.. Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model. SIAM J. Appl. Math., 6 (1990), 1663–1688.CrossRefGoogle Scholar
Ducrot, A., Travelling wave solutions for a scalar age-structured equation, Discrete Contin. Dyn. Syst. Ser. B , 7 (2007), 251–273. CrossRefGoogle Scholar
Fife, P. C., McLeod, J. B.. The approach of solutions of nonlinear diffusion equations to travelling wave solutions. Bull. Amer. Math. Soc., 81 (1975), 1076–1078.CrossRefGoogle Scholar
A. Friedman. Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, 1964.
Génieys, S., Volpert, V., Auger, P.. Pattern and waves for a model in population dynamics with nonlocal consumption of resources. Math. Model. Nat. Phenom., 1 (2006), 63–80.CrossRefGoogle Scholar
Gourley, S. A.. Travelling front solutions of a nonlocal Fisher equation. J. Math. Biol. 41 (2000), 272–284. CrossRefGoogle ScholarPubMed
Kanel, Ya. I.. 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. Google Scholar
Lefever, R., Lejeune, O.. On the origin of tiger bush. Bul. Math. Biol., 59 (1997), No. 2, 263–294. CrossRefGoogle Scholar
Volpert, A.I., Volpert, V.A.. 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.Google Scholar
A. Volpert, Vl. Volpert, Vit. Volpert. Travelling wave solutions of parabolic systems. 1994, AMS, Providence.
Volpert, V., Volpert, A., Collet, J.F.. Topological degree for elliptic operators in unbounded cylinders. Adv. Diff. Eq., 4 (1999), 777–812.Google Scholar
Volpert, V., Volpert, A.. Properness and topological degree for general elliptic operators. Abstract and Applied Analysis, 2003 (2003), 129–181.CrossRefGoogle Scholar
Volpert, A., Volpert, V.. Normal solvability of general linear elliptic problems. Abstract and Applied Analysis, 7 (2005), 733–756.CrossRefGoogle Scholar
Wang, Y., Yin, J., Travelling waves for a biological reaction diffusion model with spatio- temporal delay, J. Math. Anal. Appl., 325 (2007), 1400–1409. CrossRefGoogle Scholar
Wang, Z.C., Li, W.T., Ruan, S., Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Diff. Equations, 222 (2006), 185–232. CrossRefGoogle Scholar