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Discrete & Continuous Dynamical Systems - B

August 2020 , Volume 25 , Issue 8

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On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime
Katharina Schratz and Xiaofei Zhao
2020, 25(8): 2841-2865 doi: 10.3934/dcdsb.2020043 +[Abstract](877) +[HTML](265) +[PDF](872.14KB)
Abstract:

This work concerns the time averaging techniques for the nonlinear Klein-Gordon (KG) equation in the nonrelativistic limit regime which have recently gained a lot of attention in numerical analysis. This is due to the fact that the solution becomes highly-oscillatory in time in this regime which causes the breakdown of classical integration schemes. To overcome this numerical burden various novel numerical methods with excellent efficiency were derived in recent years. The construction of each method thereby requests essentially the averaged model of the problem. However, the averaged model of each approach is found by different kinds of asymptotic approximation techniques reaching from the modulated Fourier expansion over the multiscale expansion by frequency up to the Chapman-Enskog expansion. In this work we give a first comparison of these recently introduced asymptotic series, reviewing their approximation validity to the KG in the asymptotic limit, their smoothness assumptions as well as their geometric properties, e.g., energy conservation and long-time behaviour of the remainder.

Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection
Stephen Pankavich, Nathan Neri and Deborah Shutt
2020, 25(8): 2867-2893 doi: 10.3934/dcdsb.2020044 +[Abstract](803) +[HTML](285) +[PDF](1613.12KB)
Abstract:

Recent clinical studies have shown that HIV disease pathogenesis can depend strongly on many factors at the time of transmission, including the strength of the initial viral load and the local availability of CD4+ T-cells. In this article, a new within-host model of HIV infection that incorporates the homeostatic proliferation of T-cells is formulated and analyzed. Due to the effects of this biological process, the influence of initial conditions on the proliferation of HIV infection is further elucidated. The identifiability of parameters within the model is investigated and a local stability analysis, which displays additional complexity in comparison to previous models, is conducted. The current study extends previous theoretical and computational work on the early stages of the disease and leads to interesting nonlinear dynamics, including a parameter region featuring bistability of infectious and viral clearance equilibria and the appearance of a Hopf bifurcation within biologically relevant parameter regimes.

Pattern formation in a predator-mediated coexistence model with prey-taxis
Evan C. Haskell and Jonathan Bell
2020, 25(8): 2895-2921 doi: 10.3934/dcdsb.2020045 +[Abstract](1043) +[HTML](318) +[PDF](1479.38KB)
Abstract:

Can foraging by predators or a repulsive prey defense mechanism upset predator-mediated coexistence? This paper investigates one scenario involving a prey-taxis by a prey species. We study a system of three populations involving two competing species with a common predator. All three populations are mobile via random dispersal within a bounded spatial domain \begin{document}$ \Omega $\end{document}, but the predator's movement is influenced by one prey's gradient representing a repulsive effect on the predator. We prove existence of positive solutions, and investigate pattern formation through bifurcation analysis and numerical simulation.

Discontinuous Galerkin method for the Helmholtz transmission problem in two-level homogeneous media
Qingjie Hu, Zhihao Ge and Yinnian He
2020, 25(8): 2923-2948 doi: 10.3934/dcdsb.2020046 +[Abstract](677) +[HTML](272) +[PDF](1805.26KB)
Abstract:

In this paper, the discontinuous Galerkin (DG) method is developed and analyzed for solving the Helmholtz transmission problem (HTP) with the first order absorbing boundary condition in two-level homogeneous media. This whole domain is separated into two disjoint subdomains by an interface, where two types of transmission conditions are provided. The application of the DG method to the HTP gives the discrete formulation. A rigorous theoretical analysis demonstrates that the discrete formulation can retain absolute stability without any mesh constraint. We prove that the errors in \begin{document}$ H^{1} $\end{document} and \begin{document}$ L^{2} $\end{document} norms are bounded by \begin{document}$ C_{1}kh + C_{2}k^{4}h^{2} $\end{document} and \begin{document}$ C_{1}kh^{2} + C_{2}k^{3}h^{2} $\end{document}, respectively, where \begin{document}$ C_1 $\end{document} and \begin{document}$ C_2 $\end{document} are positive constants independent of the wave number \begin{document}$ k $\end{document} and the mesh size \begin{document}$ h $\end{document}. Numerical experiments are conducted to verify the accuracy of the theoretical results and the efficiency of the numerical method.

