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Discrete & Continuous Dynamical Systems - B

2012 , Volume 17 , Issue 8

Special Issue
on PDE Models of Biological Processes

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Preface
Avner Friedman, Sze-Bi Hsu and Yuan Lou
2012, 17(8): i-i doi: 10.3934/dcdsb.2012.17.8i +[Abstract](103) +[PDF](77.7KB)
Abstract:
Recent years have seen dramatic increase in the number and variety of new mathematical models describing biological processes. Many of these models are formulated in terms of systems of partial differential equations. Relevant biological questions give rise to interesting questions regarding properties of the solutions of these equations. The present volume includes eleven articles, each describing a set of problems and results in PDEs inspired by biology. Although in many instances the mathematical analysis may help to better understand the underlying biological processes, the emphasis here is on new mathematical ideas and new mathematical results. The goal is to demonstrate the broad spectrum of new PDE theories that are emerging on the border of two fields: biology and mathematics.

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Exact travelling wave solutions of three-species competition--diffusion systems
Chiun-Chuan Chen, Li-Chang Hung, Masayasu Mimura and Daishin Ueyama
2012, 17(8): 2653-2669 doi: 10.3934/dcdsb.2012.17.2653 +[Abstract](218) +[PDF](543.1KB)
Abstract:
We consider the problem where $W$ invades the $(U,V)$ system in the three species Lotka-Volterra competition-diffusion model. Numerical simulation indicates that the presence of $W$ can dramatically change the competitive interaction between $U$ and $V$ in some parameter range if the invading $W$ is not too small. We also construct exact travelling wave solutions with non-trivial three components and track the bifurcation branches of these solutions by AUTO.
Steady states in hierarchical structured populations with distributed states at birth
József Z. Farkas and Peter Hinow
2012, 17(8): 2671-2689 doi: 10.3934/dcdsb.2012.17.2671 +[Abstract](169) +[PDF](423.8KB)
Abstract:
We investigate steady states of a quasilinear first order hyperbolic partial integro-differential equation. The model describes the evolution of a hierarchical structured population with distributed states at birth. Hierarchical size-structured models describe the dynamics of populations when individuals experience size-specific environment. This is the case for example in a population where individuals exhibit cannibalistic behavior and the chance to become prey (or to attack) depends on the individual's size. The other distinctive feature of the model is that individuals are recruited into the population at arbitrary size. This amounts to an infinite rank integral operator describing the recruitment process. First we establish conditions for the existence of a positive steady state of the model. Our method uses a fixed point result of nonlinear maps in conical shells of Banach spaces. Then we study stability properties of steady states for the special case of a separable growth rate using results from the theory of positive operators on Banach lattices.
A three dimensional model of wound healing: Analysis and computation
Avner Friedman, Bei Hu and Chuan Xue
2012, 17(8): 2691-2712 doi: 10.3934/dcdsb.2012.17.2691 +[Abstract](240) +[PDF](1866.9KB)
Abstract:
This paper is concerned with a three-dimensional model of wound healing. The boundary of the wound is a free boundary, and the region surrounding it is viewed as a partially healed tissue, satisfying a viscoelastic constitutive law for the velocity v. In the partially healed region the densities of several types of cells and the concentrations of several chemical species satisfy a coupled system of parabolic equations, whereas the tissue density satisfies a hyperbolic equation. The parabolic equations include advection by the velocity v and chemotaxis/haptotaxis terms. We prove existence and uniqueness of a smooth solution of the free boundary problem, for some time interval $0\leq t\leq T$, $T>0$. We also simulate the model equations to demonstrate the difference in the healing rate between normal wounds and chronic (or ischemic) wounds.
Recent developments on wave propagation in 2-species competition systems
Jong-Shenq Guo and Chang-Hong Wu
2012, 17(8): 2713-2724 doi: 10.3934/dcdsb.2012.17.2713 +[Abstract](230) +[PDF](353.7KB)
Abstract:
In this paper, we shall survey some recent results on the wave propagation in 2-species competition systems with Lotka-Volterra type nonlinearity. This includes systems with continuous and discrete diffusion (or migration). We are interested in both monostable case and bistable with strong competition case. Questions on minimal speed for the monostable case, uniqueness of wave speed and propagation failure in the bistable case, monotonicity and uniqueness of wave profile for both cases are addressed. Finally, we give some open problems on wave propagation in 2-species competition systems.
