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## Discrete & Continuous Dynamical Systems - B

November 2015 , Volume 20 , Issue 9

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2015, 20(9): 2765-2791
doi: 10.3934/dcdsb.2015.20.2765

*+*[Abstract](2629)*+*[PDF](1231.2KB)**Abstract:**

In this paper, we investigate a mathematical model of hematopoietic stem cell dynamics. We take two cell populations into account, quiescent and proliferating one, and we note the difference between dividing cells that enter directly to the quiescent phase and dividing cells that return to the proliferating phase to divide again. The resulting mathematical model is a system of two age-structured partial differential equations. By integrating this system over age and using the characteristics method, we reduce it to a delay differential-difference system, and we investigate the existence and stability of the steady states. We give sufficient conditions for boundedness and unboundedness properties for the solutions of this system. By constructing a Lyapunov function, the trivial steady state, describing cell's dying out, is proven to be globally asymptotically stable when it is the only equilibrium. The stability analysis of the unique positive steady state, the most biologically meaningful one, and the existence of a Hopf bifurcation allow the determination of a stability area, which is related to a delay-dependent characteristic equation. Numerical simulations illustrate our results on the asymptotic behavior of the steady states and show very rich dynamics of this model. This study may be helpful in understanding the uncontrolled proliferation of blood cells in some hematological disorders.

2015, 20(9): 2793-2817
doi: 10.3934/dcdsb.2015.20.2793

*+*[Abstract](2000)*+*[PDF](998.4KB)**Abstract:**

The aim of this paper is to perform a theoretical and numerical study on the dynamics of vortices in Bose-Einstein condensation in the case where the trapping potential varies randomly in time. We take a deterministic vortex solution as an initial condition for the stochastically fluctuated Gross-Pitaevskii equation, and we observe the influence of the stochastic perturbation on the evolution. We theoretically prove that up to times of the order of $\epsilon^{-2}$, the solution having the same symmetry properties as the vortex decomposes into the sum of a randomly modulated vortex solution and a small remainder, and we derive the equations for the modulation parameter. In addition, we show that the first order of the remainder, as $\epsilon$ goes to zero, converges to a Gaussian process. Finally, some numerical simulations on the dynamics of the vortex solution in the presence of noise are presented.

2015, 20(9): 2819-2858
doi: 10.3934/dcdsb.2015.20.2819

*+*[Abstract](1687)*+*[PDF](541.1KB)**Abstract:**

In this paper we study some mathematical models describing evolution of population density and spread of epidemics in population systems in which spatial movement of individuals depends only on the departure and arrival locations and does not have apparent connection with the population density. We call such models as population migration models and migration epidemics models, respectively. We first apply the theories of positive operators and positive semigroups to make systematic investigation to asymptotic behavior of solutions of the population migration models as time goes to infinity, and next use such results to study asymptotic behavior of solutions of the migration epidemics models as time goes to infinity. Some interesting properties of solutions of these models are obtained.

2015, 20(9): 2859-2884
doi: 10.3934/dcdsb.2015.20.2859

*+*[Abstract](2022)*+*[PDF](7116.9KB)**Abstract:**

In this paper we discuss the stochastic differential equation (SDE) susceptible-infected-susceptible (SIS) epidemic model with demographic stoch-asticity. First we prove that the SDE has a unique nonnegative solution which is bounded above. Then we give conditions needed for the solution to become extinct. Next we use the Feller test to calculate the respective probabilities of the solution first hitting zero or the upper limit. We confirm our theoretical results with numerical simulations and then give simulations with realistic parameter values for two example diseases: gonorrhea and pneumococcus.

2015, 20(9): 2885-2931
doi: 10.3934/dcdsb.2015.20.2885

*+*[Abstract](1929)*+*[PDF](619.6KB)**Abstract:**

In this paper, we establish a priori estimates for three-dimensional compressible Euler equations with the moving physical vacuum boundary, the $\gamma$-gas law equation of state for $\gamma=2$ and the general initial density $\rho_0 \in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which plays a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$.

