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Discrete & Continuous Dynamical Systems - B

September 2017 , Volume 22 , Issue 7

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Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion
Ahmed Boudaoui, Tomás Caraballo and Abdelghani Ouahab
2017, 22(7): 2521-2541 doi: 10.3934/dcdsb.2017084 +[Abstract](449) +[HTML](11) +[PDF](415.3KB)
Abstract:

This paper is concerned with the existence and continuous dependence of mild solutions to stochastic differential equations with non-instantaneous impulses driven by fractional Brownian motions. Our approach is based on a Banach fixed point theorem and Krasnoselski-Schaefer type fixed point theorem.

Bifurcation and final patterns of a modified Swift-Hohenberg equation
Yuncherl Choi, Taeyoung Ha, Jongmin Han and Doo Seok Lee
2017, 22(7): 2543-2567 doi: 10.3934/dcdsb.2017087 +[Abstract](489) +[HTML](19) +[PDF](1309.8KB)
Abstract:

In this paper, we study the dynamical bifurcation and final patterns of a modified Swift-Hohenberg equation(MSHE). We prove that the MSHE bifurcates from the trivial solution to an \begin{document}$S^1$\end{document}-attractor as the control parameter \begin{document}$\alpha $\end{document} passes through a critical number \begin{document}$\hat{\alpha }$\end{document}. Using the center manifold analysis, we study the bifurcated attractor in detail by showing that it consists of finite number of singular points and their connecting orbits. We investigate the stability of those points. We also provide some numerical results supporting our analysis.

Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels
Yangrong Li, Renhai Wang and Jinyan Yin
2017, 22(7): 2569-2586 doi: 10.3934/dcdsb.2017092 +[Abstract](379) +[HTML](21) +[PDF](415.0KB)
Abstract:

Backward compact dynamics is deduced for a non-autonomous Benjamin-Bona-Mahony (BBM) equation on an unbounded 3D-channel. A backward compact attractor is defined by a time-dependent family of backward compact, invariant and pullback attracting sets. The theoretical existence result for such an attractor is derived from the backward flattening property, and this property is proved to be equivalent to the backward asymptotic compactness in a uniformly convex Banach space. Finally, it is shown that the BBM equation has a backward compact attractor in a Sobolev space under some suitable assumptions, such as, backward translation boundedness and backward small-tail. Both spectrum decomposition and cut-off technique are used to give all required backward uniform estimates.

On some difference equations with exponential nonlinearity
Eugenia N. Petropoulou
2017, 22(7): 2587-2594 doi: 10.3934/dcdsb.2017098 +[Abstract](397) +[HTML](23) +[PDF](355.7KB)
Abstract:

The problem of the existence of complex \begin{document}$\ell_1$\end{document} solutions of two difference equations with exponential nonlinearity is studied, one of which is nonautonomous. As a consequence, several information are obtained regarding the asymptotic stability of their equilibrium points, as well as the corresponding generating function and \begin{document}$z-$\end{document} transform of their solutions. The results, which are obtained using a general theorem based on a functional-analytic technique, provide also a rough estimate of the region of attraction of each equilibrium point for the autonomous case. When restricted to real solutions, the results are compared with other recently published results.

Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model
Tak Kuen Siu and Yang Shen
2017, 22(7): 2595-2626 doi: 10.3934/dcdsb.2017100 +[Abstract](585) +[HTML](33) +[PDF](602.6KB)
Abstract:

A risk-minimizing approach to pricing contingent claims in a general non-Markovian, regime-switching, jump-diffusion model is discussed, where a convex risk measure is used to describe risk. The pricing problem is formulated as a two-person, zero-sum, stochastic differential game between the seller of a contingent claim and the market, where the latter may be interpreted as a ''fictitious'' player. A backward stochastic differential equation (BSDE) approach is applied to discuss the game problem. Attention is given to the entropic risk measure, which is a particular type of convex risk measures. In this situation, a pricing kernel selected by an equilibrium state of the game problem is related to the one selected by the Esscher transform, which was introduced to the option-pricing world in the seminal work by [38].

Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition
Lu Yang and Meihua Yang
2017, 22(7): 2627-2650 doi: 10.3934/dcdsb.2017102 +[Abstract](443) +[HTML](23) +[PDF](547.0KB)
Abstract:

In this paper, we study the dynamic behavior of a stochastic reaction-diffusion equation with dynamical boundary condition, where the nonlinear terms \begin{document}$f$\end{document} and \begin{document}$h$\end{document} satisfy the polynomial growth condition of arbitrary order. Some higher-order integrability of the difference of the solutions near the initial time, and the continuous dependence result with respect to initial data in \begin{document}$H^1(\mathcal{O})× H^{\frac 1 2}(Γ)$\end{document} were established. As a direct application, we can obtain the existence of pullback random attractor \begin{document}$A$\end{document} in the spaces \begin{document}$L^{p}(\mathcal{O})× L^{p}(Γ)$\end{document} and \begin{document}$H^1(\mathcal{O})× H^{\frac 1 2}(Γ)$\end{document} immediately.

Extinction in stochastic predator-prey population model with Allee effect on prey
Miljana JovanoviĆ and Marija KrstiĆ
2017, 22(7): 2651-2667 doi: 10.3934/dcdsb.2017129 +[Abstract](472) +[HTML](20) +[PDF](2882.0KB)
Abstract:

This paper presents the analysis of the conditions which lead the stochastic predator-prey model with Allee effect on prey population to extinction. In order to find these conditions we first prove the existence and uniqueness of global positive solution of considered model using the comparison theorem for stochastic differential equations. Then, we establish the conditions under which extinction of predator and prey populations occur. We also find the conditions for parameters of the model under which the solution of the system is globally attractive in mean. Finally, the numerical illustration with real life example is carried out to confirm our theoretical results.

Existence and stability of periodic oscillations of a rigid dumbbell satellite around its center of mass
Jifeng Chu, Zaitao Liang, Pedro J. Torres and Zhe Zhou
2017, 22(7): 2669-2685 doi: 10.3934/dcdsb.2017130 +[Abstract](458) +[HTML](23) +[PDF](630.6KB)
Abstract:

We study the existence and stability of periodic solutions of a differential equation that models the planar oscillations of a satellite in an elliptic orbit around its center of mass. The proof is based on a suitable version of Poincaré-Birkhoff theorem and the third order approximation method.

Diffusive heat transport in Budyko's energy balance climate model with a dynamic ice line
James Walsh
2017, 22(7): 2687-2715 doi: 10.3934/dcdsb.2017131 +[Abstract](329) +[HTML](17) +[PDF](745.9KB)
Abstract:

M. Budyko and W. Sellers independently introduced seminal energy balance climate models in 1969, each with a goal of investigating the role played by positive ice albedo feedback in climate dynamics. In this paper we replace the relaxation to the mean horizontal heat transport mechanism used in the models of Budyko and Sellers with diffusive heat transport. We couple the resulting surface temperature equation with an equation for movement of the edge of the ice sheet (called the ice line), recently introduced by E. Widiasih. We apply the spectral method to the temperature-ice line system and consider finite approximations. We prove there exists a stable equilibrium solution with a small ice cap, and an unstable equilibrium solution with a large ice cap, for a range of parameter values. If the diffusive transport is too efficient, however, the small ice cap disappears and an ice free Earth becomes a limiting state. In addition, we analyze a variant of the coupled diffusion equations appropriate as a model for extensive glacial episodes in the Neoproterozoic Era. Although the model equations are no longer smooth due to the existence of a switching boundary, we prove there exists a unique stable equilibrium solution with the ice line in tropical latitudes, a climate event known as a Jormungand or Waterbelt state. As the systems introduced here contain variables with differing time scales, the main tool used in the analysis is geometric singular perturbation theory.

