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Discrete & Continuous Dynamical Systems - B

2015 , Volume 20 , Issue 4

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An energy-consistent depth-averaged Euler system: Derivation and properties
Marie-Odile Bristeau , Anne Mangeney , Jacques Sainte-Marie and  Nicolas Seguin
2015, 20(4): 961-988 doi: 10.3934/dcdsb.2015.20.961 +[Abstract](38) +[PDF](704.5KB)
In this paper, we present an original derivation process of a non-hydrostatic shallow water-type model which aims at approximating the incompressible Euler and Navier-Stokes systems with free surface. The closure relations are obtained by a minimal energy constraint instead of an asymptotic expansion. The model slightly differs from the well-known Green-Naghdi model and is confronted with stationary and analytical solutions of the Euler system corresponding to rotational flows. At the end of the paper, we give time-dependent analytical solutions for the Euler system that are also analytical solutions for the proposed model but that are not solutions of the Green-Naghdi model. We also give and compare analytical solutions of the two non-hydrostatic shallow water models.
Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment
Yongli Cai and  Weiming Wang
2015, 20(4): 989-1013 doi: 10.3934/dcdsb.2015.20.989 +[Abstract](29) +[PDF](819.8KB)
In this paper, we explore a parasite-host epidemiological model incorporating demographic and epidemiological processes in a spatially heterogeneous environment in which the individuals are subject to a random movement. We show the global stability of the extinction equilibrium in three different cases, and prove the existence, uniqueness and the global stability of the disease--free equilibrium. When the death rate in the model becomes a constant, we give the existence of the endemic equilibrium and the global stability of the endemic equilibrium in a special case. Furthermore, we perform a series of numerical simulations to display the effects of the movement of hosts and the heterogeneous environment on the disease dynamics. Our analytical and numerical results reveal that the disease extinction/outbreak can be ignited by both individual mobility and the environmental heterogeneity.
Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation
Hongmei Cheng and  Rong Yuan
2015, 20(4): 1015-1029 doi: 10.3934/dcdsb.2015.20.1015 +[Abstract](29) +[PDF](388.5KB)
This paper is concerned with the asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equation on $\mathbb{R}^n$, $n\geq 4$. Our first result states that pyramidal traveling fronts are asymptotically stable under the initial perturbations that decay at space infinity. Then we further show the existence of a solution that oscillates permanently between two pyramidal traveling fronts, which implies that pyramidal traveling fronts are not asymptotically stable under more general perturbations. Our main technique is the supersolution and subsolution method coupled with the comparison principle.
On an ODE-PDE coupling model of the mitochondrial swelling process
Sabine Eisenhofer , Messoud A. Efendiev , Mitsuharu Ôtani , Sabine Schulz and  Hans Zischka
2015, 20(4): 1031-1057 doi: 10.3934/dcdsb.2015.20.1031 +[Abstract](29) +[PDF](3497.7KB)
Mitochondrial swelling has huge impact to multicellular organisms since it triggers apoptosis, the programmed cell death. In this paper we present a new mathematical model of this phenomenon. As a novelty it includes spatial effects, which are of great importance for the in vivo process. Our model considers three mitochondrial subpopulations varying in the degree of swelling. The evolution of these groups is dependent on the present calcium concentration and is described by a system of ODEs, whereas the calcium propagation is modeled by a reaction-diffusion equation taking into account spatial effects. We analyze the derived model with respect to existence and long-time behavior of solutions and obtain a complete mathematical classification of the swelling process.
Migration and orientation of endothelial cells on micropatterned polymers: A simple model based on classical mechanics
Julie Joie , Yifeng Lei , Marie-Christine Durrieu , Thierry Colin , Clair Poignard and  Olivier Saut
2015, 20(4): 1059-1076 doi: 10.3934/dcdsb.2015.20.1059 +[Abstract](32) +[PDF](2291.3KB)
Understanding the endothelial cell migration on micropatterned polymers, as well as the cell orientation is a critical issue in tissue engineering, since it is the preliminary step towards cell polarization and that possibly leads to the blood vessel formation. In this paper, we derive a simple agent-based model to describe the migration and the orientation of endothelial cells seeded on bioactive micropatterned polymers. The aim of the modeling is to provide a simple model that corroborates quantitatively the experiments, without considering the complex phenomena inherent to cell migration. Our model is obtained thanks to a classical mechanics approach based on experimental observations. Even though its simplicity, it provides numerical results that are quantitatively in accordance with the experimental data, and thus our approach can be seen as a preliminary way towards a simple modeling of cell migration.
