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Discrete & Continuous Dynamical Systems - B

2012 , Volume 17 , Issue 7

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A Neumann Boundary Value Problem in Two-Ion Electro-Diffusion with Unequal Valencies
Pablo Amster , Man Kam Kwong and  Colin Rogers
2012, 17(7): 2299-2311 doi: 10.3934/dcdsb.2012.17.2299 +[Abstract](27) +[PDF](368.2KB)
In prior work, a series of two-point boundary value problems have been investigated for a steady state two-ion electro-diffusion model system in which the sum of the valencies $\nu_+$ and $\nu_-$ is zero. In that case, reduction is obtained to the canonical Painlevé II equation for the scaled electric field. Here, a physically important Neumann boundary value problem in the generic case when $\nu_+ + \nu_-\neq 0$ is investigated. The problem is novel in that the model equation for the electric field involves yet to be determined boundary values of the solution. A reduction of the Neumann boundary value problem in terms of elliptic functions is obtained for privileged valency ratios. A topological index argument is used to establish the existence of a solution in the general case, under the assumption $\nu_+ + \nu_- \leq 0$.
From a PDE model to an ODE model of dynamics of synaptic depression
Adam Bobrowski and  Katarzyna Morawska
2012, 17(7): 2313-2327 doi: 10.3934/dcdsb.2012.17.2313 +[Abstract](35) +[PDF](389.8KB)
We provide a link between two recent models of dynamics of synaptic depression. To this end, we specify the missing transmission conditions in the PDE model of Bielecki and Kalita, and show that if diffusion is fast and communication between pools is slow, the PDE model is well approximated by the ODE model of Aristizabal and Glavinovič. From the mathematical point of view the ODE model is obtained as a singular perturbation of the PDE model with singularities both in the operator and in the boundary and transmission conditions. The result is put in the context of degenerate convergence of semigroups of operators, where a sequence of strongly continuous semigroups approaches a semigroup that is strongly continuous only on a subspace of the original Banach space. Biologically, our approach allows a new, natural interpretation of the ODE model’s parameters.
Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability
Zhi-Min Chen and  Philip A. Wilson
2012, 17(7): 2329-2341 doi: 10.3934/dcdsb.2012.17.2329 +[Abstract](29) +[PDF](208.8KB)
The Benjamin-Feir instability describes the instability of a uniform oscillatory wave train in an irrotational flow subject to small perturbation of wave number, amplitude and frequency. Their instability analysis is based on the perturbation around the second order Stokes wave which satisfies the dynamic and kinematic free-surface boundary conditions up to the second order. In the same irrotational flow and perturbation framework of the Benjamin-Feir analysis, the perturbation in the present paper is around a nonlinear oscillatory wave train which solves exactly the dynamic free-surface boundary condition and satisfies the kinematic free-surface boundary condition up to the third order. It is shown that the nonlinear oscillatory wave train is stable with respect to the perturbation when the irrotational flow involves small Rayleigh energy dissipation.
An immersed linear finite element method with interface flux capturing recovery
So-Hsiang Chou
2012, 17(7): 2343-2357 doi: 10.3934/dcdsb.2012.17.2343 +[Abstract](39) +[PDF](349.2KB)
A flux recovery technique is introduced for the computed solution of an immersed finite element method for one dimensional second-order elliptic problems. The recovery is by a cheap formula evaluation and is carried out over a single element at a time while ensuring the continuity of the flux across the interelement boundaries and the validity of the discrete conservation law at the element level. Optimal order rates are proved for both the primary variable and its flux. For piecewise constant coefficient problems our method can capture the flux at nodes and at the interface points exactly. Moreover, it has the property that errors in the flux are all the same at all nodes and interface points for general problems. We also show second order pressure error and first order flux error at the nodes. Numerical examples are provided to confirm the theory.