Energy variational approach to study charge inversion (layering) near charged walls
YunKyong Hyon, James E. Fonseca, Bob Eisenberg and Chun Liu
2012, 17(8): 2725-2743 doi: 10.3934/dcdsb.2012.17.2725 +[Abstract](307) +[PDF](598.9KB)
Abstract:
We introduce a mathematical model, which describes the charge inversion phenomena in systems with a charged wall or boundary. This model may prove helpful in understanding semiconductor devices, ion channels, and electrochemical systems like batteries that depend on complex distributions of charge for their function. The mathematical model is derived using the energy variational approach that takes into account ion diffusion, electrostatics, finite size effects, and specific boundary behavior. In ion dynamic theory, a well-known system of equations is the Poisson-Nernst-Planck (PNP) equation that includes entropic and electrostatic energy. The PNP type of equation can also be derived by the energy variational approach. However, the PNP equations have not produced the charge inversion/layering in charged wall situations presumably because the conventional PNP does not include the finite size of ions and other physical features needed to create the charge inversion. In this paper, we investigate the key features needed to produce the charge inversion phenomena using a mathematical model, the energy variational approach. One of the key features is a finite size (finite volume) effect, which is an unavoidable property of ions important for their dynamics on small scales. The other is an interfacial constraint to capture the spatial variation of electroneutrality in systems with charged walls. The interfacial constraint is established by the diffusive interface approach that approximately describes the boundary effect produced by the charged wall. The energy variational approach gives us a mathematically self-consistent way to introduce the interfacial constraint. We mainly discuss those two key features in this paper. Employing the energy variational approach, we derive a non-local partial differential equation with a total energy consisting of the entropic energy, electrostatic energy, repulsion energy representing the excluded volume effect, and the contribution of an interfacial constraint related to overall electroneutrality between bulk/bath and wall. The resulting mathematical model produces the charge inversion phenomena near charged walls. We compare the computational results of the mathematical model to those of Monte-Carlo computations.
On limit systems for some population models with cross-diffusion
Kousuke Kuto and Yoshio Yamada
2012, 17(8): 2745-2769 doi: 10.3934/dcdsb.2012.17.2745 +[Abstract](146) +[PDF](360.2KB)
Abstract:
This paper deals with the following reaction-diffusion system $$ (SP) \begin{equation} \left\{\begin{array}{11} \Delta[(1+\alpha v)u]+u(a-u-cv)=0, \\ \Delta[(1+\beta u)v]+v(b-du-v)=0, \end{array} \right. \end{equation} $$ in a bounded domain of $\Bbb{R}^N$ with homogeneous Neumann boundary conditions or Dirichlet boundary conditions. Our main purpose is to understand the structure of positive solutions of (SP) and know the effects of cross-diffusion coefficients $\alpha$ and $\beta$. For this purpose, our strategy is to study limiting behavior of positive solutions when $\alpha$ or $\beta$ goes to $\infty$ and derive the corresponding limit systems. We will obtain a priori estimates of $u$ and $v$ independently of $\beta$ (resp. $\alpha$) with small $\alpha\ge0$ (resp. $\beta\ge0$) in case $1\le N\le 3$ under Neumann boundary conditions, while we will obtain a priori estimates of $u$ and $v$ independently of $\alpha$ and $\beta$ in case $1\le N\le 5$ under Dirichlet boundary conditions. These a priori estimates allow us to investigate limiting behavior of positive solutions. When $\alpha=0$ and $\beta\to\infty$, we can derive two limit systems for Neumann conditions and one limit system for Dirichlet conditions. We will also give some results on the structure of positive solutions for such limit systems.
On the dependence of population size upon random dispersal rate
Song Liang and Yuan Lou
2012, 17(8): 2771-2788 doi: 10.3934/dcdsb.2012.17.2771 +[Abstract](254) +[PDF](389.3KB)
Abstract:
This paper concerns the dependence of the population size for a single species on its random dispersal rate and its applications on the invasion of species. The population size of a single species often depends on its random dispersal rate in non-trivial manners. Previous results show that the population size is usually not a monotone function of the random dispersal rate. We construct some examples to illustrate that the population size, as a function of the random dispersal rate, can have at least two local maxima. As an application we illustrate that the invasion of exotic species depends upon the random dispersal rate of the resident species in complicated manners. Previous results show that the total population is maximized at some intermediate random dispersal rate for several classes of local intrinsic growth rates. We find one family of local intrinsic growth rates such that the total population is maximized exactly at zero random dispersal rate. We show that the population distribution becomes flatter in average if we increase the random dispersal rate, and the environmental gradient is always steeper than the population distribution, at least in some average sense. We also discuss the dependence of the population size on movement rates in other contexts and propose some open problems.