2015, 20(9): 2933-2947
doi: 10.3934/dcdsb.2015.20.2933

*+*[Abstract](2407)*+*[PDF](2557.9KB)**Abstract:**

In this paper, we study two stochastic SIS epidemic models: the first one with a constant population size, and the second one with a death factor. We analyze persistence and extinction behaviors for these models. The persistence time depends on the initial population size and satisfies a stationary backward Kolmogorov differential equation, which is a linear second-order partial differential equation with variable degenerate coefficients. We solve this equation numerically using a classical finite element method. We give computational evidence that the importance of understanding the dynamics of both the deterministic and the stochastic epidemic models is due to the numerical approximations to the mean persistence time. This can give more information about the model and may perhaps explain strange behaviors, such as the differences between the deterministic model and the stochastic one for long times.

2015, 20(9): 2949-2965
doi: 10.3934/dcdsb.2015.20.2949

*+*[Abstract](2040)*+*[PDF](557.4KB)**Abstract:**

We consider a system of operator equations involving play and Prandtl-Ishlinskii hysteresis operators. This system generalizes the classical mechanical models of elastoplasticity, friction and fatigue by introducing coupling between the operators. We show that under quite general assumptions the coupled system is equivalent to one effective Prandtl-Ishlinskii operator or, more precisely, to a discontinuous extension of the Prandtl-Ishlinskii operator based on the Kurzweil integral of the derivative of the state function. This effective operator is described constructively in terms of the parameters of the coupled system. Our result is based on a substitution formula which we prove for the Kurzweil integral of regulated functions integrated with respect to functions of bounded variation. This formula allows us to prove the composition rule for the generalized (discontinuous) Prandtl-Ishlinskii operators. The composition rule, which underpins the analysis of the coupled model, then establishes that a composition of generalized Prandtl-Ishlinskii operators is also a generalized Prandtl-Ishlinskii operator provided that a monotonicity condition is satisfied.

2015, 20(9): 2967-2992
doi: 10.3934/dcdsb.2015.20.2967

*+*[Abstract](2149)*+*[PDF](532.1KB)**Abstract:**

In this paper we consider an optimal control problem (OCP) for the coupled system of a nonlinear monotone Dirichlet problem with matrix-valued $L^\infty(\Omega;\mathbb{R}^{N\times N} )$-controls in coefficients and a nonlinear equation of Hammerstein type. Since problems of this type have no solutions in general, we make a special assumption on the coefficients of the state equation and introduce the class of so-called solenoidal admissible controls. Using the direct method in calculus of variations, we prove the existence of an optimal control. We also study the stability of the optimal control problem with respect to the domain perturbation. In particular, we derive the sufficient conditions of the Mosco-stability for the given class of OCPs.

2015, 20(9): 2993-3011
doi: 10.3934/dcdsb.2015.20.2993

*+*[Abstract](1843)*+*[PDF](455.8KB)**Abstract:**

In this paper we investigate the existence of invariant stable and center-stable manifolds for solutions to partial neutral functional differential equations of the form $$\begin{cases}\frac{\partial}{\partial t}Fu_t = B(t)Fu_t + \Phi(t,u_t),\quad t\in (0,\infty),\cr u_0 = \phi\in \mathcal{C}: = C([-r, 0], X) \end{cases}$$ when the family of linear partial differential operators $(B(t))_{t\ge 0}$ generates the evolution family $(U(t,s))_{t\ge s\ge 0}$ (on Banach space $X$) having an exponential dichotomy or trichotomy on the half-line and the nonlinear delay operator $\Phi$ satisfies the $\varphi$-Lipschitz condition, i.e., $\| \Phi(t,\phi) -\Phi(t,\psi)\| \le \varphi(t)\|\phi -\psi\|_{\mathcal{C}}$ for $\phi, \psi\in \mathcal{C}$, where $\varphi(t)$ belongs to some admissible function space on the half-line.