A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced
Badal Joshi
2015, 20(4): 1077-1105 doi: 10.3934/dcdsb.2015.20.1077 +[Abstract](39) +[PDF](571.1KB)
Certain chemical reaction networks (CRNs) when modeled as a deterministic dynamical system taken with mass-action kinetics have the property of reaction network detailed balance (RNDB) which is achieved by imposing network-related constraints on the reaction rate constants. Markov chains (whether arising as models of CRNs or otherwise) have their own notion of detailed balance, imposed by the network structure of the graph of the transition matrix of the Markov chain. When considering Markov chains arising from chemical reaction networks with mass-action kinetics, we will refer to this property as Markov chain detailed balance (MCDB). Finally, we refer to the stochastic analog of RNDB as Whittle stochastic detailed balance (WSDB). It is known that RNDB and WSDB are equivalent. We prove that WSDB and MCDB are also intimately related but are not equivalent. While RNDB implies MCDB, the converse is not true. The conditions on rate constants that result in networks with MCDB but without RNDB are stringent, and thus examples of this phenomenon are rare, a notable exception is a network whose Markov chain is a birth and death process. We give a new algorithm to find conditions on the rate constants that are required for MCDB.
Codimension 3 B-T bifurcations in an epidemic model with a nonlinear incidence
Chengzhi Li , Jianquan Li and  Zhien Ma
2015, 20(4): 1107-1116 doi: 10.3934/dcdsb.2015.20.1107 +[Abstract](45) +[PDF](175.8KB)
It was shown in [11] that in an epidemic model with a nonlinear incidence and two compartments some complex dynamics can appear, such as the backward bifurcation, codimension 1 Hopf bifurcation and codimension 2 Bogdanov-Takens bifurcation. In this paper we prove that for the same model the codimension of Bogdanov-Takens bifurcation can be 3 and is at most 3. Hence, more complex new phenomena, such as codimension 2 Hopf bifurcation, codimension 2 homoclinic bifurcation and semi-stable limit cycle bifurcation, exhibit. Especially, the system can have and at most have 2 limit cycles near the positive singularity.
Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response
Haiyin Li and  Yasuhiro Takeuchi
2015, 20(4): 1117-1134 doi: 10.3934/dcdsb.2015.20.1117 +[Abstract](66) +[PDF](2234.3KB)
We investigate the dynamics of a non-autonomous and density dependent predator-prey system with Beddington-DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered. First, we derive a sufficient condition of permanence by comparison theorem, at the same time we propose a weaker condition ensuring some positive bounded set to be positive invariant. Next, we obtain two existence conditions for positive periodic solution by Brouwer fixed-point theorem and by continuation theorem, where the second condition is weaker than the first and gives the existence range of periodic solution. Further we show the global attractivity of the bounded positive solution by constructing Lyapunov function. Similarly, we have sufficient condition of global attractivity of boundary periodic solution.
Global dynamics and traveling wave solutions of two predators-one prey models
Jian-Jhong Lin , Weiming Wang , Caidi Zhao and  Ting-Hui Yang
2015, 20(4): 1135-1154 doi: 10.3934/dcdsb.2015.20.1135 +[Abstract](68) +[PDF](568.9KB)
In this work, we consider an ecological system of three species with two predators-one prey type without or with diffusion. For the system without diffusion, i.e. a system of three ODEs, we clarify global dynamics of all equilibria and find an exact condition to guarantee the existence and global asymptotic stability of the positive equilibrium. However, when the corresponding condition does not hold, the prey becomes extinct due to the over exploitation. On the other hand, for the system with diffusion, using the cross iteration method we find the minimum speed $c_*$. The existence of traveling wave front connecting the trivial solution and the coexistence state with some sufficient conditions is verified if the wave speed is large than $c_*$ and we also prove the nonexistence of such solutions if the wave speed is less than $c_*$. Finally, numerical simulations of system without or with diffusion are implemented and biological meanings are discussed.