Mathematical analysis of a HIV model with quadratic logistic growth term
Xinyue Fan , Claude-Michel Brauner and  Linda Wittkop
2012, 17(7): 2359-2385 doi: 10.3934/dcdsb.2012.17.2359 +[Abstract](49) +[PDF](658.8KB)
We consider a model of disease dynamics in the modeling of Human Immunodeficiency Virus (HIV). The system consists of three ODEs for the concentrations of the target T cells, the infected cells and the virus particles. There are two main parameters, $N$, the total number of virions produced by one infected cell, and $r$, the logistic parameter which controls the growth rate. The equilibria corresponding to the infected state are asymptotically stable in a region $(\mathcal I)$, but unstable in a region $(\mathcal P)$. In the unstable region, the levels of the various cell types and virus particles oscillate, rather than converging to steady values. Hopf bifurcations occurring at the interfaces are fully investigated via several techniques including asymptotic analysis. The Hopf points are connected through a "snake" of periodic orbits [24]. Numerical results are presented.
Construction of a finite-time Lyapunov function by meshless collocation
Peter Giesl
2012, 17(7): 2387-2412 doi: 10.3934/dcdsb.2012.17.2387 +[Abstract](35) +[PDF](1315.3KB)
We consider a nonautonomous ordinary differential equation of the form $\dot{x}=f(t,x)$, $x\in \mathbb{R}^n$ over a finite-time interval $t\in [T_1,T_2]$. The basin of attraction of an attracting solution can be determined using a finite-time Lyapunov function.
    In this paper, such a finite-time Lyapunov function is constructed by Meshless Collocation, in particular Radial Basis Functions. Thereto, a finite-time Lyapunov function is characterised as the solution of a second-order linear partial differential equation with boundary values. This problem is approximately solved using Meshless Collocation, and it is shown that the approximate solution can be used to determine the basin of attraction. Error estimates are obtained and verified in examples.
Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations
Hongbin Guo and  Michael Yi Li
2012, 17(7): 2413-2430 doi: 10.3934/dcdsb.2012.17.2413 +[Abstract](29) +[PDF](588.3KB)
Population migration and immigration have greatly increased the spread and transmission of many infectious diseases at a regional, national and global scale. To investigate quantitatively and qualitatively the impact of migration and immigration on the transmission dynamics of infectious diseases, especially in heterogeneous host populations, we incorporate immigration/migration terms into all sub-population compartments, susceptible and infected, of two types of well-known heterogeneous epidemic models: multi-stage models and multi-group models for HIV/AIDS and other STDs. We show that, when migration or immigration into infected sub-population is present, the disease always becomes endemic in the population and tends to a unique asymptotically stable endemic equilibrium $P^*.$ The global stability of $P^*$ is established under general and biological meaningful conditions, and the proof utilizes a global Lyapunov function and the graph-theoretic techniques developed in Guo et al. (2008).
Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition
Jongmin Han and  Chun-Hsiung Hsia
2012, 17(7): 2431-2449 doi: 10.3934/dcdsb.2012.17.2431 +[Abstract](38) +[PDF](402.7KB)
In this article, we study the stability and dynamic bifurcation for the two dimensional Swift-Hohenberg equation with an odd periodic condition. It is shown that an attractor bifurcates from the trivial solution as the control parameter crosses the critical value. The bifurcated attractor consists of finite number of singular points and their connecting orbits. Using the center manifold theory, we verify the nondegeneracy and the stability of the singular points.
Stability conditions for a class of delay differential equations in single species population dynamics
Gang Huang , Yasuhiro Takeuchi and  Rinko Miyazaki
2012, 17(7): 2451-2464 doi: 10.3934/dcdsb.2012.17.2451 +[Abstract](55) +[PDF](453.3KB)
We consider a class of nonlinear delay differential equations,which describes single species population growth with stage structure. By constructing appropriate Lyapunov functionals, the global asymptotic stability criteria, which are independent of delay, are established. Much sharper stability conditions than known results are provided. Applications of the results to some population models show the effectiveness of the methods described in the paper.