Global existence and convergence rates of solutions for the compressible Euler equations with damping
Ruiying Wei, Yin Li and Zheng-an Yao
2020, 25(8): 2949-2969 doi: 10.3934/dcdsb.2020047 +[Abstract](661) +[HTML](250) +[PDF](460.34KB)
Abstract:

The Cauchy problem for the 3D compressible Euler equations with damping is considered. Existence of global-in-time smooth solutions is established under the condition that the initial data is small perturbations of some given constant state in the framework of Sobolev space \begin{document}$ H^3(\mathbb{R}^{3}) $\end{document} only, but we don't need the bound of \begin{document}$ L^1 $\end{document} norm. Moreover, the optimal \begin{document}$ L^{2} $\end{document}-\begin{document}$ L^{2} $\end{document} convergence rates are also obtained for the solution. Our proof is based on the benefit of the low frequency and high frequency decomposition, here, we just need spectral analysis of the low frequency part of the Green function to the linearized system, so that we succeed to avoid some complicate analysis.

Spectral theory and time asymptotics of size-structured two-phase population models
Mustapha Mokhtar-Kharroubi and Quentin Richard
2020, 25(8): 2969-3004 doi: 10.3934/dcdsb.2020048 +[Abstract](638) +[HTML](232) +[PDF](520.58KB)
Abstract:

This work provides a general spectral analysis of size-structured two-phase population models. Systematic functional analytic results are given. We deal first with the case of finite maximal size. We characterize the irreducibility of the corresponding \begin{document}$ L^{1} $\end{document} semigroup in terms of properties of the different parameters of the system. We characterize also the spectral gap property of the semigroup. It turns out that the irreducibility of the semigroup implies the existence of the spectral gap. In particular, we provide a general criterion for asynchronous exponential growth. We show also how to deal with time asymptotics in case of lack of irreducibility. Finally, we extend the theory to the case of infinite maximal size.

Periodic traveling wave solutions of periodic integrodifference systems
Guo Lin and Shuxia Pan
2020, 25(8): 3005-3031 doi: 10.3934/dcdsb.2020049 +[Abstract](610) +[HTML](244) +[PDF](466.89KB)
Abstract:

This paper is concerned with the periodic traveling wave solutions of integrodifference systems with periodic parameters. Without the assumptions on monotonicity, the existence of periodic traveling wave solutions is deduced to the existence of generalized upper and lower solutions by fixed point theorem and an operator with multi steps. The asymptotic behavior of periodic traveling wave solutions is investigated by the stability of periodic solutions in the corresponding initial value problem or the corresponding difference systems. To illustrate our conclusions, we study the periodic traveling wave solutions of two models including a scalar equation and a competitive type system, which do not generate monotone semiflows. The existence or nonexistence of periodic traveling wave solutions with all positive wave speeds is presented, which implies the minimal wave speeds of these models.

The limits of solutions of a linear delay integral equation
Kazuki Himoto and Hideaki Matsunaga
2020, 25(8): 3033-3048 doi: 10.3934/dcdsb.2020050 +[Abstract](522) +[HTML](227) +[PDF](389.78KB)
Abstract:

In this paper we classify the limits of solutions of a linear integral equation with finite delay. In particular, if the solution tends to a point or a periodic orbit, we establish the explicit expressions depending on given initial functions by using analysis of characteristic roots and the formal adjoint theory. Our results also present a necessary and sufficient condition for the exponential stability of the equation.

Modelling the evolutionary dynamics of host resistance-related traits in a susceptible-infected community with density-dependent mortality
Jian Zu, Miaolei Li, Yuexi Gu and Shuting Fu
2020, 25(8): 3049-3086 doi: 10.3934/dcdsb.2020051 +[Abstract](607) +[HTML](250) +[PDF](8771.26KB)
Abstract:

This study explores the evolutionary dynamics of host resistance based on a susceptible-infected population model with density-dependent mortality. We assume that the resistant ability of susceptible host will adaptively evolve, a different type of host differs in its susceptibility to infection, but the resistance to a pathogen involves a cost such that a less susceptible host results in a lower birth rate. By using the methods of adaptive dynamics and critical function analysis, we find that the evolutionary outcome relies mainly on the trade-off relationship between host resistance and its fertility. Firstly, we show that if the trade-off curve is globally concave, then a continuously stable strategy is predicted. In contrast, if the trade-off curve is weakly convex in the vicinity of singular strategy, then evolutionary branching of host resistance is possible. Secondly, after evolutionary branching in the host resistance has occurred, we examine the coevolutionary dynamics of dimorphic susceptible hosts and find that for a type of concave-convex-concave trade-off curve, the finally evolutionary outcome may contain a relatively higher susceptible host and a relatively higher resistant host, which can continuously stably coexist on a long-term evolutionary timescale. If the convex region of trade-off curve is relatively wider, then the finally evolutionary outcome may contain a fully resistant host and a moderately resistant host. Thirdly, through numerical simulation, we find that for a type of sigmoidal trade-off curve, after branching due to the high cost in terms of the birth rate, always the branch with stronger resistance goes extinct, the eventually evolutionary outcome includes a monomorphic host with relatively weaker resistance.

