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Discrete & Continuous Dynamical Systems - B

2015 , Volume 20 , Issue 7

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Optimal linear stability condition for scalar differential equations with distributed delay
Samuel Bernard and Fabien Crauste
2015, 20(7): 1855-1876 doi: 10.3934/dcdsb.2015.20.1855 +[Abstract](103) +[PDF](583.9KB)
Linear scalar differential equations with distributed delays appear in the study of the local stability of nonlinear differential equations with feedback, which are common in biology and physics. Negative feedback loops tend to promote oscillations around steady states, and their stability depends on the particular shape of the delay distribution. Since in applications the mean delay is often the only reliable information available about the distribution, it is desirable to find conditions for stability that are independent from the shape of the distribution. We show here that for a given mean delay, the linear equation with distributed delay is asymptotically stable if the associated differential equation with a discrete delay is asymptotically stable. We illustrate this criterion on a compartment model of hematopoietic cell dynamics to obtain sufficient conditions for stability.
Functionals-preserving cosine families generated by Laplace operators in C[0,1]
Adam Bobrowski, Adam Gregosiewicz and Małgorzata Murat
2015, 20(7): 1877-1895 doi: 10.3934/dcdsb.2015.20.1877 +[Abstract](59) +[PDF](592.5KB)
Let \( C[0,1] \) be the space of continuous functions on the unit interval \( [0,1] \). A cosine family $\{C(t), t \in \mathbb{R}\}$ in $C[0,1]$ is said to be Laplace-operator generated, if its generator is a restriction of the Laplace operator $L\colon f \mapsto f''$ to a suitable subset of $C^2[0,1].$ The family is said to preserve a functional $F \in (C[0,1])^*$ if for all $f \in C[0,1]$ and $t \in \mathbb{R}, $ $FC(t)f = Ff.$ We study a class of pairs of functionals such that for each member of this class there is a unique Laplace-operator generated cosine family that preserves both functionals in the pair.
Long-time behavior of solutions of a BBM equation with generalized damping
Jean-Paul Chehab, Pierre Garnier and Youcef Mammeri
2015, 20(7): 1897-1915 doi: 10.3934/dcdsb.2015.20.1897 +[Abstract](118) +[PDF](1287.8KB)
We study the long-time behavior of the solution of a damped BBM equation $u_t + u_x - u_{xxt} + uu_x + \mathscr{L}_{\gamma}(u) = 0$. The proposed dampings $\mathscr{L}_{\gamma}$ generalize standards ones, as parabolic ($\mathscr{L}_{\gamma}(u)=-\Delta u$) or weak damping ($\mathscr{L}_{\gamma}(u)=\gamma u$) and allows us to consider a greater range. After establish the local well-posedness in the energy space, we investigate some numerical properties.
Chaos in a model for masting
Kaijen Cheng and Kenneth Palmer
2015, 20(7): 1917-1932 doi: 10.3934/dcdsb.2015.20.1917 +[Abstract](53) +[PDF](440.1KB)
Isagi et al introduced a model for masting, that is, the intermittent production of flowers and fruit by trees. A tree produces flowers and fruit only when the stored energy exceeds a certain threshold value. If flowers and fruit are not produced, the stored energy increases by a certain fixed amount; if flowers and fruit are produced, the energy is depleted by an amount proportional to the excess stored energy. Thus a one-dimensional model is derived for the amount of stored energy. When the ratio of the amount of energy used for flowering and fruit production in a reproductive year to the excess amount of stored energy before that year is small, the stored energy approaches a constant value as time passes. However when this ratio is large, the amount of stored energy varies unpredictably and as the ratio increases the range of possible values for the stored energy increases also. In this article we describe this chaotic behavior precisely with complete proofs.
Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period
Yuncherl Choi, Jongmin Han and Chun-Hsiung Hsia
2015, 20(7): 1933-1957 doi: 10.3934/dcdsb.2015.20.1933 +[Abstract](71) +[PDF](501.4KB)
In this paper, we study bifurcation of the damped Kuramoto-Sivashinsky equation on an odd periodic interval of period $2\lambda$. We fix the control parameter $\alpha \in (0,1)$ and study how the equation bifurcates to attractors as $\lambda$ varies. Using the center manifold analysis, we prove that the bifurcated attractors are homeomorphic to $S^1$ and consist of four or eight singular points and their connecting orbits. We verify the structure of the bifurcated attractors by investigating the stability of each singular point.