Nonlinear conformation response in the finite channel: Existence of a unique solution for the dynamic PNP model
Joseph W. Jerome
2012, 17(7): 2465-2482 doi: 10.3934/dcdsb.2012.17.2465 +[Abstract](25) +[PDF](389.3KB)
The standard PNP model for ion transport in channels in cell membranes has been widely studied during the previous two decades; there is a substantial literature for both the dynamic and steady models. What is currently lacking is a generally accepted gating model, which is linked to the observed conformation changes on the protein molecule. In [SIAM J. Appl. Math. 61 (2000), no.3, 792–802], C.W. Gardner, the author, and R.S. Eisen- berg suggested a model for the net charge density in the infinite channel, which has connections to stochastic dynamical systems, and which predicted rectan- gular current pulses. The finite channel was analyzed by these authors in [J. Theoret. Biol. 219 (2002), no. 3, 291–299]. The finite channel cannot, in general, be analyzed by a traveling wave approach. In this paper, a rigorous study of the initial-boundary value problem is carried out for the deterministic version of the finite channel; an existence/uniqueness result, with a weak maximum principle, is derived on the space-time domain under assumptions on the inital and boundary data which confine the channel to certain states. Significant open problems remain and are discussed
Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion
Jin Li and  Jianhua Huang
2012, 17(7): 2483-2508 doi: 10.3934/dcdsb.2012.17.2483 +[Abstract](42) +[PDF](509.7KB)
A 2D Stochastic incompressible non-Newtonian fluid driven by fractional Brownian motion with Hurst index $H \in (\frac{1}{2},1)$ is studied. The Wiener-type stochastic integrals are introduced for infinite-dimensional fractional Brownian motion. Including the requirements of Nuclear and Hilbert-Schmidt operators, three kinds of condition, which ensure the existence and regularity of infinite-dimensional stochastic convolution for the corresponding additive linear stochastic equation, are summarized. Without the requirements of compact parameters, another condition is proposed for the existence and regularity of stochastic convolution. By any of the four kinds of condition, the existence and uniqueness of mild solution are obtained for the stochastic non-Newtonian fluid through a modified fixed point theorem in the selected intersection space. Existence of a random attractor is then obtained for the random dynamical system generated by non-Newtonian fluid.
On a reaction-diffusion model for sterile insect release method with release on the boundary
Xin Li and  Xingfu Zou
2012, 17(7): 2509-2522 doi: 10.3934/dcdsb.2012.17.2509 +[Abstract](37) +[PDF](254.4KB)
We consider a partial differential equation model that describes the sterile insect release method (SIRM) in a bounded 1-dimensional domain (interval). Unlike everywhere-releasing in the domain as considered in previous works [17] and [14] , we propose the mechanism of releasing on the boundary only. We show existence of the fertile-free steady state and prove its stability under some conditions. By using the upper-lower solution method, we also show that under some other conditions there may exist a coexistence steady state. Biological implications of our mathematical results are that the SIRM with releasing only on the boundary can successfully eradicate the fertile insects as long as the strength of the sterile releasing is reasonably large, while the method may also fail if the releasing is not sufficient.
Interface oscillations in reaction-diffusion systems above the Hopf bifurcation
Rebecca McKay , Theodore Kolokolnikov and  Paul Muir
2012, 17(7): 2523-2543 doi: 10.3934/dcdsb.2012.17.2523 +[Abstract](35) +[PDF](973.6KB)
We consider a reaction-diffusion system of the form \[ \left\{ \begin{array} \ u_{t}=\varepsilon^{2}u_{xx}+f(u,w)\\ \tau w_{t}=Dw_{xx}+g(u,w) \end{array} \right. \] with Neumann boundary conditions on a finite interval. Under certain generic conditions on the nonlinearities $f,g$ and in the singular limit $\varepsilon\ll1$ such a system may admit a steady state solution where $u$ has sharp interfaces. It is also known that such interfaces may be destabilized due to a Hopf bifurcation [Y. Nishiura and M. Mimura. SIAM J.Appl. Math., 49:481--514, 1989], as $\tau$ is increased beyond a certain threshold $\tau_{h}$. In this paper, we study what happens for $\tau>\tau _{h},$ or even $\tau\rightarrow\infty,$ for a solution that consists of either one or two interfaces. Under the additional assumption $D\gg1,$ using singular perturbation theory, we determine the existence of another threshold $\tau _{c}>\tau_{h}$ (where $\tau_{c}$ is allowed to be infinite) such that if $\tau_{h}<\tau<\tau_{c}$ then the system admits a solution consisting of periodically oscillating interfaces. On the other hand if $\tau>\tau_{c},$ the extent of the oscillation eventually exceeds the spatial domain size, even though very long transient dynamics can preceed this occurence. We make use of recently developed numerical software (that employs adaptive error control in space and time) to accurately compute an approximate solution. Excellent agreement with the analytical theory is observed.