Long-term orbit dynamics viewed through the yellow main component in the parameter space of a family of optimal fourth-order multiple-root finders
Young Hee Geum and Young Ik Kim
2020, 25(8): 3087-3109 doi: 10.3934/dcdsb.2020052 +[Abstract](545) +[HTML](235) +[PDF](2052.51KB)
Abstract:

An analysis based on an elementary theory of plane curves is presented to locate bifurcation points from a main component in the parameter space of a family of optimal fourth-order multiple-root finders. We explore the basic dynamics of the iterative multiple-root finders under the Möbius conjugacy map on the Riemann sphere. A linear stability theory on local bifurcations is developed from the viewpoint of an arbitrarily small perturbation about the fixed point of the iterative map with a control parameter. Invariant conjugacy properties are established for the fixed point and its multiplier. The parameter spaces and dynamical planes are investigated to analyze the underlying dynamics behind the iterative map. Numerical experiments support the theory of locating bifurcation points of satellite and primitive components in the parameter space.

On a non-linear size-structured population model
Yunfei Lv, Yongzhen Pei and Rong Yuan
2020, 25(8): 3111-3133 doi: 10.3934/dcdsb.2020053 +[Abstract](588) +[HTML](244) +[PDF](242.39KB)
Abstract:

This paper deals with a size-structured population model consisting of a quasi-linear first-order partial differential equation with nonlinear boundary condition. The existence and uniqueness of solutions are firstly obtained by transforming the system into an equivalent integral equation such that the corresponding integral operator forms a contraction. Furthermore, the existence of global attractor is established by proving the asymptotic smoothness and eventual compactness of the nonlinear semigroup associated with the solutions. Finally, we discuss the uniform persistence and existence of compact attractor contained inside the uniformly persistent set.

Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions
Jihoon Lee and Vu Manh Toi
2020, 25(8): 3135-3152 doi: 10.3934/dcdsb.2020054 +[Abstract](627) +[HTML](265) +[PDF](409.21KB)
Abstract:

In this paper we study the asymptotic behavior of solutions for a class of nonautonomous reaction-diffusion equations with dynamic boundary conditions possessing finite delay. Under the polynomial conditions of reaction term, suitable conditions of delay terms and a minimal conditions of time-dependent force functions, we first prove the existence and uniqueness of solutions by using the Galerkin method. Then, we ensure the existence of pullback attractors for the associated process to the problem by proving some uniform estimates and asymptotic compactness properties (via an energy method). With an additional condition of time-dependent force functions, we prove that the boundedness of pullback attractors in smoother spaces.

Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise
Xingni Tan, Fuqi Yin and Guihong Fan
2020, 25(8): 3153-3170 doi: 10.3934/dcdsb.2020055 +[Abstract](646) +[HTML](239) +[PDF](458.6KB)
Abstract:

We mainly consider the existence of a random exponential attractor (positive invariant compact measurable set with finite fractal dimension and attracting orbits exponentially) for stochastic discrete long wave-short wave resonance equation driven by multiplicative white noise. Firstly, we prove the existence of a random attractor of the considered equation by proving the existence of a uniformly tempered pullback absorbing set and making an estimate on the "tails" of solutions. Secondly, we show the Lipschitz property of the solution process generated by the considered equation. Finally, we prove the existence of a random exponential attractor of the considered equation, which implies the finiteness of fractal dimension of random attractor.

Global attraction in a system of delay differential equations via compact and convex sets
Sebastián Buedo-Fernández
2020, 25(8): 3171-3181 doi: 10.3934/dcdsb.2020056 +[Abstract](513) +[HTML](227) +[PDF](388.77KB)
Abstract:

We provide sufficient conditions for a concrete type of systems of delay differential equations (DDEs) to have a global attractor. The principal idea is based on a particular type of global attraction in difference equations in terms of nested, convex and compact sets. We prove that the solutions of the system of DDEs inherit the convergence to the equilibrium from an associated discrete dynamical system.

