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Discrete & Continuous Dynamical Systems - B

2017 , Volume 22 , Issue 4

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Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation
Georgy L. Alfimov, Pavel P. Kizin and Dmitry A. Zezyulin
2017, 22(4): 1207-1229 doi: 10.3934/dcdsb.2017059 +[Abstract](200) +[HTML](4) +[PDF](915.8KB)
Abstract:

The paper is devoted to nonlinear localized modes (“gap solitons”) for the spatially one-dimensional Gross-Pitaevskii equation (1D GPE) with a periodic potential and repulsive interparticle interactions. It has been recently shown (G. L. Alfimov, A. I. Avramenko, Physica D, 254, 29 (2013)) that under certain conditions all the stationary modes for the 1D GPE can be coded by bi-infinite sequences of symbols of some finite alphabet (called “codes” of the solutions). We present and justify a numerical method which allows to reconstruct the profile of a localized mode by its code. As an example, the method is applied to compute the profiles of gap solitons for 1D GPE with a cosine potential.

Stability analysis of an enteropathogen population growing within a heterogeneous group of animals
Carles Barril and Àngel Calsina
2017, 22(4): 1231-1252 doi: 10.3934/dcdsb.2017060 +[Abstract](77) +[HTML](2) +[PDF](532.4KB)
Abstract:

An autonomous semi-linear hyperbolic pde system for the proliferation of bacteria within a heterogeneous population of animals is presented and analysed. It is assumed that bacteria grow inside the intestines and that they can be either attached to the epithelial wall or as free particles in the lumen. A condition involving ecological parameters is given, which can be used to decide the existence of endemic equilibria as well as local stability properties of the non-endemic one. Some implications on phage therapy are addressed.

Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals
Tobias Black
2017, 22(4): 1253-1272 doi: 10.3934/dcdsb.2017061 +[Abstract](93) +[HTML](2) +[PDF](475.3KB)
Abstract:

We consider the two-species-two-chemical chemotaxis system

where \begin{document} $Ω\subset\mathbb{R}^n$ \end{document} is a bounded domain with smooth boundary. The system models Lotka-Volterra competition of two species coupled with an additional chemotactic influence. In this model each species is attracted by the signal produced by the other.

We firstly show that if \begin{document} $n=2$ \end{document} and the parameters in the system above are positive, the solution to the corresponding Neumann initial-boundary value problem, emanating from appropriately regular and nonnegative initial data, is global and bounded.

Furthermore, we prove asymptotic stabilization of arbitrary global bounded solutions for any \begin{document} $n≥q2$ \end{document}, in the sense that:

• If \begin{document} $a_1<1$ \end{document}, \begin{document} $a_2<1$ \end{document} and both \begin{document} $\frac{μ_1}{χ_1^2}$ \end{document} and \begin{document} $\frac{μ_2}{χ_2^2}$ \end{document} are sufficiently large, then any global solution satisfying \begin{document} $u\not\equiv0\not\equiv w$ \end{document} converges towards the unique positive spatially homogeneous equilibrium of the system given above.

and

• If \begin{document} $a_1≥q 1$ \end{document}, \begin{document} $a_2<1$ \end{document} and \begin{document} $\frac{μ_2}{χ_2^2}$ \end{document} is sufficiently large any global solution satisfying \begin{document} $w\not\equiv0$ \end{document} tends to \begin{document} $(0,1,1,0)$ \end{document} as \begin{document} $t\to∞$ \end{document}.