Boundedness in a two-species chemotaxis parabolic system with two chemicals
Xie Li and Yilong Wang
2017, 22(7): 2717-2729 doi: 10.3934/dcdsb.2017132 +[Abstract](459) +[HTML](30) +[PDF](418.8KB)
Abstract:

This paper is devoted to the chemotaxis system

which models the interaction between two species in presence of two chemicals, where \begin{document}$χ, \, ξ∈\mathbb{R}$\end{document}, \begin{document}$Ω\subset\mathbb{R}^2$\end{document} are bounded domains with smooth boundary. It is shown that under the homogeneous Neumann boundary conditions the system possesses a unique global classical solution which is bounded whenever both \begin{document}$\int_Ω u_0dx$\end{document} and \begin{document}$\int_Ω w_0dx$\end{document} are appropriately small. In particular, we extend the recent results obtained by Tao and Winkler (2015, Disc. Cont. Dyn. Syst. B) to the fully parabolic case, i.e., the case of \begin{document}$τ=1$\end{document}.

Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory
Christopher Rackauckas and Qing Nie
2017, 22(7): 2731-2761 doi: 10.3934/dcdsb.2017133 +[Abstract](868) +[HTML](47) +[PDF](2093.2KB)
Abstract:

Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs.

A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment
Jing Ge, Ling Lin and Lai Zhang
2017, 22(7): 2763-2776 doi: 10.3934/dcdsb.2017134 +[Abstract](389) +[HTML](16) +[PDF](690.8KB)
Abstract:

To explore the impact of media coverage and spatial heterogeneity of environment on the prevention and control of infectious diseases, a spatial-temporal SIS reaction-diffusion model with the nonlinear contact transmission rate is proposed. The nonlinear contact transmission rate is spatially dependent and introduced to describe the impact of media coverage on the transmission dynamics of disease. The basic reproduction number associated with the disease in the heterogeneous environment is established. Our results show that the degree of mass media attention plays an important role in preventing the spreading of infectious diseases. Numerical simulations further confirm our analytical findings.

Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems
Michael Winkler
2017, 22(7): 2777-2793 doi: 10.3934/dcdsb.2017135 +[Abstract](401) +[HTML](24) +[PDF](453.8KB)
Abstract:

We consider the no-flux initial-boundary value problem for Keller-Segel-type chemotaxis growth systems of the form

in a ball \begin{document} $Ω\subset\mathbb{R}^n$ \end{document}, \begin{document} $n≥ 3$ \end{document}, with parameters \begin{document} $χ>0, ρ≥ 0$ \end{document} and \begin{document} $μ>0$ \end{document}.

By means of an argument based on a conditional quasi-energy inequality, it is firstly shown that if \begin{document} $χ=1$ \end{document} is fixed, then for any given \begin{document} $K>0$ \end{document} and \begin{document} $T>0$ \end{document} one can find radially symmetric initial data, possibly depending on \begin{document} $K$ \end{document} and \begin{document} $T$ \end{document}, such that for arbitrary \begin{document} $μ∈ (0, 1)$ \end{document} the corresponding local-in-time classical solution \begin{document} $(u, v)$ \end{document} satisfies

with some \begin{document} $x∈Ω$ \end{document} and \begin{document} $t∈ (0, T)$ \end{document}; in fact, this growth phenomenon is actually identified as being generic in the sense that the set of all initial data having this property is dense in the set of all suitably regular radial initial data in a certain topology.

Secondly, turning a focus on possible effects of large chemotactic sensitivities, on the basis of the above it is shown that when \begin{document} $ρ≥ 0$ \end{document} and \begin{document} $μ>0$ \end{document} are fixed, then for all \begin{document} $L>0, T>0$ \end{document}and \begin{document} $χ>μ$ \end{document} one can fix radial initial data \begin{document} $(u_{0, χ}, v_{0, χ})$ \end{document} which decay in \begin{document} $L^∞(Ω)× W^{1, ∞}(Ω)$ \end{document} as \begin{document} $χ\to∞$ \end{document}, and which are such that for the respective solution \begin{document} $(u_χ, v_χ)$ \end{document} there exist \begin{document} $x∈Ω$ \end{document} and \begin{document} $t∈ (0, T)$ \end{document} fulfilling

Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility
Jinliang Wang, Xianning Liu, Toshikazu Kuniya and Jingmei Pang
2017, 22(7): 2795-2812 doi: 10.3934/dcdsb.2017151 +[Abstract](361) +[HTML](19) +[PDF](486.0KB)
Abstract:

In this paper, we investigate the global asymptotic stability of multi-group SIR and SEIR age-structured models. These models allow the infectiousness and the death rate of susceptible individuals to vary and depend on the susceptibility, with which we can consider the heterogeneity of population. We establish global dynamics and demonstrate that the heterogeneity does not alter the dynamical structure of the basic SIR and SEIR with age-dependent susceptibility. Our results also demonstrate that, for age structured multi-group models considered, the graph-theoretic approach can be successfully applied by choosing an appropriate weighted matrix as well.

Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling
Lianzhang Bao and Wenjie Gao
2017, 22(7): 2813-2829 doi: 10.3934/dcdsb.2017152 +[Abstract](317) +[HTML](22) +[PDF](401.3KB)
Abstract:

This work deals with the properties of the traveling wave solutions of a double degenerate cross-diffusion model

where \begin{document} $p≥q 0, q>1, l>1$ \end{document}. This system accounts for degenerate diffusion at the population density \begin{document} $n=b=0$ \end{document} and \begin{document} $b=1$ \end{document} modeling the growth of certain bacteria colony with volume filling. The existence of the finite traveling wave solutions is proven which provides partial answers to the spatial patterns of the colony. In order to overcome the difficulty of traditional phase plane analysis on higher dimension, we use Schauder fixed point theorem and shooting arguments in our paper.

New convergence analysis for assumed stress hybrid quadrilateral finite element method
Binjie Li, Xiaoping Xie and Shiquan Zhang
2017, 22(7): 2831-2856 doi: 10.3934/dcdsb.2017153 +[Abstract](346) +[HTML](18) +[PDF](512.0KB)
Abstract:

New error estimates are established for Pian and Sumihara's (PS) 4-node assumed stress hybrid quadrilateral element [T.H.H. Pian, K. Sumihara, Rational approach for assumed stress finite elements, Int. J. Numer. Methods Engrg., 20 (1984), 1685-1695], which is widely used in engineering computation. Based on an equivalent displacement-based formulation to the PS element, we show that the numerical strain and a postprocessed numerical stress are uniformly convergent with respect to the Lamé constant \begin{document} $λ$ \end{document} on the meshes produced through the uniform bisection procedure. Within this analysis framework, we also show that both the numerical strain and stress are uniformly convergent on meshes which are stable for the \begin{document} $Q_1-P_0$ \end{document} Stokes element.

The stabilized semi-implicit finite element method for the surface Allen-Cahn equation
Xufeng Xiao, Xinlong Feng and Jinyun Yuan
2017, 22(7): 2857-2877 doi: 10.3934/dcdsb.2017154 +[Abstract](450) +[HTML](27) +[PDF](1459.5KB)
Abstract:

Two semi-implicit numerical methods are proposed for solving the surface Allen-Cahn equation which is a general mathematical model to describe phase separation on general surfaces. The spatial discretization is based on surface finite element method while the temporal discretization methods are first-and second-order stabilized semi-implicit schemes to guarantee the energy decay. The stability analysis and error estimate are provided for the stabilized semi-implicit schemes. Furthermore, the first-and second-order operator splitting methods are presented to compare with stabilized semi-implicit schemes. Some numerical experiments including phase separation and mean curvature flow on surfaces are performed to illustrate stability and accuracy of these methods.

Asymptotic behaviour of an age and infection age structured model for the propagation of fungal diseases in plants
Jean-Baptiste Burie, Arnaud Ducrot and Abdoul Aziz Mbengue
2017, 22(7): 2879-2905 doi: 10.3934/dcdsb.2017155 +[Abstract](327) +[HTML](13) +[PDF](511.8KB)
Abstract:

A mathematical model describing the propagation of fungal diseases in plants is proposed. The model takes into account both chronological age and age since infection. We investigate and fully characterize the large time behaviour of the solutions. Existence of a unique endemic stationary state is ensured by a threshold condition: \begin{document}$\mathcal R_0>1$\end{document}. Then using Lyapounov arguments, we prove that if \begin{document}$\mathcal R_0 ≤ 1$\end{document} the disease free stationary state is globally stable while when \begin{document}$\mathcal R_0>1$\end{document}, the unique endemic stationary state is globally stable with respect to a suitable set of initial data.

