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Discrete & Continuous Dynamical Systems - B

2012 , Volume 17 , Issue 1

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Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions
Grégoire Allaire , Yves Capdeboscq and  Marjolaine Puel
2012, 17(1): 1-31 doi: 10.3934/dcdsb.2012.17.1 +[Abstract](49) +[PDF](717.8KB)
We study the asymptotic behavior of the first eigenvalue and eigenfunction of a one-dimensional periodic elliptic operator with Neumann boundary conditions. The second order elliptic equation is not self-adjoint and is singularly perturbed since, denoting by $\epsilon$ the period, each derivative is scaled by an $\epsilon$ factor. The main difficulty is that the domain size is not an integer multiple of the period. More precisely, for a domain of size $1$ and a given fractional part $0\leq\delta<1$, we consider a sequence of periods $\epsilon_n=1/(n+\delta)$ with $n\in \mathbb{N}$. In other words, the domain contains $n$ entire periodic cells and a fraction $\delta$ of a cell cut by the domain boundary. According to the value of the fractional part $\delta$, different asymptotic behaviors are possible: in some cases an homogenized limit is obtained, while in other cases the first eigenfunction is exponentially localized at one of the extreme points of the domain.
Linear programming based Lyapunov function computation for differential inclusions
Robert Baier , Lars Grüne and  Sigurđur Freyr Hafstein
2012, 17(1): 33-56 doi: 10.3934/dcdsb.2012.17.33 +[Abstract](142) +[PDF](522.7KB)
We present a numerical algorithm for computing Lyapunov functions for a class of strongly asymptotically stable nonlinear differential inclusions which includes spatially switched systems and systems with uncertain parameters. The method relies on techniques from nonsmooth analysis and linear programming and constructs a piecewise affine Lyapunov function. We provide necessary background material from nonsmooth analysis and a thorough analysis of the method which in particular shows that whenever a Lyapunov function exists then the algorithm is in principle able to compute it. Two numerical examples illustrate our method.
Exponential decay in non-uniform porous-thermo-elasticity model of Lord-Shulman type
Zhong-Jie Han and  Gen-Qi Xu
2012, 17(1): 57-77 doi: 10.3934/dcdsb.2012.17.57 +[Abstract](42) +[PDF](535.8KB)
The spectrum and asymptotic behavior of the non-uniform porous-thermo-elasticity of Lord-Shulman type is considered in this paper. It is shown that the corresponding system operator generates a $C_0$ semigroup of contractions in an appropriate Hilbert space setting. By a detailed spectral analysis, the asymptotic expressions of the spectrum of the system is gotten. Based on the spectral property, the Riesz basis property of the (generalized) eigenvectors is proved, which implies that the system satisfies the spectrum-determined-growth condition. Then the exponential stability of this system is deduced from the distribution of the spectrum.
On computing heteroclinic trajectories of non-autonomous maps
Thorsten Hüls and  Yongkui Zou
2012, 17(1): 79-99 doi: 10.3934/dcdsb.2012.17.79 +[Abstract](41) +[PDF](828.5KB)
We propose an adequate notion of a heteroclinic trajectory in non-autonomous systems that generalizes the notion of a heteroclinic orbit of an autonomous system. A heteroclinic trajectory connects two families of semi-bounded trajectories that are bounded in backward and forward time. We apply boundary value techniques for computing one representative of each family. These approximations allow the construction of projection boundary conditions that enable the calculation of a heteroclinic trajectory with high accuracy. The resulting algorithm is applied to non-autonomous toy models as well as to an example from mathematical biology.
Periodic solutions of a non-divergent diffusion equation with nonlinear sources
Chunhua Jin and  Jingxue Yin
2012, 17(1): 101-126 doi: 10.3934/dcdsb.2012.17.101 +[Abstract](57) +[PDF](306.2KB)
This paper is concerned with the existence of nontrivial and nonnegative periodic solutions of a doubly degenerate and singular parabolic equation in non-divergent form with nonlinear sources. We will determine exact classification for the exponent values of the source, and so, for the nonexistence of nontrivial periodic solutions, as well as the existence of those solutions with compact support, and the existence of positive periodic solutions.
Qualitative analysis of a diffusive prey-predator model with trophic interactions of three levels
Huiling Li , Peter Y. H. Pang and  Mingxin Wang
2012, 17(1): 127-152 doi: 10.3934/dcdsb.2012.17.127 +[Abstract](31) +[PDF](458.6KB)
In this paper, we consider a mathematical model for a prey-predator dynamical system with diffusion and trophic interactions of three levels. In this model, a general trophic function based on the ratio between the prey and a linear function of the predator is used at each level. At the two limits of this trophic function, one recovers the classical prey-dependent and ratio-dependent predation models, respectively. We offer a complete discussion of the dynamical behavior of the model under the homogeneous Neumann boundary condition (the same behavior is also seen in the absence of diffusion). We also discuss existence, uniqueness, stability and bifurcation of equilibrium behavior corresponding to positive steady state solutions under the homogeneous Dirichlet boundary condition. Finally, we give interpretations of some of these results in the context of different predation models.
