# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

December 2019 , Volume 24 , Issue 12

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2019, 24(12): 6325-6347 doi: 10.3934/dcdsb.2019141 +[Abstract](1387) +[HTML](318) +[PDF](1802.51KB)
Abstract:

The objective of this paper is to propose some high-order compact schemes for two-dimensional Ginzburg-Landau equation. The space is approximated by high-order compact methods to improve the computational efficiency. Based on Crank-Nicolson method in time, several temporal approximations are used starting from different viewpoints. The numerical characters of the new schemes, such as the existence and uniqueness, stability, convergence are investigated. Some numerical illustrations are reported to confirm the advantages of the new schemes by comparing with other existing works. In the numerical experiments, the role of some parameters in the model is considered and tested.

2019, 24(12): 6349-6365 doi: 10.3934/dcdsb.2019142 +[Abstract](1073) +[HTML](243) +[PDF](455.16KB)
Abstract:

The generalized Hunter-Saxton system comprises several well-kno-wn models from fluid dynamics and serves as a tool for the study of fluid convection and stretching in one-dimensional evolution equations. In this work, we examine the global regularity of periodic smooth solutions of this system in \begin{document}$L^p$\end{document}, \begin{document}$p \in [1,\infty)$\end{document}, spaces for nonzero real parameters \begin{document}$(\lambda,\kappa)$\end{document}. Our results significantly improve and extend those by Wunsch et al. [29,30,31] and Sarria [23]. Furthermore, we study the effects that different boundary conditions have on the global regularity of solutions by replacing periodicity with a homogeneous three-point boundary condition and establish finite-time blowup of a local-in-time solution of the resulting system for particular values of the parameters.

2019, 24(12): 6367-6385 doi: 10.3934/dcdsb.2019143 +[Abstract](1521) +[HTML](483) +[PDF](642.22KB)
Abstract:

In this paper, stochastic differential equations that model the dynamics of a hepatitis C virus are derived from a system of ordinary differential equations. The stochastic model incorporates the host immunity. Firstly, the existence of a unique ergodic stationary distribution is derived by using the theory of Hasminskii. Secondly, sufficient conditions are obtained for the destruction of hepatocytes and the convergence of target cells. Moreover based on realistic parameters, numerical simulations are carried out to show the analytical results. These results highlight the role of environmental noise in the spread of hepatitis C viruses. The theoretical work extend the results of the corresponding deterministic system.

2019, 24(12): 6387-6417 doi: 10.3934/dcdsb.2019144 +[Abstract](1530) +[HTML](388) +[PDF](705.65KB)
Abstract:

The second-order implicit integration factor method (IIF2) is effective at solving stiff reaction–diffusion equations owing to its nice stability condition. IIF has previously been applied primarily to systems in which the reaction contained no explicitly time-dependent terms and the boundary conditions were homogeneous. If applied to a system with explicitly time-dependent reaction terms, we find that IIF2 requires prohibitively small time-steps, that are relative to the square of spatial grid sizes, to attain its theoretical second-order temporal accuracy. Although the second-order implicit exponential time differencing (iETD2) method can accurately handle explicitly time-dependent reactions, it is more computationally expensive than IIF2. In this paper, we develop a hybrid approach that combines the advantages of both methods, applying IIF2 to reaction terms that are not explicitly time-dependent and applying iETD2 to those which are. The second-order \begin{document}$\underline {\text{h}} {\text{ybrid}}$\end{document} \begin{document}${\text{I}}\underline {{\text{IF}}} - \underline {\text{E}} {\text{TD}}$\end{document} method (hIFE2) inherits the lower complexity of IIF2 and the ability to remain second-order accurate in time for large time-steps from iETD2. Also, it inherits the unconditional stability from IIF2 and iETD2 methods for dealing with the stiffness in reaction–diffusion systems. Through a transformation, hIFE2 can handle nonhomogeneous boundary conditions accurately and efficiently. In addition, this approach can be naturally combined with the compact and array representations of IIF and ETD for systems in higher spatial dimensions. Various numerical simulations containing linear and nonlinear reactions are presented to demonstrate the superior stability, accuracy, and efficiency of the new hIFE method.

