September  2006, 6(5): 1175-1189. doi: 10.3934/dcdsb.2006.6.1175

Diffusion-aggregation processes with mono-stable reaction terms

1. 

Centre for Mathematical Biology, Mathematical Institute, University of Oxford, OX1 3LB Oxford

2. 

Dept. of Engineering Sciences and Methods, University of Modena and Reggio Emilia, I-42100, Italy

3. 

Dept. of Mathematical Sciences, Polytechnic University of Marche, Ancona, I-60131, Italy

4. 

Dept. of Electronic and Telecommunications, University of Florence, Florence, I-50139, Italy

Received  August 2005 Revised  January 2006 Published  June 2006

This paper analyses front propagation of the equation

$\upsilon_{\tau}=[D(\upsilon)\upsilon_{x}]_{x}+f(\upsilon) \tau\ge 0, x\in R,$

where $f$ is a monostable (i.e. Fisher-type) nonlinear reaction term and $D(\upsilon)$ changes its sign once, from positive to negative values, in the interval $\upsilon \in [0, 1]$ where the process is studied. This model equation accounts for simultaneous diffusive and aggregative behaviors of a population dynamic depending on the population density $\upsilon$ at time $\tau$ and position $x$. The existence of infinitely many traveling wave solutions is proven. These fronts are parameterized by their wave speed and monotonically connect the stationary states $\upsilon \equiv 0$ and $\upsilon \equiv 1$. In the degenerate case, i.e. when $D(0) = 0$ and/or $D(1) = 0$, sharp profiles appear, corresponding to the minimum wave speed. They also have new behaviors, in addition to those already observed in diffusive models, since they can be right compactly supported, left compactly supported, or both. The dynamics can exhibit, respectively, the phenomena of finite speed of propagation, finite speed of saturation, or both.

Citation: Philip K. Maini, Luisa Malaguti, Cristina Marcelli, Serena Matucci. Diffusion-aggregation processes with mono-stable reaction terms. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1175-1189. doi: 10.3934/dcdsb.2006.6.1175
[1]

Mikhail Kuzmin, Stefano Ruggerini. Front propagation in diffusion-aggregation models with bi-stable reaction. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 819-833. doi: 10.3934/dcdsb.2011.16.819

[2]

Yana Nec, Vladimir A Volpert, Alexander A Nepomnyashchy. Front propagation problems with sub-diffusion. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 827-846. doi: 10.3934/dcds.2010.27.827

[3]

Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1769-1781. doi: 10.3934/dcdsb.2014.19.1769

[4]

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

[5]

Yuming Paul Zhang. On a class of diffusion-aggregation equations. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 907-932. doi: 10.3934/dcds.2020066

[6]

Luisa Malaguti, Cristina Marcelli, Serena Matucci. Continuous dependence in front propagation of convective reaction-diffusion equations. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1083-1098. doi: 10.3934/cpaa.2010.9.1083

[7]

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

[8]

Mohar Guha, Keith Promislow. Front propagation in a noisy, nonsmooth, excitable medium. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 617-638. doi: 10.3934/dcds.2009.23.617

[9]

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

[10]

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

[11]

Sungrim Seirin Lee. Dependence of propagation speed on invader species: The effect of the predatory commensalism in two-prey, one-predator system with diffusion. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 797-825. doi: 10.3934/dcdsb.2009.12.797

[12]

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

[13]

Benoît Perthame, P. E. Souganidis. Front propagation for a jump process model arising in spacial ecology. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1235-1246. doi: 10.3934/dcds.2005.13.1235

[14]

Emeric Bouin. A Hamilton-Jacobi approach for front propagation in kinetic equations. Kinetic and Related Models, 2015, 8 (2) : 255-280. doi: 10.3934/krm.2015.8.255

[15]

Bo Su and Martin Burger. Global weak solutions of non-isothermal front propagation problem. Electronic Research Announcements, 2007, 13: 46-52.

[16]

Jong-Shenq Guo, Chang-Hong Wu. Front propagation for a two-dimensional periodic monostable lattice dynamical system. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 197-223. doi: 10.3934/dcds.2010.26.197

[17]

Margarita Arias, Juan Campos, Cristina Marcelli. Fastness and continuous dependence in front propagation in Fisher-KPP equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 11-30. doi: 10.3934/dcdsb.2009.11.11

[18]

Mia Jukić, Hermen Jan Hupkes. Curvature-driven front propagation through planar lattices in oblique directions. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2189-2251. doi: 10.3934/cpaa.2022055

[19]

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

[20]

Chufen Wu, Peixuan Weng. Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 867-892. doi: 10.3934/dcdsb.2011.15.867

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (17)

[Back to Top]