LaSalle type stationary oscillation theorems for Affine-Periodic Systems
Hongren Wang, Xue Yang, Yong Li and Xiaoyue Li
2017, 22(7): 2907-2921 doi: 10.3934/dcdsb.2017156 +[Abstract](447) +[HTML](30) +[PDF](377.2KB)
Abstract:

The paper concerns the existence of affine-periodic solutions for affine-periodic (functional) differential systems, which is a new type of quasi-periodic solutions if they are bounded. Some more general criteria than LaSalle's one on the existence of periodic solutions are established. Some applications are also given.

Exponential stability of solutions for retarded stochastic differential equations without dissipativity
Min Zhu, Panpan Ren and Junping Li
2017, 22(7): 2923-2938 doi: 10.3934/dcdsb.2017157 +[Abstract](347) +[HTML](23) +[PDF](397.0KB)
Abstract:

This work focuses on a class of retarded stochastic differential equations that need not satisfy dissipative conditions. The principle technique of our investigation is to use variation-of-constants formula to overcome the difficulties due to the lack of the information at the current time. By using variation-of-constants formula and estimating the diffusion coefficients we give sufficient conditions for $p$-th moment exponential stability, almost sure exponential stability and convergence of solutions from different initial value. Finally, we provide two examples to illustrate the effectiveness of the theoretical results.

Minimization of carbon abatement cost: Modeling, analysis and simulation
Xiaoli Yang, Jin Liang and Bei Hu
2017, 22(7): 2939-2969 doi: 10.3934/dcdsb.2017158 +[Abstract](546) +[HTML](22) +[PDF](1179.0KB)
Abstract:

In this paper, we consider a problem of minimizing the carbon abatement cost of a country. Two models are built within the stochastic optimal control framework based on two types of abatement policies. The corresponding HJB equations are deduced, and the existence and uniqueness of their classical solutions are established by PDE methods. Using parameters in the models obtained from real data, we carried out numerical simulations via semi-implicit method. Then we discussed the properties of the optimal policies and minimal costs. Our results suggest that a country needs to keep a relatively low economy and population growth rate and keep a stable economy in order to reduce the total carbon abatement cost. In the long run, it's better for a country to seek for more efficient carbon abatement techniques and an environmentally friendly way of economic development.

Interaction between water and plants: Rich dynamics in a simple model
Xiaoli Wang, Junping Shi and Guohong Zhang
2017, 22(7): 2971-3006 doi: 10.3934/dcdsb.2017159 +[Abstract](757) +[HTML](23) +[PDF](1303.1KB)
Abstract:

An ordinary differential equation model describing interaction of water and plants in ecosystem is proposed. Despite its simple looking, it is shown that the model possesses surprisingly rich dynamics including multiple stable equilibria, backward bifurcation of positive equilibria, supercritical or subcritical Hopf bifurcations, bubble loop of limit cycles, homoclinic bifurcation and Bogdanov-Takens bifurcation. We classify bifurcation diagrams of the system using the rain-fall rate as bifurcation parameter. In the transition from global stability of bare-soil state for low rain-fall to the global stability of high vegetation state for high rain-fall rate, oscillatory states or multiple equilibrium states can occur, which can be viewed as a new indicator of catastrophic environmental shift.

Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations
Hongmei Cheng and Rong Yuan
2017, 22(7): 3007-3022 doi: 10.3934/dcdsb.2017160 +[Abstract](370) +[HTML](20) +[PDF](418.3KB)
Abstract:

In this paper, we mainly discuss the existence and asymptotic stability of traveling fronts for the nonlocal evolution equations. With the monostable assumption, we obtain that there exists a constant \begin{document}$c^*>0$\end{document}, such that the equation has no traveling fronts for \begin{document}$0<c<c^*$\end{document} and a traveling front for each cc*. For \begin{document}$c>c^*$\end{document}, we will further show that the traveling front is globally asymptotic stable and is unique up to translation. If we applied to some differential equations or integro-differential equations, our results recover and/or complement a number of existing ones.

2017  Impact Factor: 0.972

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