Global phase portrait of a degenerate Bogdanov-Takens system with symmetry
Hebai Chen, Xingwu Chen and Jianhua Xie
2017, 22(4): 1273-1293 doi: 10.3934/dcdsb.2017062 +[Abstract](107) +[HTML](5) +[PDF](882.9KB)
Abstract:

In this paper we study the global phase portrait of the normal form of a degenerate Bogdanov-Takens system with symmetry, i.e., a class of van der Pol-Duffing oscillators. This normal form is two-parametric and its parameters are considered in the whole parameter space, i.e., not viewed as a perturbation of some Hamiltonian system. We discuss the existence of limit cycles and prove its uniqueness if it exists. Moreover, by constructing a distance function we not only give the necessary and sufficient condition for the existence of heteroclinic loops connecting two saddles, but also prove its monotonicity and smoothness. Finally, we obtain a complete classification on the global phase portraits in the Poincaré disc as well as the complete global bifurcation diagram in the parameter space and find more plentiful phase portraits than the case that parameters are just sufficiently small.

A continuum model for nematic alignment of self-propelled particles
Pierre Degond, Angelika Manhart and Hui Yu
2017, 22(4): 1295-1327 doi: 10.3934/dcdsb.2017063 +[Abstract](124) +[HTML](347) +[PDF](639.7KB)
Abstract:

A continuum model for a population of self-propelled particles interacting through nematic alignment is derived from an individual-based model. The methodology consists of introducing a hydrodynamic scaling of the corresponding mean field kinetic equation. The resulting perturbation problem is solved thanks to the concept of generalized collision invariants. It yields a hyperbolic but non-conservative system of equations for the nematic mean direction of the flow and the densities of particles flowing parallel or anti-parallel to this mean direction. Diffusive terms are introduced under a weakly non-local interaction assumption and the diffusion coefficient is proven to be positive. An application to the modeling of myxobacteria is outlined.

A general decay result for a multi-dimensional weakly damped thermoelastic system with second sound
Salah Drabla, Salim A. Messaoudi and Fairouz Boulanouar
2017, 22(4): 1329-1339 doi: 10.3934/dcdsb.2017064 +[Abstract](88) +[HTML](5) +[PDF](352.0KB)
Abstract:

In this article we consider an n-dimensional system of thermoelasticity with second sound in the presence of a weak frictional damping. We establish an explicit and general decay rate result, using some properties of convex functions. Our result is obtained without imposing any restrictive growth assumption on the frictional damping term.

Competition in periodic media:Ⅰ-Existence of pulsating fronts
Léo Girardin
2017, 22(4): 1341-1360 doi: 10.3934/dcdsb.2017065 +[Abstract](63) +[HTML](1) +[PDF](427.5KB)
Abstract:

This paper is concerned with the existence of pulsating front solutions in space-periodic media for a bistable two-species competition-diffusion Lotka-Volterra system. Considering highly competitive systems, a simple-high frequency or small amplitudes" algebraic sufficient condition for the existence of pulsating fronts is stated. This condition is in fact sufficient to guarantee that all periodic coexistence states vanish and become unstable as the competition becomes large enough.

Vanishing capillarity limit of the non-conservative compressible two-fluid model
Jin Lai, Huanyao Wen and Lei Yao
2017, 22(4): 1361-1392 doi: 10.3934/dcdsb.2017066 +[Abstract](70) +[HTML](1) +[PDF](486.5KB)
Abstract:

In this paper, we consider the non-conservative compressible two-fluid model with constant viscosity coefficients and unequal pressure function in $\mathbb{R}^3$, we study the vanishing capillarity limit of the smooth solution to the initial value problem. We first establish the uniform estimates of global smooth solution with respect to the capillary coefficients $σ^+$ and $σ^-$, then by the Lion-Aubin lemma, we can obtain the unique smooth solution of the 3D non-conservative compressible two-fluid model with the capillary coefficients converges globally in time to the smooth solution of the 3D generic two-fluid model as $σ^+$ and $σ^-$ tend to zero. Also, we give the convergence rate estimates with respect to the capillary coefficients $σ^+$ and $σ^-$ for any given positive time.

Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays
Shangzhi Li and Shangjiang Guo
2017, 22(4): 1393-1423 doi: 10.3934/dcdsb.2017067 +[Abstract](106) +[HTML](1) +[PDF](543.4KB)
Abstract:

This paper is devoted to a cooperative model composed of two species withstage structure and state-dependent maturation delays. Firstly, positivity and boundedness of solutions are addressed to describe the population survival and the natural restriction of limited resources. It is shown that for a given pair of positive initial functions, the two mature populations are uniformly bounded away from zero and that the two mature populations are bounded above only if the the coupling strength is small enough. Moreover, if the coupling strength is large enough then the two mature populations tend to infinity as the time tends to infinity. In particular, the positivity of the two immature populations has been established under some additional conditions. Secondly, the existence and patterns of equilibria are investigated by means of degree theory and Lyapunov-Schmidt reduction. Thirdly, the local stability of the equilibria is also discussed through a formal linearization. Fourthly, the global behavior of solutions is discussed and the explicit bounds for the eventual behaviors of the two mature populations and two immature populations are obtained. Finally, global asymptotical stability is investigated by using the comparison principle of the state-dependent delay equations.

Feedback controllability for blowup points of semilinear heat equations
Ping Lin
2017, 22(4): 1425-1434 doi: 10.3934/dcdsb.2017068 +[Abstract](118) +[HTML](1) +[PDF](383.9KB)
Abstract:

This paper studies a controllability problem for blowup points of two classes of semilinear heat equations.Our goal to act controls on the systems we studied is to make the corresponding solutions blow upat given points. This differs with the controllability problem of equations with the property of blowup in the references, where the purpose of using controls is to prevent blowupby controls. We obtain the feedback controls for our controllability problem of blowup points.

Saddle-node bifurcations of multiple homoclinic solutions in ODES
Xiao-Biao Lin and Changrong Zhu
2017, 22(4): 1435-1460 doi: 10.3934/dcdsb.2017069 +[Abstract](104) +[HTML](1) +[PDF](509.3KB)
Abstract:

We study codimension 3 degenerate homoclinic bifurcations under periodic perturbations. Assume that among the 3 bifurcation equations, one is due to the homoclinic tangecy along the orbital direction. To the lowest order, the bifurcation equations become 3 quadratic equations. Under generic conditions on perturbations of the normal and tangential directions of the homoclinic orbit, up to 8 homoclinic orbits can be created through saddle-node bifurcations. Our results generate the homoclinic tangency bifurcation in Guckenheimer and Holmes [8].

Global existence for a thin film equation with subcritical mass
Jian-Guo Liu and Jinhuan Wang
2017, 22(4): 1461-1492 doi: 10.3934/dcdsb.2017070 +[Abstract](67) +[HTML](2) +[PDF](633.0KB)
Abstract:

In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin film equation in one-dimensional case

where \begin{document}$n≥q 1$\end{document}. There exists a critical mass \begin{document}$M_c=\frac{2\sqrt{6}π}{3}$\end{document} found by Witelski et al.(2004 Euro. J. of Appl. Math. 15,223-256) for \begin{document}$n=1$\end{document}. We obtain global existence of a non-negative entropy weak solution if initial mass is less than \begin{document}$M_c$\end{document}. For \begin{document}$n≥q 4$\end{document}, entropy weak solutions are positive and unique. For \begin{document}$n=1$\end{document}, a finite time blow-up occurs for solutions with initial mass larger than \begin{document}$M_c$\end{document}. For the Cauchy problem with \begin{document}$n=1$\end{document} and initial mass less than \begin{document}$M_c$\end{document}, we show that at least one of the following long-time behavior holds:the second moment goes to infinity as the time goes to infinity or \begin{document}$ h(·, t_k)\rightharpoonup 0$\end{document} in \begin{document}$L^1(\mathbb{R})$\end{document} for some subsequence \begin{document}${t_k} \to \infty $\end{document}.