Stability of an efficient Navier-Stokes solver with Navier boundary condition
Jie Liao and  Xiao-Ping Wang
2012, 17(1): 153-171 doi: 10.3934/dcdsb.2012.17.153 +[Abstract](51) +[PDF](401.9KB)
In this paper, we study the stability of an efficient numerical scheme based on pressure separation for the Navier-Stokes equations with Navier slip boundary condition in a bounded domain with smooth boundary. The method was introduced in [7, 8] for the Navier-Stokes equation with no-slip boundary condition which decouples the updates of pressure and velocity through explicit time stepping for pressure. The scheme was shown to be very efficient and unconditionally stable. In this paper, we extend this pressure separation method to the problem with Navier slip boundary condition and prove the unconditional stability of the resulting numerical scheme under certain condition on the curvature of the boundary.
Traveling wave solutions of a competitive recursion
Guo Lin and  Wan-Tong Li
2012, 17(1): 173-189 doi: 10.3934/dcdsb.2012.17.173 +[Abstract](44) +[PDF](408.5KB)
This paper is concerned with the traveling wave solutions of a competitive recursion. By using a cross iteration scheme, we first establish the existence of traveling wave solutions, which are the invasion waves of two competitive invaders. These wave solutions are useful in understanding the long time behavior of solution of the corresponding Cauchy type problem where the initial distribution is a perturbation of the wave profile of a traveling wave solution that may be nonmonotone.
Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension
Rebecca McKay and  Theodore Kolokolnikov
2012, 17(1): 191-220 doi: 10.3934/dcdsb.2012.17.191 +[Abstract](33) +[PDF](807.7KB)
We consider a class of one-dimensional reaction-diffusion systems, \[ \left\{ \begin{array} [ u_{t}=\varepsilon^{2}u_{xx}+f(u,w)\\ \tau w_{t}=Dw_{xx}+g(u,w) \end{array} \right. \] with homogeneous Neumann boundary conditions on a one dimensional interval. Under some generic conditions on the nonlinearities $f,g$ and in the singular limit $\varepsilon\rightarrow0,$ such a system admits a steady state for which $u$ consists of sharp back-to-back interfaces. For a sufficiently large $D$ and for sufficiently small $\tau$, such a steady state is known to be stable in time. On the other hand, it is also known that in the so-called shadow limit $D\rightarrow\infty,$ patterns having more than one interface are unstable. In this paper we analyse in detail the transition between the stable patterns when $D=O(1)$ and the shadow system when $D\rightarrow\infty$. We show that this transition occurs when $D$ is exponentially large in $\varepsilon$ and we derive instability thresholds $D_{1}\gg D_{2}\gg D_{3}\gg\ldots$ such that a periodic pattern with $2K$ interfaces is stable if $D < D_{K}$ and is unstable when $D > D_{K}$. We also study the dynamics of the interfaces when $D$ is exponentially large; this allows us to describe in detail the mechanism leading to the instability. Direct numerical computations of stability and dynamics are performed, and these results are in excellent agreement with corresponding results as predicted by the asymptotic theory.
On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients
Linfeng Mei and  Xiaoyan Zhang
2012, 17(1): 221-243 doi: 10.3934/dcdsb.2012.17.221 +[Abstract](42) +[PDF](173.1KB)
We investigate the steady state solutions of a phytoplankton-nutrient system proposed by Huisman et al. in [14] that models the dynamics of a single phytoplankton species whose growth is limited by light and nutrients in a vertical water column. We first study the existence and nonexistence problem of the model and prove there is at least one positive solution to the system when the parameters involved are in a suitable range. We then analyze the limiting profiles of the positive solutions as the specific phytoplankton loss rate approaches zero and as the diffusion coefficient of the system tends to zero, respectively. In the small diffusion case, we show that the phytoplankton species all die out regardless of how large the nutrient supply is, and the nutrients distribution approaches a linear function determined by the parameters of the system. This phenomenon is in sharp contrast to that of the model studied by Du and Hsu [3, 4], where the phytoplankton species can concentrate at the bottom, at the surface or at a specific depth of the water column decided by the amplitude of the nutrient supply. We also study the asymptotic profile of the positive solution when the diffusion coefficient is very large. Our results reveal that in such a case the phytoplankton and the nutrients distribute evenly in the water column.
    Our concentration results also reveal that passive diffusion and active movement (sinking or floating) should be in proportion to the oscillation phenomena showed in [14, 24] to occur.
Shooting and numerical continuation methods for computing time-minimal and energy-minimal trajectories in the Earth-Moon system using low propulsion
Gautier Picot
2012, 17(1): 245-269 doi: 10.3934/dcdsb.2012.17.245 +[Abstract](33) +[PDF](2572.7KB)
In this article we describe the principle of computations of optimal transfers between quasi-Keplerian orbits in the Earth-Moon system using low-propulsion. The spacecraft's motion is modelled by the equations of the control restricted 3-body problem and we base our work on previous studies concerning the orbit transfer in the two-body problem where geometric and numeric methods were developed to compute optimal solutions. Using numerical simple shooting and continuation methods connected with fundamental results from control theory, such as the Pontryagin Maxium Principle and the second order optimality conditions related to the concept of conjugate points, we compute time-minimal and energy-minimal trajectories between the geostationary initial orbit and a final circular orbit around the Moon, passing through the neighborhood of the libration point $L_1$. Our computations give simple trajectories, obtained by referring to numerical values of the SMART-1 mission.