Quasi-effective stability for a nearly integrable volume-preserving mapping
Fuzhong Cong and Hongtian Li
2015, 20(7): 1959-1970 doi: 10.3934/dcdsb.2015.20.1959 +[Abstract](93) +[PDF](372.9KB)
This paper is concerned with the stability of the orbits for a nearly integrable volume-preserving mapping. We prove that the nearly integrable volume-preserving mapping possesses quasi-effective stability under the classical KAM-type nondegeneracy, that is, there is an open subset of the phase space whose measure is nearly full, such that the considered mapping is effective stable on this subset. This announces a connection between the Nekhoroshev theory and KAM theory.
Lyapunov functions and global stability for a discretized multigroup SIR epidemic model
Deqiong Ding, Wendi Qin and Xiaohua Ding
2015, 20(7): 1971-1981 doi: 10.3934/dcdsb.2015.20.1971 +[Abstract](100) +[PDF](353.3KB)
In this paper, a discretized multigroup SIR epidemic model is constructed by applying a nonstandard finite difference schemes to a class of continuous time multigroup SIR epidemic models. This discretization scheme has the same dynamics with the original differential system independent of the time step, such as positivity of the solutions and the stability of the equilibria. Discrete-time analogue of Lyapunov functions is introduced to show that the global asymptotic stability is fully determined by the basic reproduction number $R_0$.
Reorientation of smectic a liquid crystals by magnetic fields
Carlos J. García-Cervera and Sookyung Joo
2015, 20(7): 1983-2000 doi: 10.3934/dcdsb.2015.20.1983 +[Abstract](62) +[PDF](11824.3KB)
We consider the de Gennes' smectic A free energy with a complex order parameter in order to study the influence of magnetic fields on the smectic layers in the strong field limit as well as near the critical field. In previous work by the authors [6], the critical field and a description of the layer undulations at the instability were obtained using $\Gamma$-convergence and bifurcation theory. It was proved that the critical field is lowered by a factor of $\sqrt{\pi}$ compared to the classical Helfrich Hurault theory by using natural boundary conditions for the complex order parameter, but still with strong anchoring condition for the director. In this paper, we present numerical simulations for undulations at the critical field as well as the layer and director configurations well above the critical field. We show that the estimate of the critical field and layer configuration at the critical field agree with the analysis in [6]. Furthermore, the changes in smectic order density as well as layer and director will be illustrated numerically as the field increases well above the critical field. This provides the smectic layers' melting along the bounding plates where the layers are fixed. In the natural case, at a high field, we prove that the directors align with the applied field and the layers are homeotropically aligned in the domain, keeping the smectic order density at a constant in $L^2$.
Analysis of a model for bent-core liquid crystals columnar phases
Tiziana Giorgi and Feras Yousef
2015, 20(7): 2001-2026 doi: 10.3934/dcdsb.2015.20.2001 +[Abstract](62) +[PDF](1313.8KB)
We consider a model originally introduced to study layer-undulated structures in bent-core molecule liquid crystals. We first prove existence of minimizers, then analyze a simplified version used to study how in columnar phases the width of the column affects the type of switching, which occurs under an applied electric field. We show via $\Gamma$-convergence that as the width of the column tends to infinity, rotation around the tilt cone is favored, provided the coefficient of the coupling term, between the polar parameter, the nematic parameter, and the layer normal is large.
Protection zone in a modified Lotka-Volterra model
Xiao He and Sining Zheng
2015, 20(7): 2027-2038 doi: 10.3934/dcdsb.2015.20.2027 +[Abstract](92) +[PDF](369.6KB)
This paper studies the dynamic behavior of solutions to a modified Lotka-Volterra reaction-diffusion system with homogeneous Neumann boundary conditions, for which a protection zone should be created to prevent the extinction of the prey only if the prey's growth rate is small. We find a critical size of the protection zone, determined by the ratio of the predation rate and the refuge ability, to ensure the existence, uniqueness and global asymptotic stability of positive steady states for general predator's growth rate $\mu>0$. Bellow the critical size the dynamics of the model would be similar to the case without protection zones. The known uniqueness results for the protection problems with other functional responses, e.g., Holling II model, Leslie model, Beddington-DeAngelis model, were all required that the predator's growth rate $\mu>0$ is large enough. Such a large $\mu$ assumption is not needed for the uniqueness and asymptotic results to the modified Lotka-Volterra reaction-diffusion system considered in this paper.
