American Institute of Mathematical Sciences

Advanced
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,", Animal Aggregations, (1931). Google Scholar

[2]

W. Alt, Models for mutual attraction and aggregation of motile individuals,, in, 57 (1985), 33. Google Scholar

[3]

D. G. Aronson, The role of diffusion in mathematical population biology: Skellam revisited,, Lecture Notes in Biomathematics, 57 (1985), 2. 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. 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. Google Scholar

[6]

B. H. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection-Reaction,", velling Waves in Nonlinear Diffusion-Convection-Reaction, (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. 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. 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. 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. Google Scholar

[11]

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

show all references

References:
[1]

W. C. Allee, "Animal Aggregations,", Animal Aggregations, (1931). Google Scholar

[2]

W. Alt, Models for mutual attraction and aggregation of motile individuals,, in, 57 (1985), 33. Google Scholar

[3]

D. G. Aronson, The role of diffusion in mathematical population biology: Skellam revisited,, Lecture Notes in Biomathematics, 57 (1985), 2. 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. 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. Google Scholar

[6]

B. H. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection-Reaction,", velling Waves in Nonlinear Diffusion-Convection-Reaction, (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. 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. 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. 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. Google Scholar

[11]

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

[1]

Philip K. Maini, Luisa Malaguti, Cristina Marcelli, Serena Matucci. Diffusion-aggregation processes with mono-stable reaction terms. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1175-1189. doi: 10.3934/dcdsb.2006.6.1175
[2]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242
[3]

Antoine Benoit. Finite speed of propagation for mixed problems in the $WR$ class. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2351-2358. doi: 10.3934/cpaa.2014.13.2351
[4]

Lihua Min, Xiaoping Yang. Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 543-566. doi: 10.3934/cpaa.2014.13.543
[5]

Jean-Daniel Djida, Juan J. Nieto, Iván Area. Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4031-4053. doi: 10.3934/dcdsb.2019049
[6]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 41-67. doi: 10.3934/dcds.2008.21.41
[7]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 19-61. doi: 10.3934/dcds.2009.25.19
[8]

Yong Zhou, Zhengguang Guo. Blow up and propagation speed of solutions to the DGH equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 657-670. doi: 10.3934/dcdsb.2009.12.657
[9]

S. Bonafede, G. R. Cirmi, A.F. Tedeev. Finite speed of propagation for the porous media equation with lower order terms. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 305-314. doi: 10.3934/dcds.2000.6.305
[10]

Shangbing Ai, Wenzhang Huang, Zhi-An Wang. Reaction, diffusion and chemotaxis in wave propagation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 1-21. doi: 10.3934/dcdsb.2015.20.1
[11]

Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057
[12]

L. M. Abia, O. Angulo, J.C. López-Marcos. Size-structured population dynamics models and their numerical solutions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1203-1222. doi: 10.3934/dcdsb.2004.4.1203
[13]

Tiberiu Harko, Man Kwong Mak. Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach. Mathematical Biosciences & Engineering, 2015, 12 (1) : 41-69. doi: 10.3934/mbe.2015.12.41
[14]

Chiun-Chuan Chen, Li-Chang Hung, Masayasu Mimura, Daishin Ueyama. Exact travelling wave solutions of three-species competition--diffusion systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2653-2669. doi: 10.3934/dcdsb.2012.17.2653
[15]

R. Kowalczyk, A. Gamba, L. Preziosi. On the stability of homogeneous solutions to some aggregation models. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 203-220. doi: 10.3934/dcdsb.2004.4.203
[16]

Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541
[17]

Andrea L. Bertozzi, Dejan Slepcev. Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1617-1637. doi: 10.3934/cpaa.2010.9.1617
[18]

Faustino Sánchez-Garduño, Philip K. Maini, Judith Pérez-Velázquez. A non-linear degenerate equation for direct aggregation and traveling wave dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 455-487. doi: 10.3934/dcdsb.2010.13.455
[19]

Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707
[20]

Martin Burger, Marco Di Francesco. Large time behavior of nonlocal aggregation models with nonlinear diffusion. Networks & Heterogeneous Media, 2008, 3 (4) : 749-785. doi: 10.3934/nhm.2008.3.749

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences