Front propagation in diffusion-aggregation models with bi-stable reaction

Mikhail Kuzmin; Stefano Ruggerini

doi:10.3934/dcdsb.2011.16.819

Article Contents

2011, Volume 16, Issue 3: 819-833. Doi: 10.3934/dcdsb.2011.16.819

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Front propagation in diffusion-aggregation models with bi-stable reaction

Mikhail Kuzmin^1, and
Stefano Ruggerini^2,

1.
Research Mathematical Institute, Voronezh State University, 394006 Voronezh
2.
Department of Pure and Applied Mathematics, University of Modena and Reggio Emilia, 41125 Reggio Emilia

Received: February 28, 2010

Revised: December 31, 2010

Published: September 2011

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Abstract

In this paper, necessary and sufficient conditions are given for the existence of travelling wave solutions of the reaction-diffusion-aggregation equation

$v_\tau=(D(v)v_x)_{x}+f(v), $

where the diffusivity $D$ changes sign twice in the interval $(0,1)$ (from positive to negative and again to positive) and the reaction $f$ is bi-stable. We show that classical travelling waves with decreasing profile do exist for a single admissible value of their speed of propagation which can be either positive or negative, according to the behavior of $f$ and $D$. An example is given, illustrating the employed techniques. The results are then generalized to a diffusivity $D$ with $2n$ sign changes.

Keywords:

Mathematics Subject Classification: Primary: 35K25, 92D25; Secondary: 34B40, 34B16.

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References

[1]	W. C. Allee, "Animal Aggregations," University of Chicago Press, Chicago, 1931.
[2]	W. Alt, Models for mutual attraction and aggregation of motile individuals, in "Lecture Notes in Biomathematics," 57, Springer-Verlag, (1985), 33-38.
[3]	D. G. Aronson, The role of diffusion in mathematical population biology: Skellam revisited, in "Lecture Notes in Biomathematics," 57, Springer-Verlag, (1985), 2-6.
[4]	V. Capasso, D. Morale and K. Oelschläger, An interacting particle system modelling aggregation behaviour: from individuals to populations, J. Math. Biology, 50 (2005), 49-66.doi: 10.1007/s00285-004-0279-1.
[5]	L. Ferracuti, C. Marcelli and F. Papalini, Travelling waves in some reaction-diffusion-aggregation equations, Adv. Dyn. Syst. Appl., 4 (2009), 19-33.
[6]	B. H. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection-Reaction," Birkhäuser Verlag, Basel, 2004.
[7]	P. K. Maini, L. Malaguti, C. Marcelli and S. Matucci, Diffusion-aggregation processes with mono-stable reaction terms, Discrete Continuous Dynam. Systems - B, 6 (2006), 1175-1189.
[8]	P. K. Maini, L. Malaguti, C. Marcelli and S. Matucci, Aggregative movement and front propagation for bi-stable population models, Math. Models and Methods in Appl. Sciences, 17 (2007), 1351-1368.doi: 10.1142/S0218202507002303.
[9]	L. Malaguti and C. Marcelli, Sharp profiles in degenerate and doubly-degenerate Fisher-KPP equations, J. Diff. Eqs, 195 (2003), 471-496.doi: 10.1016/j.jde.2003.06.005.
[10]	Víctor Padrón, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, Trans. of the American Math. Society, 7 (2004), 2737-2756.
[11]	Peter Turchin, Population consequences of aggregative movements, The Journal of Animal Ecology, 1 (1989), 75-100.doi: 10.2307/4987.