2019, 24(12): 6419-6444 doi: 10.3934/dcdsb.2019145 +[Abstract](961) +[HTML](254) +[PDF](518.66KB)
Abstract:

The well-posedness of a chemotaxis system with indirect signal production in a two-dimensional domain is shown, all solutions being global unlike for the classical Keller-Segel chemotaxis system. Nevertheless, there is a threshold value \begin{document}$M_c$\end{document} of the mass of the first component which separates two different behaviours: solutions are bounded when the mass is below \begin{document}$M_c$\end{document} while there are unbounded solutions starting from initial conditions having a mass exceeding \begin{document}$M_c$\end{document}. This result extends to arbitrary two-dimensional domains a previous result of Tao & Winkler (2017) obtained for radially symmetric solutions to a simplified version of the model in a ball and relies on a different approach involving a Liapunov functional.

2019, 24(12): 6445-6464 doi: 10.3934/dcdsb.2019146 +[Abstract](1440) +[HTML](277) +[PDF](767.03KB)
Abstract:

In this paper, we develop four energy-preserving algorithms for the regularized long wave (RLW) equation. On the one hand, we combine the discrete variational derivative method (DVDM) in time and the modified finite volume method (mFVM) in space to derive a fully implicit energy-preserving scheme and a linear-implicit conservative scheme. On the other hand, based on the (invariant) energy quadratization technique, we first reformulate the RLW equation to an equivalent form with a quadratic energy functional. Then we discretize the reformulated system by the mFVM in space and the linear-implicit Crank-Nicolson method and the leap-frog method in time, respectively, to arrive at two new linear structure-preserving schemes. All proposed fully discrete schemes are proved to preserve the corresponding discrete energy conservation law. The proposed linear energy-preserving schemes not only possess excellent nonlinear stability, but also are very cheap because only one linear equation system needs to be solved at each time step. Numerical experiments are presented to show the energy conservative property and efficiency of the proposed methods.

2019, 24(12): 6465-6480 doi: 10.3934/dcdsb.2019147 +[Abstract](1157) +[HTML](322) +[PDF](517.9KB)
Abstract:

Traveling wave solutions of a chemotaxis model with a reaction term are studied. We investigate the existence and non-existence of traveling wave solutions in certain ranges of parameters. Particularly for a positive rate of chemical growth, we prove the existence of a heteroclinic orbit by constructing a positively invariant set in the three dimensional space. The monotonicity of traveling waves is also analyzed in terms of chemotaxis, reaction and diffusion parameters. Finally, the traveling wave solutions are shown to be linearly unstable.

Lin Niu and
2019, 24(12): 6481-6494 doi: 10.3934/dcdsb.2019148 +[Abstract](938) +[HTML](251) +[PDF](433.3KB)
Abstract:

A nonlinear dynamical system is called eventually competitive (or cooperative) provided that it preserves a partial order in backward (or forward) time only after some reasonable initial transient. We present in this paper the Non-oscillation Principle for eventually competitive or cooperative systems, by which the non-ordering of (both \begin{document}$\omega$\end{document}- and \begin{document}$\alpha$\end{document}-) limit sets is obtained for such systems; and moreover, we established the Poincaré-Bendixson Theorem and structural stability for three-dimensional eventually competitive and cooperative systems.

2019, 24(12): 6495-6509 doi: 10.3934/dcdsb.2019150 +[Abstract](1247) +[HTML](266) +[PDF](462.49KB)
Abstract:

In this paper we investigate the center problem for the discontinuous piecewise smooth quasi–homogeneous but non–homogeneous polynomial differential systems. First, we provide sufficient and necessary conditions for the existence of a center in the discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Moreover, these centers are global, and the period function of their periodic orbits is monotonic. Second, we characterize the centers of the discontinuous piecewise smooth quasi–homogeneous cubic and quartic polynomial differential systems.

2019, 24(12): 6511-6531 doi: 10.3934/dcdsb.2019151 +[Abstract](1017) +[HTML](266) +[PDF](481.51KB)
Abstract:

The aim of this paper is the derivation of general two-scale compactness results for coupled bulk-surface problems. Such results are needed for example for the homogenization of elliptic and parabolic equations with boundary conditions of second order in periodically perforated domains. We are dealing with Sobolev functions with more regular traces on the oscillating boundary, in the case when the norm of the traces and their surface gradients are of the same order. In this case, the two-scale convergence results for the traces and their gradients have a similar structure as for perforated domains, and we show the relation between the two-scale limits of the bulk-functions and their traces. Additionally, we apply our results to a reaction diffusion problem of elliptic type with a Wentzell-boundary condition in a multi-component domain.