The reaction-diffusion system for an SIR epidemic model with a free boundary
Haomin Huang and Mingxin Wang
2015, 20(7): 2039-2050 doi: 10.3934/dcdsb.2015.20.2039 +[Abstract](344) +[PDF](397.3KB)
The reaction-diffusion system for an $SIR$ epidemic model with a free boundary is studied. This model describes a transmission of diseases. The existence, uniqueness and estimates of the global solution are discussed first. Then some sufficient conditions for the disease vanishing are given. With the help of investigating the long time behavior of solution to the initial and boundary value problem in half space, the long time behavior of the susceptible population $S$ is obtained for the disease vanishing case.
Dynamics of stochastic fractional Boussinesq equations
Jianhua Huang, Tianlong Shen and Yuhong Li
2015, 20(7): 2051-2067 doi: 10.3934/dcdsb.2015.20.2051 +[Abstract](65) +[PDF](423.2KB)
The current paper is devoted to the asymptotic behavior of the stochastic fractional Boussinesq equations (SFBE). The global well-posedness of SFBE is proved, and the existence of a random attractor for the random dynamical system generalized by the SFBE are also provided.
Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion
Dan Li, Jing'an Cui and Yan Zhang
2015, 20(7): 2069-2088 doi: 10.3934/dcdsb.2015.20.2069 +[Abstract](70) +[PDF](439.1KB)
Using stochastic differential equations with Lévy jumps, this paper studies the effect of environmental stochasticity and random catastrophes on the permanence of Lotka-Volterra facultative systems. Under certain simple assumptions, we establish the sufficient conditions for weak permanence in the mean and extinction of the non-autonomous system, respectively. In particular, a necessary and sufficient condition for permanence and extinction of autonomous system with jump-diffusion are obtained. We generalize some former results under weaker assumptions. Finally, we discuss the biological implications of the main results.
The spreading fronts in a mutualistic model with advection
Mei Li and Zhigui Lin
2015, 20(7): 2089-2105 doi: 10.3934/dcdsb.2015.20.2089 +[Abstract](76) +[PDF](487.3KB)
This paper is concerned with a system of semilinear parabolic equations with two free boundaries, which describe the spreading fronts of the invasive species in a mutualistic ecological model. The advection term is introduced to model the behavior of the invasive species in one dimension space. The local existence and uniqueness of a classical solution are obtained and the asymptotic behavior of the free boundary problem is studied. Our results indicate that for small advection, two free boundaries tend monotonically to finite limits or infinities at the same time, and a spreading-vanishing dichotomy holds, namely, either the expanding environment is limited and the invasive species dies out, or the invasive species spreads to all new environment and establishes itself in a long run. Moreover, some rough estimates of the spreading speed are also given when spreading happens.
Existence and uniqueness of steady flows of nonlinear bipolar viscous fluids in a cylinder
Allen Montz, Hamid Bellout and Frederick Bloom
2015, 20(7): 2107-2128 doi: 10.3934/dcdsb.2015.20.2107 +[Abstract](55) +[PDF](436.6KB)
The existence and uniqueness of solutions to the boundary-value problem for steady Poiseuille flow of an isothermal, incompressible, nonlinear bipolar viscous fluid in a cylinder of arbitrary cross-section is established. Continuous dependence of solutions, in an appropriate norm, is also established with respect to the constitutive parameters of the bipolar fluid model, as these parameters converge to zero, under the additional assumption that the cylinder has a circular cross-section.
Competition for one nutrient with recycling and allelopathy in an unstirred chemostat
Hua Nie and Feng-Bin Wang
2015, 20(7): 2129-2155 doi: 10.3934/dcdsb.2015.20.2129 +[Abstract](63) +[PDF](797.6KB)
In this paper, we study a PDE model of two species competing for a single limiting nutrient resource in a chemostat in which one microbial species excretes a toxin that increases the mortality of another. Our goal is to understand the role of spatial heterogeneity and allelopathy in blooms of harmful algae. We first demonstrate that the two-species system and its single species subsystem satisfy a mass conservation law that plays an important role in our analysis. We investigate the possibilities of bistability and coexistence for the two-species system by appealing to the method of topological degree in cones and the theory of uniform persistence. Numerical simulations confirm the theoretical results.
Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion
Yong Ren, Xuejuan Jia and Lanying Hu
2015, 20(7): 2157-2169 doi: 10.3934/dcdsb.2015.20.2157 +[Abstract](91) +[PDF](386.9KB)
In this paper, we establish the $p$-th moment exponential stability and quasi sure exponential stability of the solutions to impulsive stochastic differential equations driven by $G$-Brownian motion (IGSDEs in short) by means of $G$-Lyapunov function method. An example is presented to illustrate the efficiency of the obtained results.