Optimal harvesting and planting control in stochastic logistic population models
Hiroaki Morimoto
2012, 17(7): 2545-2559 doi: 10.3934/dcdsb.2012.17.2545 +[Abstract](33) +[PDF](358.2KB)
We consider the optimal harvesting and planting control problem to maximize the expected total net benefits in the stochastic logistic population model. The variational inequality associated with this problem is given by the degenerate form of elliptic type with quadratic coefficients. Using the viscosity solutions technique, we solve the corresponding penalty equation and show the existence of a solution to the variational inequality. The optimal harvesting and planting policy is characterized in terms of two thresholds for the variational inequality.
Kolmogorov's normal form for equations of motion with dissipative effects
Letizia Stefanelli and  Ugo Locatelli
2012, 17(7): 2561-2593 doi: 10.3934/dcdsb.2012.17.2561 +[Abstract](32) +[PDF](672.3KB)
We focus on the equations of motion related to the “dissipative spin–orbit model”, which is commonly studied in Celestial Mechanics. We consider them in the more general framework of a 2$n$–dimensional action–angle phase space. Since the friction terms are assumed to be linear and isotropic with respect to the action variables, the Kolmogorov’s normalization algorithm for quasi-integrable Hamiltonians can be easily adapted to the dissipative system considered here. This allows us to prove the existence of quasi-periodic invariant tori that are local attractors.
Stability of boundary layers for the inflow compressible Navier-Stokes equations
Jing Wang and  Lining Tong
2012, 17(7): 2595-2613 doi: 10.3934/dcdsb.2012.17.2595 +[Abstract](47) +[PDF](400.2KB)
In this paper, we consider the boundary layer stability of the one-dimensional isentropic compressible Navier-Stokes equations with an inflow boundary condition. We assume only one of the two characteristics to the corresponding Euler equations is negative up to some small time. We prove the existence of the boundary layers, then instead of using the skew symmetric matrix, we give a higher convergence rate of the approximate solution than the previous results by a standard energy method as long as the strength of the boundary layers is suitably small.
A vector-bias malaria model with incubation period and diffusion
Zhiting Xu and  Xiao-Qiang Zhao
2012, 17(7): 2615-2634 doi: 10.3934/dcdsb.2012.17.2615 +[Abstract](86) +[PDF](429.5KB)
This paper is devoted to the study of the global dynamics of a vector-bias malaria model with incubation period and diffusion. The global attractivity of the disease-free or endemic equilibrium is first proved for the spatially homogeneous system. Then the threshold dynamics is established for the spatially heterogeneous system in terms of the basic reproduction ratio. A set of sufficient conditions is further obtained for the global attractivity of the positive steady state.
Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions
Lu Yang , Meihua Yang and  Peter E. Kloeden
2012, 17(7): 2635-2651 doi: 10.3934/dcdsb.2012.17.2635 +[Abstract](52) +[PDF](459.1KB)
The existence of a unique minimal pullback attractor is established for the evolutionary process associated with a non-autonomous quasi-linear parabolic equations with a dynamical boundary condition in $L^{r_1}(\Omega)\times L^{r_1}(\Gamma)$ under that assumption that the external forcing term satisfies a weak integrability condition, where $r_1$ $>$ $2$ is determined by the order of the nonlinearity.

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