Approximation of stationary statistical properties of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation
Bo You and Chunxiang Zhao
2020, 25(8): 3183-3198 doi: 10.3934/dcdsb.2020057 +[Abstract](493) +[HTML](226) +[PDF](488.1KB)
Abstract:

The main objective of this paper is to study the semi-implicit semi-discrete scheme in time of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation. We prove the global attractor and stationary statistical properties of the semi-implicit semi-discrete scheme in time of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation converge to those of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation as the time step goes to zero.

Exit problem for Ornstein-Uhlenbeck processes: A random walk approach
Samuel Herrmann and Nicolas Massin
2020, 25(8): 3199-3215 doi: 10.3934/dcdsb.2020058 +[Abstract](542) +[HTML](239) +[PDF](591.07KB)
Abstract:

In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm so-called Walk on Moving Spheres was already introduced in the Brownian context. The aim is therefore to generalize this numerical approach to the Ornstein-Uhlenbeck process and to describe the efficiency of the method.

Razumikhin-type theorems on polynomial stability of hybrid stochastic systems with pantograph delay
Wei Mao, Liangjian Hu and Xuerong Mao
2020, 25(8): 3217-3232 doi: 10.3934/dcdsb.2020059 +[Abstract](592) +[HTML](231) +[PDF](399.52KB)
Abstract:

The main aim of this paper is to investigate the polynomial stability of hybrid stochastic systems with pantograph delay (HSSwPD). By using the Razumikhin technique and Lyapunov functions, we establish several Razumikhin-type theorems on the \begin{document}$ p $\end{document}th moment polynomial stability and almost sure polynomial stability for HSSwPD. For linear HSSwPD, sufficient conditions for polynomial stability are presented.

Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise
Chi Phan
2020, 25(8): 3233-3256 doi: 10.3934/dcdsb.2020060 +[Abstract](619) +[HTML](237) +[PDF](567.79KB)
Abstract:

The longtime and global pullback dynamics of stochastic Hind-marsh-Rose equations with multiplicative noise on a three-dimensional bounded domain in neurodynamics is investigated in this work. The existence of a random attractor for this random dynamical system is proved through the exponential transformation and uniform estimates showing the pullback absorbing property and the pullback asymptotically compactness of this cocycle in the \begin{document}$ L^2 $\end{document} Hilbert space.

On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures
Zhiyuan Wen and Meirong Zhang
2020, 25(8): 3257-3274 doi: 10.3934/dcdsb.2020061 +[Abstract](539) +[HTML](260) +[PDF](467.26KB)
Abstract:

In this paper, we will first establish the necessary and sufficient conditions for the existence of the principal eigenvalues of second-order measure differential equations with indefinite weighted measures subject to the Neumann boundary condition. Then we will show the principal eigenvalues are continuously dependent on the weighted measures when the weak$^*$ topology is considered for measures. As applications, we will finally solve several optimization problems on principal eigenvalues, including some isospectral problems.

Null controllability for distributed systems with time-varying constraint and applications to parabolic-like equations
Larbi Berrahmoune
2020, 25(8): 3275-3303 doi: 10.3934/dcdsb.2020062 +[Abstract](360) +[HTML](133) +[PDF](597.21KB)
Abstract:

We consider the null controllability problem fo linear systems of the form \begin{document}$ y'(t) = Ay(t)+Bu(t) $\end{document} on a Hilbert space \begin{document}$ Y $\end{document}. We suppose that the control operator \begin{document}$ B $\end{document} is bounded from the control space \begin{document}$ U $\end{document} to a larger extrapolation space containing \begin{document}$ Y $\end{document}. The control \begin{document}$ u $\end{document} is constrained to lie in a time-varying bounded subset \begin{document}$ \Gamma(t) \subset U $\end{document}. From a general existence result based on a selection theorem, we obtain various properties on local and global constrained null controllability. The existence of the time optimal control is established in a general framework. When the constraint set \begin{document}$ \Gamma (t) $\end{document} contains the origin in its interior at each \begin{document}$ t>0 $\end{document}, the local constrained property turns out to be equivalent to a weighted dual observability inequality of \begin{document}$ L^{1} $\end{document} type with respect to the time variable. We treat also the problem of determining a steering control for general constraint sets \begin{document}$ \Gamma (t) $\end{document} in nonsmooth convex analysis context. Applications to the heat equation are treated for distributed and boundary controls under the assumptions that \begin{document}$ \Gamma (t) $\end{document} is a closed ball centered at the origin and its radius is time-varying.

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