A Lattice model on somitogenesis of zebrafish
Kang-Ling Liao and Chih-Wen Shih
2012, 17(8): 2789-2814 doi: 10.3934/dcdsb.2012.17.2789 +[Abstract](178) +[PDF](910.9KB)
Abstract:
Somitogenesis is the process of the development of somites which are segmental structure in vertebrate embryos. This process depends on the expression of segmentation clock genes. In this investigation, we consider lattice systems which describe the kinetics of the chief segmentation clock genes in zebrafish under negative feedback regulation with delay through interaction with the Delta-Notch signaling among neighboring cells. We first derive the analytical theories for the oscillation-arrested and synchronous oscillation in an autonomous lattice model. Based on the parameter regimes in the theories, we design suitable gradients of degradation rates and delays in a non-autonomous lattice model. Such a lattice system can generate synchronous oscillations, oscillatory traveling waves, oscillation slowing down, oscillation-arrested, and high-low expression levels. We further distinguish between different gradient structures which lead to normal and abnormal segmentations respectively and connect these structures to the dynamical regimes in the cell-cell model.
Vegetation patterns and desertification waves in semi-arid environments: Mathematical models based on local facilitation in plants
Jonathan A. Sherratt and Alexios D. Synodinos
2012, 17(8): 2815-2827 doi: 10.3934/dcdsb.2012.17.2815 +[Abstract](165) +[PDF](305.6KB)
Abstract:
In semi-arid regions, infiltration of rain water into the soil is significantly higher in vegetated areas than for bare ground. However, quantitative data on the dependence of infiltration capacity on plant biomass is very limited. In this paper, we use a simple reaction-diffusion-advection model to investigate the effects of varying the strength of this dependence. We begin by studying the formation of banded vegetation patterns on gentle slopes ("tiger bush"), which is a hallmark of semi-deserts. We calculate the range of rainfall parameter values over which such patterns occur, using numerical continuation methods. We then consider interfaces between vegetation and bare ground, showing that the vegetated region either expands or contracts depending on whether the rainfall parameter is above or below a critical value. We conclude by discussing the mathematical questions raised by our work.
Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment
Naveen K. Vaidya, Feng-Bin Wang and Xingfu Zou
2012, 17(8): 2829-2848 doi: 10.3934/dcdsb.2012.17.2829 +[Abstract](262) +[PDF](299.1KB)
Abstract:
In this paper, we propose a mathematical model to describe the avian influenza dynamics in wild birds with bird mobility and heterogeneous environment incorporated. In addition to establishing the basic properties of solutions to the model, we also prove the threshold dynamics which can be expressed either by the basic reproductive number or by the principal eigenvalue of the linearization at the disease free equilibrium. When the environment factor in the model becomes a constant (homogeneous environment), we are able to find explicit formulas for the basic reproductive number and the principal eigenvalue. We also perform numerical simulation to explore the impact of the heterogeneous environment on the disease dynamics. Our analytical and numerical results reveal that the avian influenza dynamics in wild birds is highly affected by both bird mobility and environmental heterogeneity.
Wavefront of an angiogenesis model
Zhi-An Wang
2012, 17(8): 2849-2860 doi: 10.3934/dcdsb.2012.17.2849 +[Abstract](203) +[PDF](528.8KB)
Abstract:
In this paper, we show the existence of traveling wave solutions to a chemotaxis model describing the initiation of angiogenesis. By a change of dependent variable, we transform the wave equation of the angiogenesis model to a Fisher type wave equation. Then we make use of the methods of analyzing the Fisher wave equation to obtain the existence of traveling wave solutions to the angiogenesis model. In virtue of the asymptotic behavior of the traveling wave solution at infinity, we find the explicit wave speed for cases of both zero and nonzero chemical diffusion. Finally based on the fact that the wave speed is convergent with respect to the chemical diffusion, we rigorously establish the zero chemical diffusion limit of traveling wave solutions by the energy estimates.

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