Optimal harvesting of a stochastic delay competitive model
Meng Liu and Chuanzhi Bai
2017, 22(4): 1493-1508 doi: 10.3934/dcdsb.2017071 +[Abstract](83) +[HTML](1) +[PDF](374.9KB)
Abstract:

In this paper an $n$-species stochastic delay competitive model with harvesting is proposed. Some dynamical properties of the model are considered. We first establish sufficient conditions for persistence in the mean of the species. Then asymptotic stability in distribution of the harvesting model is studied. Next the optimal harvesting effort and the maximum harvesting yield are given by using the ergodic approach. Finally the analytical results are illustrated through simulation figures using MATLAB followed by discussions and conclusions.

Morse indices and symmetry breaking for the Gelfand equation in expanding annuli
Linfeng Mei and Zongming Guo
2017, 22(4): 1509-1523 doi: 10.3934/dcdsb.2017072 +[Abstract](123) +[HTML](1) +[PDF](416.4KB)
Abstract:

Bifurcation of nonradial solutions from radial solutions of

\begin{document}$-Δ u=λ e^u$ \end{document}

in expanding annuli of ${\mathbb{R}^N}$ with $3 ≤q N ≤q 9$ is studied. To obtain the main results, we use a blow-up argument via Morse indices of the regular entire solutions of (0.1).

An almost periodic malaria transmission model with time-delayed input of vector
Lizhong Qiang and Bin-Guo Wang
2017, 22(4): 1525-1546 doi: 10.3934/dcdsb.2017073 +[Abstract](69) +[HTML](3) +[PDF](754.1KB)
Abstract:

An almost periodic malaria transmission model with the time-delayed input of vector is considered. It is shown that the disease is uniformly persistent when the basic reproduction ratio $R_{0}>1$, and it will die out when $R_{0} < 1$ under the assumption that there exists a small invasion. Furthermore, the global stability of the disease-free almost periodic state is obtained provided that the disease-induced death rate is null. Finally, we illustrate the above results by numerical simulations and show that the periodic epidemic models may overestimate or underestimate the malaria risk.

Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses
Alexander Rezounenko
2017, 22(4): 1547-1563 doi: 10.3934/dcdsb.2017074 +[Abstract](224) +[HTML](4) +[PDF](491.7KB)
Abstract:

A virus dynamics model with intracellular state-dependent delay and nonlinear infection rate of Beddington-DeAngelis functional response is studied. The technique of Lyapunov functionals is used to analyze stability of the main interior infection equilibrium which describes the case of both CTL and antibody immune responses activated. We consider first a particular biologically motivated class of discrete state-dependent delays. The general case is investigated next. The stability of the infection-free and the immune-exhausted equilibria is also discussed.

Stability of equilibriums of stochastically perturbed delay differential neoclassical growth model
Leonid Shaikhet
2017, 22(4): 1565-1573 doi: 10.3934/dcdsb.2017075 +[Abstract](64) +[HTML](1) +[PDF](1030.1KB)
Abstract:

The known nonlinear delay differential neoclassical growth model is considered. It is assumed that this model is influenced by stochastic perturbations of the white noise type and these perturbations are directly proportional to the deviation of the system state from the zero or a positive equilibrium. Sufficient conditions for stability in probability of the positive equilibrium and for exponential mean square stability of the zero equilibrium are obtained. Numerical calculations and figures illustrate the obtained stability regions and behavior of stable and unstable solutions of the considered model. The proposed investigation procedure can be applied for arbitrary nonlinear stochastic delay differential equations with the order of nonlinearity higher than one.

Global dynamics of a coupled epidemic model
Hongying Shu and Xiang-Sheng Wang
2017, 22(4): 1575-1585 doi: 10.3934/dcdsb.2017076 +[Abstract](131) +[HTML](3) +[PDF](339.9KB)
Abstract:

In this paper, we propose a novel epidemic model coupling direct and indirect transmission of disease and study the global dynamic of the model system. Despite the nonlinearity and complexity of the system, the basic reproduction number exhibits a nice linear property: it is simply the sum of two basic reproduction numbers for direct and indirect disease transmissions respectively. We further demonstrate that the local and global dynamics of the system are related to the basic reproduction number. The new model has the advantage that it generalizes or connects to various disease models on HIV, Zika virus, avian influenza, H1N1 and so on.

Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions
Ji Shu
2017, 22(4): 1587-1599 doi: 10.3934/dcdsb.2017077 +[Abstract](83) +[HTML](2) +[PDF](363.8KB)
Abstract:

In this paper, we study the asymptotic behavior of the stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. With the properties of fractional Brownian motions, we prove the existence of a singleton sets random attractor.

Stability of traveling waves for autocatalytic reaction systems with strong decay
Yaping Wu and Niannian Yan
2017, 22(4): 1601-1633 doi: 10.3934/dcdsb.2017033 +[Abstract](82) +[HTML](1) +[PDF](1214.1KB)
Abstract:

This paper is concerned with the spatial decay and stability of travelling wave solutions for a reaction diffusion system \begin{document}$u_{t}=d u_{xx}-uv$\end{document}, \begin{document}$v_{t}=v_{xx}+uv-Kv^{q}$\end{document} with \begin{document}$q>1$\end{document}. By applying Centre Manifold Theorem and detailed asymptotic analysis, we get the precise spatial decaying rate of the travelling waves with noncritical speeds. Further by applying spectral analysis, Evans function method and some numerical simulation, we proved the spectral stability and the linear exponential stability of the waves with noncritical speeds in some weighted spaces.

Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity
Hao Yu, Wei Wang and Sining Zheng
2017, 22(4): 1635-1644 doi: 10.3934/dcdsb.2017078 +[Abstract](82) +[HTML](1) +[PDF](388.5KB)
Abstract:

This paper deals with the global boundedness of solutions to a fully parabolic Keller-Segel system \begin{document}$u_t=Δ u-\nabla (u^α \nabla v)$\end{document}, \begin{document}$v_t=Δ v-v+u$\end{document} under non-flux boundary conditions in a smooth bounded domain \begin{document}$Ω\subset\mathbb{R}^{n}$\end{document}. The case of \begin{document}$α≥ \max\{1,\frac{2}{n}\}$\end{document} with \begin{document}$n≥1$\end{document} was considered in a previous paper of the authors [Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. B, 21 (2016), 1317-1327]. In the present paper we prove for the other case \begin{document}$α∈(\frac{2}{3},1)$\end{document} that if \begin{document}$\|u_0\|_{L^\frac{nα}{2}(Ω)}$\end{document} and \begin{document}$\|\nabla v_0\|_{L^{nα}(Ω)}$\end{document} are small enough with \begin{document}$n≥q3$\end{document}, then the solutions are globally bounded with both \begin{document}$u$\end{document} and \begin{document}$v$\end{document} decaying to the same constant steady state \begin{document}$\bar{u}_0=\frac{1}{|Ω|}∈t_Ω u_0(x) dx$\end{document} exponentially in the \begin{document}$L^∞$\end{document}-norm as \begin{document}$t? ∞$\end{document}. Moreover, the above conclusions still hold for all \begin{document}$α≥q2$\end{document} and \begin{document}$n≥q1$\end{document}, provided \begin{document}$\|u_0\|_{L^{nα-n}(Ω)}$\end{document} and \begin{document}$\|\nabla v_0\|_{L^{∞}(Ω)}$\end{document} sufficiently small.

Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces
Gaocheng Yue
2017, 22(4): 1645-1671 doi: 10.3934/dcdsb.2017079 +[Abstract](92) +[HTML](1) +[PDF](582.5KB)
Abstract:

In this paper, we first prove the well-posedness for the nonautonomous reaction-diffusion equations with fractional diffusion in the locally uniform spaces framework. Under very minimal assumptions, then we study the asymptotic behavior of solutions of such equation and show the existence of \begin{document}$(H^{2(\alpha -ε),q}_U(\mathbb{R}^N),H^{2(\alpha -ε),q}_φ(\mathbb{R}^N))(0<ε<\alpha <1)$\end{document}-uniform(w.r.t.\begin{document}$g∈\mathcal{H}_{L^q_U(\mathbb{R}^N)}(g_0)$\end{document}) attractor \begin{document}$\mathcal{A}_{\mathcal{H}_{L^q_U(\mathbb{R}^N)}(g_0)}$\end{document} with locally uniform external forces being translation uniform bounded but not translation compact in \begin{document}$L_b^p(\mathbb{R};L^q_U(\mathbb{R}^N)).$\end{document} The key to that extensions is a new the space-time estimates in locally uniform spaces for the linear fractional power dissipative equation.

On global boundedness of the Chen system
Fuchen Zhang, Xiaofeng Liao, Chunlai Mu, Guangyun Zhang and Yi-An Chen
2017, 22(4): 1673-1681 doi: 10.3934/dcdsb.2017080 +[Abstract](87) +[HTML](1) +[PDF](334.1KB)
Abstract:

This paper deals with the open problem of the global boundedness of the Chen system based on Lyapunov stability theory, which was proposed by Qin and Chen (2007). The innovation of the paper is that this paper not only proves the Chen system is global bounded for a certain range of the parameters according to stability theory of dynamical systems but also gives a family of mathematical expressions of global exponential attractive sets for the Chen system with respect to the parameters of this system. Furthermore, the exponential rate of the trajectories is also obtained.

Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises
Min Zhao and Shengfan Zhou
2017, 22(4): 1683-1717 doi: 10.3934/dcdsb.2017081 +[Abstract](107) +[HTML](1) +[PDF](690.1KB)
Abstract:

In this paper, we study the long-term dynamical behavior of stochastic Boisso nade systems with time-dependent deterministic forces, additive white noise and multiplicative white noise. We first prove the existence of random attrac tor for the considered systems. And then we establish the upper semi-continui ty of random attractors for the systems as the coefficient of quadratic term tends to zero and intensities of the noises approach zero, respectively. At last, we obtain an upper bound of fractal dimension of the random attractors for both systems without quadratic term.

Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay
Jinling Zhou and Yu Yang
2017, 22(4): 1719-1741 doi: 10.3934/dcdsb.2017082 +[Abstract](111) +[HTML](5) +[PDF](488.6KB)
Abstract:

In this paper, we study a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. By Schauder's fixed point theorem and Laplace transform, we show that the existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the minimal wave speed. Some examples are listed to illustrate the theoretical results. Our results generalize some known results.

Instantaneous shrinking and extinction for a non-Newtonian polytropic filtration equation with orientated convection
Hailong Ye and Jingxue Yin
2017, 22(4): 1743-1755 doi: 10.3934/dcdsb.2017083 +[Abstract](70) +[HTML](2) +[PDF](397.2KB)
Abstract:

This paper is concerned with the instantaneous shrinking and extinction for a non-Newtonian polytropic filtration equation with orientated convection

where \begin{document} $p>1, m,q>0, N≥1$ \end{document} and \begin{document} $\overrightarrow{β}(x)$ \end{document} is a vector field defined on \begin{document} $\mathbb{R}^{N}$ \end{document}. Here, the orientation of the convection is specified to that with counteracting diffusion, that is \begin{document} $\overrightarrow{β}(x)·(-x)≥0$ \end{document}, \begin{document} $x∈\mathbb{R}^N$ \end{document}. Sufficient conditions are established for the instantaneous shrinking property of solutions with decayed initial datum of supports. For a certain class of initial datum, it is shown that there exists a critical time \begin{document} $τ^*>0$ \end{document} such that the supports of solutions are unbounded above for any \begin{document} $t < τ^*$ \end{document}, whilst the opposite is the case for any \begin{document} $t>τ^*$ \end{document}. In addition, we prove that once the supports of solutions shrink instantaneously, the solutions will vanish in finite time.

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