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October  2011, 16(3): 819-833. doi: 10.3934/dcdsb.2011.16.819

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, Russian Federation
2. 

Department of Pure and Applied Mathematics, University of Modena and Reggio Emilia, 41125 Reggio Emilia, Italy

Received  March 2010 Revised  January 2011 Published  June 2011

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: population dynamics, finite speed of propagation, extinction., Diffusion-aggregation models, travelling wave solutions.
Mathematics Subject Classification: Primary: 35K25, 92D25; Secondary: 34B40, 34B1.
Citation: Mikhail Kuzmin, Stefano Ruggerini. Front propagation in diffusion-aggregation models with bi-stable reaction. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 819-833. doi: 10.3934/dcdsb.2011.16.819
References:
[1]

W. C. Allee, "Animal Aggregations," University of Chicago Press, Chicago, 1931. Google Scholar

[2]

W. Alt, Models for mutual attraction and aggregation of motile individuals, in "Lecture Notes in Biomathematics," 57, Springer-Verlag, (1985), 33-38.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. Ferracuti, C. Marcelli and F. Papalini, Travelling waves in some reaction-diffusion-aggregation equations, Adv. Dyn. Syst. Appl., 4 (2009), 19-33.  Google Scholar

[6]

B. H. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection-Reaction," Birkhäuser Verlag, Basel, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Peter Turchin, Population consequences of aggregative movements, The Journal of Animal Ecology, 1 (1989), 75-100. doi: 10.2307/4987.  Google Scholar

show all references

References:
[1]

W. C. Allee, "Animal Aggregations," University of Chicago Press, Chicago, 1931. Google Scholar

[2]

W. Alt, Models for mutual attraction and aggregation of motile individuals, in "Lecture Notes in Biomathematics," 57, Springer-Verlag, (1985), 33-38.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. Ferracuti, C. Marcelli and F. Papalini, Travelling waves in some reaction-diffusion-aggregation equations, Adv. Dyn. Syst. Appl., 4 (2009), 19-33.  Google Scholar

[6]

B. H. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection-Reaction," Birkhäuser Verlag, Basel, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Peter Turchin, Population consequences of aggregative movements, The Journal of Animal Ecology, 1 (1989), 75-100. doi: 10.2307/4987.  Google Scholar

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