# American Institute of Mathematical Sciences

2010, 7(1): 37-49. doi: 10.3934/mbe.2010.7.37

## Structured populations with diffusion in state space

 1 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, United States

Received  May 2009 Revised  October 2009 Published  January 2010

The classical models for populations structured by size have two features which may cause problems in biologically realistic modeling approaches: the structure variable always increases, and individuals in an age cohort that are identical initially stay identical throughout their lives. Here a diffusion term is introduced in the partial differential equation which mathematically amounts to adding viscosity. This approach solves both problems but it requires to identify appropriate boundary (recruitment) conditions. The method is applied to size-structured populations, metapopulations, infectious diseases, and vector-transmitted diseases.
Citation: Karl Peter Hadeler. Structured populations with diffusion in state space. Mathematical Biosciences & Engineering, 2010, 7 (1) : 37-49. doi: 10.3934/mbe.2010.7.37
 [1] Rong Liu, Feng-Qin Zhang, Yuming Chen. Optimal contraception control for a nonlinear population model with size structure and a separable mortality. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3603-3618. doi: 10.3934/dcdsb.2016112 [2] Jibin Li, Yi Zhang. On the traveling wave solutions for a nonlinear diffusion-convection equation: Dynamical system approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1119-1138. doi: 10.3934/dcdsb.2010.14.1119 [3] Inwon C. Kim, Helen K. Lei. Degenerate diffusion with a drift potential: A viscosity solutions approach. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 767-786. doi: 10.3934/dcds.2010.27.767 [4] Xing Liang, Lei Zhang. The optimal distribution of resources and rate of migration maximizing the population size in logistic model with identical migration. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2055-2065. doi: 10.3934/dcdsb.2020280 [5] Philippe Laurençot, Christoph Walker. The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021032 [6] Iryna Pankratova, Andrey Piatnitski. Homogenization of convection-diffusion equation in infinite cylinder. Networks & Heterogeneous Media, 2011, 6 (1) : 111-126. doi: 10.3934/nhm.2011.6.111 [7] Vitali Vougalter, Vitaly Volpert. On the solvability conditions for the diffusion equation with convection terms. Communications on Pure & Applied Analysis, 2012, 11 (1) : 365-373. doi: 10.3934/cpaa.2012.11.365 [8] Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095 [9] Md. Rabiul Haque, Takayoshi Ogawa, Ryuichi Sato. Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space. Communications on Pure & Applied Analysis, 2020, 19 (2) : 677-697. doi: 10.3934/cpaa.2020031 [10] Iryna Pankratova, Andrey Piatnitski. On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 935-970. doi: 10.3934/dcdsb.2009.11.935 [11] Suman Kumar Sahoo, Manmohan Vashisth. A partial data inverse problem for the convection-diffusion equation. Inverse Problems & Imaging, 2020, 14 (1) : 53-75. doi: 10.3934/ipi.2019063 [12] Liviu I. Ignat, Ademir F. Pazoto. Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3575-3589. doi: 10.3934/dcds.2014.34.3575 [13] Chunpeng Wang, Yanan Zhou, Runmei Du, Qiang Liu. Carleman estimate for solutions to a degenerate convection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4207-4222. doi: 10.3934/dcdsb.2018133 [14] Dongxue Yan, Xianlong Fu. Long-time behavior of a size-structured population model with diffusion and delayed birth process. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021030 [15] Song Liang, Yuan Lou. On the dependence of population size upon random dispersal rate. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2771-2788. doi: 10.3934/dcdsb.2012.17.2771 [16] Jacek Banasiak, Wilson Lamb. Coagulation, fragmentation and growth processes in a size structured population. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 563-585. doi: 10.3934/dcdsb.2009.11.563 [17] Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735 [18] Abdelaziz Rhandi, Roland Schnaubelt. Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 663-683. doi: 10.3934/dcds.1999.5.663 [19] Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007 [20] Zhijie Cao, Lijun Zhang. Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2703-2717. doi: 10.3934/dcdss.2020218

2018 Impact Factor: 1.313