# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

June 2014 , Volume 19 , Issue 4

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2014, 19(4): 867-881 doi: 10.3934/dcdsb.2014.19.867 +[Abstract](2120) +[PDF](905.3KB)
Abstract:
RS feedback models have been successful in explaining the observed phenomenon of clustering in autonomous oscillation in yeast, but current models do not include the biological reality of dynamical delay and do not have the related property of quorum sensing. Here an RS type ODE model for cell cycle feedback, including an explicit term modeling a chemical feedback mediating agent, is analyzed. New dynamics include population dependent effects: subcritical pitchfork bifurcations, and quorum sensing occur. The model suggests new experimental directions in autonomous oscillation in yeast.
2014, 19(4): 883-959 doi: 10.3934/dcdsb.2014.19.883 +[Abstract](2850) +[PDF](1241.3KB)
Abstract:
This survey focuses on the most important aspects of the mathematical theory of population genetic models of selection and migration between discrete niches. Such models are most appropriate if the dispersal distance is short compared to the scale at which the environment changes, or if the habitat is fragmented. The general goal of such models is to study the influence of population subdivision and gene flow among subpopulations on the amount and pattern of genetic variation maintained. Only deterministic models are treated. Because space is discrete, they are formulated in terms of systems of nonlinear difference or differential equations. A central topic is the exploration of the equilibrium and stability structure under various assumptions on the patterns of selection and migration. Another important, closely related topic concerns conditions (necessary or sufficient) for fully polymorphic (internal) equilibria. First, the theory of one-locus models with two or multiple alleles is laid out. Then, mostly very recent, developments about multilocus models are presented. Finally, as an application, analysis and results of an explicit two-locus model emerging from speciation theory are highlighted.
2014, 19(4): 961-961 doi: 10.3934/dcdsb.2014.19.961 +[Abstract](2125) +[PDF](42.6KB)
Abstract:
This article was accidentally posted online but only to be discovered that the same article had been published (see [1]) in the previous issue of the same journal. Thus this publication is retracted. The Editorial Office offers apologies for the confusion and inconvenience it might have caused.
2014, 19(4): 979-998 doi: 10.3934/dcdsb.2014.19.979 +[Abstract](2576) +[PDF](426.9KB)
Abstract:
We consider sensor array imaging for simultaneous noise blended sources. We study a migration imaging functional and we analyze its sensitivity to singular perturbations of the speed of propagation of the medium. We consider two kinds of random sources: randomly delayed pulses and stationary random processes, and three possible kinds of perturbations. Using high frequency analysis we prove the statistical stability (with respect to the realization of the noise blending) of the scheme and obtain quantitative results on the image contrast provided by the imaging functional, which strongly depends on the type of perturbations.
2014, 19(4): 999-1025 doi: 10.3934/dcdsb.2014.19.999 +[Abstract](3234) +[PDF](1244.7KB)
Abstract:
In this paper, we consider a class of epidemic models described by five nonlinear ordinary differential equations. The population is divided into susceptible, vaccinated, exposed, infectious, and recovered subclasses. One main feature of this kind of models is that treatment and vaccination are introduced to control and prevent infectious diseases. The existence and local stability of the endemic equilibria are studied. The occurrence of backward bifurcation is established by using center manifold theory. Moveover, global dynamics are studied by applying the geometric approach. We would like to mention that in the case of bistability, global results are difficult to obtain since there is no compact absorbing set. It is the first time that higher (greater than or equal to four) dimensional systems are discussed. We give sufficient conditions in terms of the system parameters by extending the method in Arino et al. [2]. Numerical simulations are also provided to support our theoretical results. By carrying out sensitivity analysis of the basic reproduction number in terms of some parameters, some effective measures to control infectious diseases are analyzed.
2014, 19(4): 1027-1045 doi: 10.3934/dcdsb.2014.19.1027 +[Abstract](2653) +[PDF](483.8KB)
Abstract:
A damped stochastic beam equation driven by a Non-Gaussian Lévy process is studied. Under appropriate conditions, the existence theorem for a unique global weak solution is given. Moreover, we also show the existence of a unique invariant measure associated with the transition semigroup under mild conditions.
2014, 19(4): 1047-1085 doi: 10.3934/dcdsb.2014.19.1047 +[Abstract](2629) +[PDF](692.5KB)
Abstract:
We study in this article the stochastic Zakharov-Kuznetsov equation driven by a multiplicative noise. We establish, in space dimensions two and three the global existence of martingale solutions, and in space dimension two the global pathwise uniqueness and the existence of pathwise solutions. New methods are employed to deal with a special type of boundary conditions and to verify the pathwise uniqueness of martingale solutions with a lack of regularity, where both difficulties arise due to the partly hyperbolic feature of the model.
2014, 19(4): 1087-1103 doi: 10.3934/dcdsb.2014.19.1087 +[Abstract](2527) +[PDF](465.5KB)
Abstract:
A virus dynamics model for HIV or HBV is studied, which incorporates saturation effects of immune responses and an intracellular time delay. With the aid of persistence theory and Liapunov method, it is shown that the global stability of the model is totally determined by the reproductive numbers for viral infection, for CTL immune response, for antibody immune response, for antibody invasion and for CTL immune invasion. The results preclude the complicated behaviors such as the backward bifurcations and Hopf bifurcations which may be induced by saturation factors and a time delay.
2014, 19(4): 1105-1118 doi: 10.3934/dcdsb.2014.19.1105 +[Abstract](2671) +[PDF](391.3KB)
Abstract:
In this paper, using an approach of Lyapunov functional, we establish the complete global stability of a multi-group SIS epidemic model in which the effect of population migration among different regions is considered. We prove the global asymptotic stability of the disease-free equilibrium of the model for $R_0\leq 1$, and that of an endemic equilibrium for $R_0>1$. Here $R_0$ denotes the well-known basic reproduction number defined by the spectral radius of an irreducible nonnegative matrix called the next generation matrix. We emphasize that the graph-theoretic approach, which is typically used for multi-group epidemic models, is not needed in our proof.
2014, 19(4): 1119-1128 doi: 10.3934/dcdsb.2014.19.1119 +[Abstract](2680) +[PDF](366.3KB)
Abstract:
We justify some characterizations of the ground states of spin-1 Bose-Einstein condensates exhibited from numerical simulations. For ferromagnetic systems, we show the validity of the single-mode approximation (SMA). For an antiferromagnetic system with nonzero magnetization, we prove the vanishing of the $m_F=0$ component. In the end of the paper some remaining degenerate situations are also discussed. The proofs of the main results are all based on a simple observation, that a redistribution of masses among different components will reduce the kinetic energy.
2014, 19(4): 1129-1136 doi: 10.3934/dcdsb.2014.19.1129 +[Abstract](2335) +[PDF](328.0KB)
Abstract:
We provide sufficient conditions for the existence of limit cycles for the Floquet differential equations $\dot {\bf x}(t) = A{\bf x}(t)+ε(B(t){\bf x}(t)+b(t))$, where ${\bf x}(t)$ and $b(t)$ are column vectors of length $n$, $A$ and $B(t)$ are $n\times n$ matrices, the components of $b(t)$ and $B(t)$ are $T$--periodic functions, the differential equation $\dot {\bf x}(t)= A{\bf x}(t)$ has a plane filled with $T$--periodic orbits, and $ε$ is a small parameter. The proof of this result is based on averaging theory but only uses linear algebra.
2014, 19(4): 1137-1154 doi: 10.3934/dcdsb.2014.19.1137 +[Abstract](2046) +[PDF](1354.0KB)
Abstract:
We consider a finite discrete nonlinear Schrödinger equation with localized forcing, damping, and nonautonomous perturbations. In the autonomous case these systems are shown numerically to have multiple attracting spatially localized solutions. In the nonautonomous case we study analytically some properties of the pullback attractor of the system, assuming that the origin of the corresponding autonomous system is hyberbolic. We also see numerically the persistence of multiple localized attracting states under different types of nonautonomous perturbations.
2014, 19(4): 1155-1170 doi: 10.3934/dcdsb.2014.19.1155 +[Abstract](3094) +[PDF](453.0KB)
Abstract:
Much recent work has focused on persistence for epidemic models with periodic coefficients. But the case where the infected compartments satisfy a delay differential equation or a partial differential equation does not seem to have been considered so far. The purpose of this paper is to provide a framework for proving persistence in such a case. Some examples are presented, such as a periodic SIR model structured by time since infection and a periodic SIS delay model.
2014, 19(4): 1171-1195 doi: 10.3934/dcdsb.2014.19.1171 +[Abstract](2152) +[PDF](460.1KB)
Abstract:
In this paper, we consider a time-delayed and nonlocal population model with migration and relax the monotone assumption for the birth function. We study the global dynamics of the model system when the spatial domain is bounded. If the spatial domain is unbounded, we investigate the spreading speed $c^*$, the non-existence of traveling wave solutions with speed $c\in(0,c^*)$, the existence of traveling wave solutions with $c\geq c^*$, and the uniqueness of traveling wave solutions with $c>c^*$. It is shown that the spreading speed coincides with the minimal wave speed of traveling waves.
2014, 19(4): 1197-1212 doi: 10.3934/dcdsb.2014.19.1197 +[Abstract](3393) +[PDF](461.5KB)
Abstract:
Stochastic averaging for a class of stochastic differential equations (SDEs) with fractional Brownian motion, of the Hurst parameter $H$ in the interval $(\frac{1}{2},1)$, is investigated. An averaged SDE for the original SDE is proposed, and their solutions are quantitatively compared. It is shown that the solution of the averaged SDE converges to that of the original SDE in the sense of mean square and also in probability. It is further demonstrated that a similar averaging principle holds for SDEs under stochastic integral of pathwise backward and forward types. Two examples are presented and numerical simulations are carried out to illustrate the averaging principle.
2014, 19(4): 1213-1226 doi: 10.3934/dcdsb.2014.19.1213 +[Abstract](2241) +[PDF](437.5KB)
Abstract:
This paper is devoted to the existence of pullback attractors for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation. We first prove the existence of pullback absorbing sets in $H$ and $V$ for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with (1)-(8), and then we prove the existence of a pullback attractor in $H$ by the Sobolev compactness embedding theorem. Finally, we obtain the existence of a pullback attractor in $V$ for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with (1)-(8) by verifying the pullback $\mathcal{D}$ condition $(PDC)$.

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