Quasi-periodic motions in a special class of dynamical equations with dissipative effects: A pair of detection methods
Ugo Locatelli and  Letizia Stefanelli
2015, 20(4): 1155-1187 doi: 10.3934/dcdsb.2015.20.1155 +[Abstract](32) +[PDF](2116.1KB)
We consider a particular class of equations of motion, generalizing to $n$ degrees of freedom the ``dissipative spin--orbit problem'',   commonly studied in Celestial Mechanics. Those equations are formulated in a pseudo-Hamiltonian framework with action-angle coordinates; they contain a quasi-integrable conservative part and friction terms, assumed to be linear and isotropic with respect to the action variables. In such a context, we transfer two methods determining quasi-periodic solutions, which were originally designed to analyze purely Hamiltonian quasi-integrable problems.
    First, we show how the frequency map analysis can be adapted to this kind of dissipative models. Our approach is based on a key remark: the method can work as usual, by studying the behavior of the angular velocities of the motions as a function of the so called ``external frequencies'', instead of the actions.
    Moreover, we explicitly implement the Kolmogorov's normalization algorithm for the dissipative systems considered here. In a previous article, we proved a theoretical result: such a constructing procedure is convergent under the hypotheses usually assumed in KAM theory. In the present work, we show that it can be translated to a code making algebraic manipulations on a computer, so to calculate effectively quasi-periodic solutions on invariant tori (and the attracting dynamics in their neighborhoods).
    Both the methods are carefully tested, by checking that their predictions are in agreement, in the case of the so called ``dissipative forced pendulum''. Furthermore, the results obtained by applying our adaptation of the frequency analysis method to the dissipative standard map are compared with some existing ones in the literature.
A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations
Zhendong Luo
2015, 20(4): 1189-1212 doi: 10.3934/dcdsb.2015.20.1189 +[Abstract](35) +[PDF](836.9KB)
In this article, we employ proper orthogonal decomposition (POD) technique to establish a POD-based reduced-order stabilized mixed finite volume element (SMFVE) extrapolation algorithm based on two local Gaussian integrals, parameter-free, and Crank-Nicolson (CN) method with fewer degrees of freedom for the non-stationary Navier-Stokes equations. The error estimates between the POD-based reduced-order SMFVE solutions and the classical SMFVE solutions and the implementation for the POD-based reduced-order SMFVE extrapolation algorithm are provided. A numerical example is used to illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the POD-based reduced-order SMFVE extrapolation algorithm is feasible and efficient for finding numerical solutions for the non-stationary Navier-Stokes equations.
Pullback attractors for a class of nonlinear lattices with delays
Yejuan Wang and  Kuang Bai
2015, 20(4): 1213-1230 doi: 10.3934/dcdsb.2015.20.1213 +[Abstract](35) +[PDF](401.7KB)
We consider a class of nonlinear delay lattices $$ \ddot{u}_i(t)+(-1)^p\triangle^pu_i(t)+\lambda u_i(t)+\dot{u}_i(t)=h_i(u_i(t-\rho(t)))+f_i(t),~~~i \in \mathbb{Z}, $$ where $\lambda$ is a real positive constant, $p$ is any positive integer and $\triangle$ is the discrete one-dimensional Laplace operator. Under suitable conditions on $h$ and $f$ we prove the existence of pullback attractors for the multi-valued process associated with the system for which the uniqueness of solutions need not hold.
Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity
Qi Wang
2015, 20(4): 1231-1250 doi: 10.3934/dcdsb.2015.20.1231 +[Abstract](39) +[PDF](608.6KB)
In this paper, we study the nonconstant positive steady states of a Keller-Segel chemotaxis system over a bounded domain $\Omega\subset \mathbb{R}^N$, $N\geq 1$. The sensitivity function is chosen to be $\phi(v)=\ln (v+c)$ where $c$ is a positive constant. For the chemical diffusion rate being small, we construct positive solutions with a boundary spike supported on a platform. Moreover, this spike approaches the most curved part of the boundary of the domain as the chemical diffusion rate shrinks to zero. We also conduct extensive numerical simulations to illustrate the formation of stable boundary and interior spikes of the system. These spiky solutions can be used to model the self--organized cell aggregation phenomenon in chemotaxis.
Navier--Stokes equations on a rapidly rotating sphere
D. Wirosoetisno
2015, 20(4): 1251-1259 doi: 10.3934/dcdsb.2015.20.1251 +[Abstract](23) +[PDF](366.4KB)
We extend our earlier $\beta$-plane results [al-Jaboori and Wirosoetisno, 2011, DCDS-B 16:687--701] to a rotating sphere. Specifically, we show that the solution of the Navier--Stokes equations on a sphere rotating with angular velocity $1/\epsilon$ becomes zonal in the long time limit, in the sense that the non-zonal component of the energy becomes bounded by $\epsilon M$. Central to our proof is controlling the behaviour of the nonlinear term near resonances. We also show that the global attractor reduces to a single stable steady state when the rotation is fast enough.
New results of the ultimate bound on the trajectories of the family of the Lorenz systems
Fuchen Zhang , Chunlai Mu , Shouming Zhou and  Pan Zheng
2015, 20(4): 1261-1276 doi: 10.3934/dcdsb.2015.20.1261 +[Abstract](47) +[PDF](595.6KB)
In this paper, the global exponential attractive sets of a class of continuous-time dynamical systems defined by $\dot x = f\left( x \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {R^3},$ are studied. The elements of main diagonal of matrix $A$ are both negative numbers and zero, where matrix $A$ is the Jacobian matrix $\frac{{df}}{{dx}}$ of a continuous-time dynamical system defined by $\dot x = f\left( x \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {R^3},$ evaluated at the origin ${x_0} = \left( {0,0,0} \right).$ The former equations [1-6] that we are searching for a global bounded region have a common characteristic: The elements of main diagonal of matrix $A$ are all negative, where matrix $A$ is the Jacobian matrix $\frac{{df}}{{dx}}$ of a continuous-time dynamical system defined by $\dot x = f\left( x \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {R^n},$ evaluated at the origin ${x_0} = {\left( {0,0, \cdots ,0} \right)_{1 \times n}}.$ For the reason that the elements of main diagonal of matrix $A$ are both negative numbers and zero for this class of dynamical systems , the method for constructing the Lyapunov functions that applied to the former dynamical systems does not work for this class of dynamical systems. We overcome this difficulty by adding a cross term $xy$ to the Lyapunov functions of this class of dynamical systems and get a perfect result through many integral inequalities and the generalized Lyapunov functions.
The threshold of a stochastic SIRS epidemic model in a population with varying size
Yanan Zhao , Daqing Jiang , Xuerong Mao and  Alison Gray
2015, 20(4): 1277-1295 doi: 10.3934/dcdsb.2015.20.1277 +[Abstract](48) +[PDF](547.6KB)
In this paper, a stochastic susceptible-infected-removed-susceptible (SIRS) epidemic model in a population with varying size is discussed. A new threshold $\tilde{R}_0$ is identified which determines the outcome of the disease. When the noise is small, if $\tilde{R}_0<1$, the infected proportion of the population disappears, so the disease dies out, whereas if $\tilde{R}_0>1$, the infected proportion persists in the mean and we derive that the disease is endemic. Furthermore, when ${R}_0 > 1$ and subject to a condition on some of the model parameters, we show that the solution of the stochastic model oscillates around the endemic equilibrium of the corresponding deterministic system with threshold ${R}_0$, and the intensity of fluctuation is proportional to that of the white noise. On the other hand, when the noise is large, we find that a large noise intensity has the effect of suppressing the epidemic, so that it dies out. These results are illustrated by computer simulations.

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