The modeling error of well treatment for unsteady flow in porous media
Ting Zhang
2015, 20(7): 2171-2185 doi: 10.3934/dcdsb.2015.20.2171 +[Abstract](48) +[PDF](380.0KB)
In petroleum engineering, the well is usually treated as a point or line source, since its radius is much smaller than the scale of the whole reservoir. In this paper, we consider the modeling error of this treatment for unsteady flow in porous media.
On the Budyko-Sellers energy balance climate model with ice line coupling
James Walsh and Christopher Rackauckas
2015, 20(7): 2187-2216 doi: 10.3934/dcdsb.2015.20.2187 +[Abstract](132) +[PDF](981.2KB)
Over 40 years ago, M. Budyko and W. Sellers independently introduced low-order climate models that continue to play an important role in the mathematical modeling of climate. Each model has one spatial variable, and each was introduced to investigate the role ice-albedo feedback plays in influencing surface temperature. This paper serves in part as a tutorial on the Budyko-Sellers model, with particular focus placed on the coupling of this model with an ice sheet that is allowed to respond to changes in temperature, as introduced in recent work by E. Widiasih. We review known results regarding the dynamics of this coupled model, with both continuous (``Sellers-type") and discontinuous (``Budyko-type") equations. We also introduce two new Budyko-type models that are highly effective in modeling the extreme glacial events of the Neoproterozoic Era. We prove in each case the existence of a stable equilibrium solution for which the ice sheet edge rests in tropical latitudes. Mathematical tools used in the analysis include geometric singular perturbation theory and Filippov's theory of differential inclusions.
Global stability of the dengue disease transmission models
Jing-Jing Xiang, Juan Wang and Li-Ming Cai
2015, 20(7): 2217-2232 doi: 10.3934/dcdsb.2015.20.2217 +[Abstract](105) +[PDF](412.3KB)
In this paper, we further investigate the global stability of the dengue transmission models. By using persistence theory, it is showed that the disease of system uniformly persists when the basic reproduction number is larger than unity. By constructing suitable Lyapunov function methods and LaSalle Invariance Principle, we show that the unique endemic equilibrium of the model is always globally asymptotically stable as long as it exists.
Strong averaging principle for slow-fast SPDEs with Poisson random measures
Jie Xu, Yu Miao and Jicheng Liu
2015, 20(7): 2233-2256 doi: 10.3934/dcdsb.2015.20.2233 +[Abstract](56) +[PDF](451.6KB)
This work concerns the problem associated with an averaging principle for two-time-scales stochastic partial differential equations (SPDEs) driven by cylindrical Wiener processes and Poisson random measures. Under suitable dissipativity conditions, the existence of an averaging equation eliminating the fast variable for the coupled system is proved, and as a consequence, the system can be reduced to a single SPDE with a modified coefficient. Moreover, it is shown that the slow component mean-square strongly converges to the solution of the corresponding averaging equation.
Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion
Yong Xu, Bin Pei and Rong Guo
2015, 20(7): 2257-2267 doi: 10.3934/dcdsb.2015.20.2257 +[Abstract](220) +[PDF](366.3KB)
This paper investigates the stochastic averaging of slow-fast dynamical systems driven by fractional Brownian motion with the Hurst parameter $H$ in the interval $(\frac{1}{2},1)$. We establish an averaging principle by which the obtained simplified systems (the so-called averaged systems) will be applied to replace the original systems approximately through their solutions. Here, the solutions to averaged equations of slow variables which are unrelated to fast variables can converge to the solutions of slow variables to the original slow-fast dynamical systems in the sense of mean square. Therefore, the dimension reduction is realized since the solutions of uncoupled averaged equations can substitute that of coupled equations of the original slow-fast dynamical systems, namely, the asymptotic solutions dynamics will be obtained by the proposed stochastic averaging approach.
Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses
Wen-Bin Yang, Yan-Ling Li, Jianhua Wu and Hai-Xia Li
2015, 20(7): 2269-2290 doi: 10.3934/dcdsb.2015.20.2269 +[Abstract](69) +[PDF](511.1KB)
The paper is concerned with a diffusive food chain model subject to homogeneous Robin boundary conditions, which models the trophic interactions of three levels. Using the fixed point index theory, we obtain the existence and uniqueness for coexistence states. Moreover, the existence of the global attractor and the extinction for the time-dependent model are established under certain assumptions. Some numerical simulations are done to complement the analytical results.

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