2019, 24(12): 6533-6540 doi: 10.3934/dcdsb.2019152 +[Abstract](1065) +[HTML](255) +[PDF](320.97KB)
Abstract:

Uncertain heat equation is a type of uncertain partial differential equations driven by Liu processes. As an important part in uncertain heat equation, stability analysis has not been researched as yet. This paper first introduces a concept of stability in measure for uncertain heat equation, and proves a stability theorem under strong Lipschitz condition that provides a sufficient for an uncertain heat equation being stable in measure. Moreover, some examples are given.

2019, 24(12): 6541-6552 doi: 10.3934/dcdsb.2019153 +[Abstract](1080) +[HTML](234) +[PDF](370.47KB)
Abstract:

Hilbert's 16th Problem suggests a concern to the cyclicity of planar polynomial differential systems, but it is known that a key step to the answer is finding the cyclicity of center-focus equilibria of polynomial differential systems (even of order 2 or 3). Correspondingly, the same question for polynomial discontinuous differential systems is also interesting. Recently, it was proved that the cyclicity of \begin{document}$(1, 2)$\end{document}-switching FF type equilibria is at least 5. In this paper we prove that the cyclicity of \begin{document}$(1, 3)$\end{document}-switching FF type equilibria with homogeneous cubic nonlinearities is at least 3.

2019, 24(12): 6553-6605 doi: 10.3934/dcdsb.2019155 +[Abstract](2815) +[HTML](434) +[PDF](1146.38KB)
Abstract:

A decomposition principle for nonlinear dynamic compartmental systems is introduced in the present paper. This theory is based on the mutually exclusive and exhaustive, analytical and dynamic, novel system and subsystem partitioning methodologies. A deterministic mathematical method is developed for the dynamic analysis of nonlinear compartmental systems based on the proposed theory. The dynamic method enables tracking the evolution of all initial stocks, external inputs, and arbitrary intercompartmental flows as well as the associated storages derived from these stocks, inputs, and flows individually and separately within the system. The transient and the dynamic direct, indirect, acyclic, cycling, and transfer ($\texttt{diact}$) flows and associated storages transmitted along a particular flow path or from one compartment-directly or indirectly-to any other are then analytically characterized, systematically classified, and mathematically formulated. Thus, the dynamic influence of one compartment, in terms of flow and storage transfer, directly or indirectly on any other compartment is ascertained. Consequently, new mathematical system analysis tools are formulated as quantitative system indicators. The proposed mathematical method is then applied to various models from literature to demonstrate its efficiency and wide applicability.

2019, 24(12): 6607-6620 doi: 10.3934/dcdsb.2019158 +[Abstract](1230) +[HTML](280) +[PDF](373.46KB)
Abstract:

A system modeling bacteriophage treatments with coinfections in a noisy context is analysed. We prove that in a small noise regime, the system converges in the long term to a bacteria-free equilibrium. Moreover, we compare the treatment with coinfection with the treatment without coinfection, showing how coinfection affects the convergence to the bacteria-free equilibrium.

2019, 24(12): 6621-6631 doi: 10.3934/dcdsb.2019159 +[Abstract](1109) +[HTML](251) +[PDF](618.81KB)
Abstract:

Disequilibria phenomenon appears in the economic model of durable stocks proposed by A. Panchuck and T. Puu in [7]. In this paper, assuming that agents have the same utility functions, we give not only bounds of the disequilibrium but also prove the existence of a compact set of no-trade points such that it does not depend on the initial stock distribution. We also give a description of the nature of \begin{document}$\omega$\end{document}–limit sets in the general case proving that disequilibrium points can be attained as limit points of orbits.