2015, 20(9): 3013-3027
doi: 10.3934/dcdsb.2015.20.3013

*+*[Abstract](1728)*+*[PDF](2636.9KB)**Abstract:**

This paper is devoted to study the existence of solutions of hydrodynamic model for systems of self-propelled particles subject to alignment and volume exclusion interactions. On one hand, we prove the existence of solutions by using the modified Garlerkin method for quasi-linear parabolic simulations. On the other hand, we also perform simulations to compare theoretical and numerical results. The numerical results show that the numerical solutions exist for short time in some cases of coefficients.

2015, 20(9): 3029-3055
doi: 10.3934/dcdsb.2015.20.3029

*+*[Abstract](2142)*+*[PDF](561.4KB)**Abstract:**

We study the asymptotic behavior for a system consisting of a clamped flexible beam that carries a tip payload, which is attached to a nonlinear damper and a nonlinear spring at its end. Characterizing the $\omega$-limit sets of the trajectories, we give a sufficient condition under which the system is asymptotically stable. In the case when this condition is not satisfied, we show that the beam deflection approaches a non-decaying time-periodic solution.

2015, 20(9): 3057-3091
doi: 10.3934/dcdsb.2015.20.3057

*+*[Abstract](2964)*+*[PDF](689.8KB)**Abstract:**

In this paper, we investigate the global stability of a delayed multi-group SIRS epidemic model which includes not only nonlinear incidence rates but also rates of immunity loss and relapse of infection. The model analysis can be regarded as an extension to a multi-group epidemic analysis in [Muroya, Li and Kuniya, Complete global analysis of an SIRS epidemic model with graded cure rate and incomplete recovery rate,

*J. Math. Anal. Appl.*

**410**(2014) 719-732] is studied. Applying a Lyapunov functional approach, we prove that a disease-free equilibrium of the model, is globally asymptotically stable, if a threshold parameter $R_0 \leq 1$. For the global stability of an endemic equilibrium of the model, we establish a sufficient condition for small recovery rates $\delta_k \geq 0$, $k=1,2,\ldots,n$, if $R_0>1$. Further, by a monotone iterative approach, we obtain another sufficient condition for large $\delta_k$, $k=1,2,\ldots,n$. Both results generalize several known results obtained for, e.g., SIS, SIR and SIRS models in the recent literature. We also offer a new proof on permanence which is applicable to other multi-group epidemic models.

2015, 20(9): 3093-3114
doi: 10.3934/dcdsb.2015.20.3093

*+*[Abstract](2079)*+*[PDF](445.8KB)**Abstract:**

We present a systematic method to construct Lyapunov functionals of delay differential equation models of infectious diseases in vivo. For generality we construct Lyapunov functionals of models with infinitely distributed delay. We begin with simpler models without delay and construct Lyapunov functionals for the complex models progressively. We construct those functionals using our result obtained previously instead of constructing each functional independently. Additionally we discuss some problems that arise from the mathematical requirements caused by the infinitely distributed delay.

2015, 20(9): 3115-3129
doi: 10.3934/dcdsb.2015.20.3115

*+*[Abstract](2059)*+*[PDF](700.9KB)**Abstract:**

Time delay in insulin secretion, its absorption and action is a point of consideration in artificial pancreas as it may prove fatal in the extreme situation. The present mathematical model deals with two time delays out of which one occur in insulin secretion and second in its absorption and action. The model assess the change in glucose - insulin dynamics after the induction of different values of these time delays in their respective range. Also, simulation is performed over the model to quantify the amount of two time delays to avoid diabetic comma, which has not been explored much.

