Article Contents
Article Contents

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

• 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.

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

 Citation:

•  [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.