# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - B, 2006, 6 (5) : 1175-1189. doi: 10.3934/dcdsb.2006.6.1175
 [1] Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116 [2] Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103 [3] Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426 [4] Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316 [5] Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464 [6] Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325 [7] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 [8] Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301 [9] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [10] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 [11] Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270 [12] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [13] Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339 [14] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351 [15] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [16] Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458 [17] Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354 [18] A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441 [19] José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376 [20] Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $\mathcal{W}(a, b, r)$. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

2019 Impact Factor: 1.27