2015, 20(9): 3131-3163
doi: 10.3934/dcdsb.2015.20.3131

*+*[Abstract](1847)*+*[PDF](1110.9KB)**Abstract:**

This work concerns dynamics and bifurcations properties of a new class of continuous-defined one-dimensional maps: Tsoularis-Wallace's functions. This family of functions naturally incorporates a major focus of ecological research: the Allee effect. We provide a necessary condition for the occurrence of this phenomenon of extinction. To establish this result we introduce the notions of Allee's functions, Allee's effect region and Allee's bifurcation curve. Another central point of our investigation is the study of bifurcation structures for this class of functions, in a three-dimensional parameter space. We verified that under some sufficient conditions, Tsoularis-Wallace's functions have particular bifurcation structures: the big bang and the double big bang bifurcations of the so-called ``box-within-a-box'' type. The double big bang bifurcations are related to the existence of flip codimension--2 points. Moreover, it is verified that these bifurcation cascades converge to different big bang bifurcation curves, where for the corresponding parameter values are associated distinct kinds of boxes. This work contributes to clarify the big bang bifurcation analysis for continuous maps and understand their relationship with explosion birth and extinction phenomena.

2015, 20(9): 3165-3183
doi: 10.3934/dcdsb.2015.20.3165

*+*[Abstract](2477)*+*[PDF](520.8KB)**Abstract:**

We consider a model for two species interacting through chemotaxis in such a way that each species produces a signal which directs the respective motion of the other. Specifically, we shall be concerned with nonnegative solutions of the Neumann problem, posed in bounded domains $\Omega\subset \mathbb{R}^n$ with smooth boundary, for the system $$\begin{cases} u_t= \Delta u - \chi \nabla \cdot (u\nabla v), & x\in \Omega, \, t>0, \\ 0=\Delta v-v+w, & x\in \Omega, \, t>0, \qquad (\star)\\ w_t= \Delta w - \xi \nabla \cdot (w\nabla z), & x\in \Omega, \, t>0, \\ 0=\Delta z-z+u, & x\in \Omega, \, t>0, \end{cases}$$

with parameters $\chi \in \{\pm 1\}$ and $\xi\in \{\pm 1\}$, thus allowing the interaction of either attraction-repulsion, or attraction-attraction, or repulsion-repulsion type.

It is shown that

$\bullet$ in the attraction-repulsion case $\chi=1$ and $\xi=-1$, if $n\le 3$ then for any nonnegative initial data $u_0\in C^0(\bar{\Omega})$ and $ w_0\in C^0 (\bar{\Omega})$, there exists a unique global classical solution which is bounded;

$\bullet$ in the doubly repulsive case when $\chi=\xi=-1$, the same holds true;

$\bullet$ in the attraction-attraction case $\chi=\xi=1$,

$-$ if either $n=2$ and $\int_\Omega u_0 + \int_\Omega w_0$ lies below some threshold, or $n\ge 3$ and $\|u_0\|_{L^\infty(\Omega)}$ and $\|w_0\|_{L^\infty(\Omega)}$ are sufficiently small, then solutions exist globally and remain bounded, whereas

$-$ if either $n=2$ and $m$ is suitably large, or $n\ge 3$ and $m>0$ is arbitrary, then there exist smooth initial data $u_0$ and $w_0$ such that $\int_\Omega u_0 + \int_\Omega w_0=m$ and such that the corresponding solution blows up in finite time.

In particular, these results demonstrate that the circular chemotaxis mechanism underlying ($\star$) goes along with essentially the same destabilizing features as known for the classical Keller-Segel system in the doubly attractive case, but totally suppresses any blow-up phenomenon when only one, or both, taxis directions are repulsive.