2019, 24(12): 6633-6652 doi: 10.3934/dcdsb.2019160 +[Abstract](1561) +[HTML](260) +[PDF](506.2KB)
Abstract:

This paper focuses on the Cauchy problem of the rotation-two-component Camassa-Holm(R2CH) system, which is a model of equatorial water waves that includes the effect of the Coriolis force. It has been shown that the R2CH system is well-posed in Sobolev spaces \begin{document}$H^s(\mathbb{R})\times H^{s-1}(\mathbb{R})$\end{document} with \begin{document}$s>3/2$\end{document}. Using the method of approximate solutions in conjunction with well-posedness estimates, we further proved that the dependence on initial data is sharp, i.e., the data-to-solution map is continuous but not uniformly continuous. Moreover, we obtain that the solution map for the R2CH system is Hölder continuous in \begin{document}$H^\theta(\mathbb{R})\times H^{\theta-1}(\mathbb{R})$\end{document}-topology for all \begin{document}$0\leq\theta<s$\end{document} with exponent \begin{document}$\gamma$\end{document} depending on \begin{document}$s$\end{document} and \begin{document}$\theta$\end{document}. The Coriolis term and higher nonlinear term in the R2CH system bring challenges to construct the counter-approximate solutions.

2019, 24(12): 6653-6673 doi: 10.3934/dcdsb.2019161 +[Abstract](1165) +[HTML](239) +[PDF](501.43KB)
Abstract:

In this paper, we consider nonlocal nonlinear renewal equation (Markov chain, Ordinary differential equation and Partial Differential Equation). We show that the General Relative Entropy [29] can be extend to nonlinear problems and under some assumptions on the nonlinearity we prove the convergence of the solution to its steady state as time tends to infinity.

2019, 24(12): 6675-6691 doi: 10.3934/dcdsb.2019162 +[Abstract](1368) +[HTML](257) +[PDF](480.33KB)
Abstract:

In this work we establish conditions which guarantee the existence of (strictly) positive steady states of a nonlinear structured population model. In our framework, the steady state formulation amounts to recasting the nonlinear problem as a family of eigenvalue problems, combined with a fixed point problem. Amongst other things, our formulation requires us to control the growth behaviour of the spectral bound of a family of linear operators along positive rays. For the specific class of model we consider here this presents a considerable challenge. We are going to show that the spectral bound of the family of operators, arising from the steady state formulation, can be controlled by perturbations in the domain of the generators (only). These new boundary perturbation results are particularly important for models exhibiting fertility controlled dynamics. As an important by-product of the application of the boundary perturbation results we employ here, we recover (using a recent theorem by H. R. Thieme) the familiar net reproduction number (or function) for models with single state at birth, which include for example the classic McKendrick (linear) and Gurtin-McCamy (non-linear) age-structured models.

2019, 24(12): 6693-6724 doi: 10.3934/dcdsb.2019163 +[Abstract](1342) +[HTML](299) +[PDF](519.84KB)
Abstract:

We consider a system of \begin{document}$N$\end{document} competing species, each of which can access a different resources distribution and who can disperse at different speeds. We fully characterize the existence and stability of steady-states for large diffusivities. Indeed, we prove that the resources distribution yielding the largest total population size at equilibrium is, broadly speaking, always the winner when species disperse quickly. The criterion also uses the different dispersal rates. The methods used rely on an expansion of the solutions of the Lotka-Volterra sytem for large diffusivities, and is an extension of the "slowest diffuser always wins" principle.

Using this method, we also study the case of an equation modelling a trait structured population, with small mutations. We assume that each trait is characterized by its diffusivity and the resources it can access. We similarly derive a criterion mixing these diffusivities and the total population size functional for the single species model to show that for rare mutations and large diffusivities, the population concentrates in a neighbourhood of a trait maximizing this criterion.

2019, 24(12): 6725-6743 doi: 10.3934/dcdsb.2019164 +[Abstract](1370) +[HTML](245) +[PDF](408.66KB)
Abstract:

This paper addresses the Cauchy problem of the three-dimensional inhomogeneous incompressible micropolar equations. We prove the global existence and exponential decay-in-time of strong solution with vacuum over the whole space \begin{document}$\mathbb{R}^{3}$\end{document} provided that the initial data are sufficiently small. The initial vacuum is allowed.

2019, 24(12): 6745-6770 doi: 10.3934/dcdsb.2019165 +[Abstract](1376) +[HTML](263) +[PDF](1005.0KB)
Abstract:

We consider the problem of approximating numerically the moments and the supports of measures which are invariant with respect to the dynamics of continuous- and discrete-time polynomial systems, under semialgebraic set constraints. First, we address the problem of approximating the density and hence the support of an invariant measure which is absolutely continuous with respect to the Lebesgue measure. Then, we focus on the approximation of the support of an invariant measure which is singular with respect to the Lebesgue measure.