2015, 20(9): 3185-3213
doi: 10.3934/dcdsb.2015.20.3185

*+*[Abstract](2104)*+*[PDF](590.4KB)**Abstract:**

In this paper, we give an asymptotic analysis of the phase field Allen-Cahn and Cahn-Hilliard models of free surfaces with surface tension. Unlike the traditional approach that approximates the solution by the so-called

*matched asymptotic expansion*involving outer expansion, inner expansion and matching, our new approach utilizes a uniform

*double asymptotic expansion*to expand the whole phase field function directly. Although the main result is not new, we would like to emphasize that we derive the result under a uniform double asymptotic expansion. Thus, in this paper the detailed structure of the phase field functions in the equilibrium state is obtained, and the consistency of the phase field models with the corresponding sharp interface models is discussed, including the free surface Allen-Cahn model, Cahn-Hilliard model, and the Allen-Cahn model with volume constraint. The explicit asymptotic expansion of the phase field function reveals rich details of its structures. Moreover, it nicely explains some unusual phenomena we observed in numerical experiments. The theory introduced in this paper can be applied to guide the future modeling and simulation of other moving boundary problems by phase field models.

2015, 20(9): 3215-3233
doi: 10.3934/dcdsb.2015.20.3215

*+*[Abstract](2092)*+*[PDF](476.5KB)**Abstract:**

In this paper, we formulate a viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. The model can describe the in vivo infection dynamics of many viruses such as HIV-I, HCV, and HBV, where the infected cells of eclipse stage can revert to the uninfected class. Under certain parameters range, we establish that the global stability of equilibria is completely determined by the basic reproduction number $\mathfrak{R}_0$, which give us a complete picture on their global dynamics.

2015, 20(9): 3235-3254
doi: 10.3934/dcdsb.2015.20.3235

*+*[Abstract](1910)*+*[PDF](475.9KB)**Abstract:**

This paper considers the chemotaxis-Stokes system $$\begin{cases} \displaystyle n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\cdot\nabla c), &(x,t)\in \Omega\times (0,T),\\ \displaystyle c_t+u\cdot\nabla c=\Delta c-nc, &(x,t)\in\Omega\times (0,T),\qquad(\star)\\ \displaystyle u_t=\Delta u+\nabla P+n\nabla\phi , &(x,t)\in\Omega\times (0,T),\\ \nabla\cdot u=0,&(x,t)\in\Omega\times (0,T). \end{cases}$$ under no-flux boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^3$ with smooth boundary. Here $S$ is a matrix-valued sensitivity satisfying $|S(x,n,c)|<\tilde{C}(1+n)^{-\alpha}$ with some $\tilde{C}>0$ and $\alpha>0$. Although $(\star)$ does not possess the natural gradient-like functional structure available when $S$ reduces to a scalar function, we can still establish a new energy type inequality. Based on this inequality we achieve a coupled estimate for arbitrarily high Lebesgue norms of $n$ and $\nabla c$. This helps us to finally obtain the existence of a global classical solution when $\alpha$ is bigger than $\frac{1}{6}$.

2015, 20(9): 3255-3266
doi: 10.3934/dcdsb.2015.20.3255

*+*[Abstract](1848)*+*[PDF](601.5KB)**Abstract:**

We shall obtain the parameter region that ensures the global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. The parameter region can be illustrated graphically and examples of such regions are presented. Our result partially answers an open problem proposed by Elaydi and Luís [3] and complements the very recent work by Balreira, Elaydi and Luís [1].

2015, 20(9): 3267-3299
doi: 10.3934/dcdsb.2015.20.3267

*+*[Abstract](1699)*+*[PDF](743.3KB)**Abstract:**

Viscoelastic fluids represent a major challenge both from an engineering and from a mathematical point of view. Recently, we have shown that viscoelasticity induces chaos in closed-loop thermosyphons. This induced behavior might interfere with the engineering choice of using a specific fluid. In this work we show that the addition of a solute to the fluid can, under some conditions, stabilize the system due to thermodiffusion (also known as the Soret effect). Unexpectedly, the role of viscoelasticity is opposite to the case of single-element fluids, where it (generically) induces chaos. Our results are derived by combining analytical results based on the projection of the dynamics on an inertial manifold as well as numerical simulations characterized by the calculation of Lyapunov exponents.

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