Each problem is handled through an appropriate reformulation into a conic optimization problem over measures, solved in practice with two hierarchies of finite-dimensional semidefinite moment-sum-of-square relaxations, also called Lasserre hierarchies.

Under specific assumptions, the first Lasserre hierarchy allows to approximate the moments of an absolutely continuous invariant measure as close as desired and to extract a sequence of polynomials converging weakly to the density of this measure.

The second Lasserre hierarchy allows to approximate as close as desired in the Hausdorff metric the support of a singular invariant measure with the level sets of the Christoffel polynomials associated to the moment matrices of this measure.

We also present some application examples together with numerical results for several dynamical systems admitting either absolutely continuous or singular invariant measures.

2019, 24(12): 6771-6782 doi: 10.3934/dcdsb.2019166 +[Abstract](1697) +[HTML](278) +[PDF](411.55KB)
Abstract:

In this paper, we extend the theory of basic reproduction ratios \begin{document}$\mathcal{R}_0$\end{document} in [Liang, Zhang, Zhao, JDDE], which concerns with abstract functional differential systems in a time-periodic environment. We prove the threshold dynamics, that is, the sign of \begin{document}$\mathcal{R}_0-1$\end{document} determines the dynamics of the associated linear system. We also propose a direct and efficient numerical method to calculate \begin{document}$\mathcal{R}_0$\end{document}.

2019, 24(12): 6783-6815 doi: 10.3934/dcdsb.2019167 +[Abstract](1399) +[HTML](275) +[PDF](7747.06KB)
Abstract:

In this paper, we study the local bifurcations of an enzyme-catalyzed reaction system with positive parameters \begin{document}$\alpha$\end{document}, \begin{document}$\beta$\end{document}, \begin{document}$\gamma$\end{document} and integer \begin{document}$n\geq 2$\end{document}. This system is orbitally equivalent to a polynomial differential system with order \begin{document}$n+2$\end{document}. Although not all coordinates of equilibria can be computed because of the high degree of polynomial, parameter conditions for the coexistence of equilibria and their qualitative properties are obtained. Furthermore, it is proved that this system has various bifurcations, including saddle-node bifurcation, transcritical bifurcation, pitchfork bifurcation and Hopf bifurcation. Based on Lyapunov quantities, the order of weak focus is proved to be at most 3. Furthermore, parameter conditions of the exact order of weak focus are obtained. Finally, numerical simulations are employed to illustrate our results.

2019, 24(12): 6817-6835 doi: 10.3934/dcdsb.2019168 +[Abstract](1533) +[HTML](308) +[PDF](571.7KB)
Abstract:

In this paper, we consider the flocking problem of modified continu- ous-time and discrete-time Cucker-Smale models where every agent has its own intrinsic dynamics with Lipschitz property. The dynamics of the models are governed by the interplay between agents' own intrinsic dynamics and Cucker-Smale coupling dynamics. Based on the explicit construction of Lyapunov functionals, we show that conditional flocking would occur. And then we study the relationship between the Lipschitz constant \begin{document}$L$\end{document} of the intrinsic dynamics and exponent \begin{document}$\beta$\end{document} measuring the strength of the interaction between agents when flocking occurs. We also give two examples to show flocking might not occur for enough large \begin{document}$L$\end{document} or unconditionally for \begin{document}$\beta>0$\end{document}. At last, we provide several numerical simulations to illustrate our theoretical results.

2019, 24(12): 6837-6854 doi: 10.3934/dcdsb.2019169 +[Abstract](1466) +[HTML](235) +[PDF](471.74KB)
Abstract:

In this paper, we consider the well-posedness of the weakly damped stochastic nonlinear Schrödinger(NLS) equation driven by multiplicative noise. First, we show the global existence of the unique solution for the damped stochastic NLS equation in critical case. Meanwhile, the exponential integrability of the solution is proved, which implies the continuous dependence on the initial data. Then, we analyze the effect of the damped term and noise on the blow-up phenomenon. By modifying the associated energy, momentum and variance identity, we deduce a sharp blow-up condition for damped stochastic NLS equation in supercritical case. Moreover, we show that when the damped effect is large enough, the damped effect can prevent the blow-up of the solution with high probability.

2019  Impact Factor: 1.27