A simple regulatory circuit that can simultaneously generate excitability of two different mechanisms
Changhong Shi , Han-Xiong Li and  Tianshou Zhou
2012, 17(1): 271-282 doi: 10.3934/dcdsb.2012.17.271 +[Abstract](48) +[PDF](581.5KB)
Coupled positive and negative feedback loops occur in many cellular signaling systems. We show that a two-component circuit with this kind of network structure can simultaneously generate excitability of two different mechanisms that is similar to either integrator or resonator in neuroscience. Moreover, we find that there is an opposite tendency between switching frequencies in the two excitable mechanisms, and the duration and amplitude of the response spike are more resistant to noise in the integrate system than in the resonate system. In addition, we discuss, combining the Bacillis subtilis model organism, some possible biological implications of these differences.
A periodic reaction-diffusion model with a quiescent stage
Feng-Bin Wang
2012, 17(1): 283-295 doi: 10.3934/dcdsb.2012.17.283 +[Abstract](37) +[PDF](367.4KB)
In this paper, we investigate the asymptotic behaviour for a periodic reaction-diffusion model with a quiescent stage. By appealing to the theory of asymptotic speeds of spread and traveling waves for monotone periodic semiflow, we establish the existence of the spreading speed and show that it coincides with the minimal wave speed for monotone periodic traveling waves. Finally, we consider the case where the spatial domain is bounded. A threshold result on the global attractivity of either zero or a positive periodic solution are established.
Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay
Jinliang Wang and  Lijuan Guan
2012, 17(1): 297-302 doi: 10.3934/dcdsb.2012.17.297 +[Abstract](50) +[PDF](281.0KB)
A recent paper [H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete and Continuous Dynamical Systems - Series B, 12(2009), 511--524] presented a mathematical model for HIV-1 infection with intracellular delay and cell-mediated immune response. By combining the analysis of the characteristic equation and the Lyapunov-LaSalle method, they obtain a necessary and sufficient condition for the global stability of the infection-free equilibrium and give sufficient conditions for the local stability of the two infection equilibria: one without CTLs being activated and the other with. In the present paper, we show that the global dynamics are fully determined for $\Re_1<1<\Re_0$ and $\Re_1>1$ (Theorem 4.2 and Theorem 4.3) without other additional conditions. The approach used here, is to use a direct Lyapunov functional and Lyapunov-LaSalle invariance principle.
A passivity-based stability criterion for reaction diffusion systems with interconnected structure
Liming Wang
2012, 17(1): 303-323 doi: 10.3934/dcdsb.2012.17.303 +[Abstract](28) +[PDF](664.3KB)
In this paper, stability of a class of reaction diffusion systems is studied. Conditions on global asymptotic stability of the homogeneous equilibrium are derived based on the diagonal stability of a dissipativity matrix. This work extends previous result on global asymptotic stability from cyclic systems to general systems with interconnected structure. In addition, it reformulates the approach using an "input-output" formalism that makes the results easier to understand and apply. A biological example from the Mitogen-Activated Protein Kinase (MAPK) system is provided at the end to illustrate the new approach and the main result.
Generalized Jacobi rational spectral methods with essential imposition of Neumann boundary conditions in unbounded domains
Zhong-Qing Wang and  Jing-Xia Wu
2012, 17(1): 325-346 doi: 10.3934/dcdsb.2012.17.325 +[Abstract](25) +[PDF](460.9KB)
In this paper, we develop several generalized Jacobi rational spectral methods with essential imposition of Neumann boundary conditions for one/two dimensional Neumann problems. Some basic results on the generalized Jacobi rational approximations for Neumann problems are established, which play important roles in the related spectral methods. Three model problems are considered. The convergence of proposed schemes is proved. Numerical results demonstrate their spectral accuracy and efficiency.
Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay
Shi-Liang Wu , Wan-Tong Li and  San-Yang Liu
2012, 17(1): 347-366 doi: 10.3934/dcdsb.2012.17.347 +[Abstract](52) +[PDF](424.2KB)
This paper is concerned with the exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with nonlocal delay. The existence and comparison theorem of solutions of the corresponding Cauchy problem in a weighted Sobolev space are first established for the systems on $\mathbb{R}$ by appealing to the theory of semigroup and abstract functional differential equations. The exponential stability of traveling fronts is then proved by the comparison principle and the (technical) weighted energy method. Comparing with the previous results, our results recovers and/or improves a number of existing ones. Finally, we apply our results to some biological and epidemic models and obtain some new results.
Stability and Hopf bifurcations for a delayed diffusion system in population dynamics
Xiang-Ping Yan and  Wan-Tong Li
2012, 17(1): 367-399 doi: 10.3934/dcdsb.2012.17.367 +[Abstract](44) +[PDF](1200.6KB)
A generalized two-species Lotka-Volterra reaction-diffusion system with a discrete delay and subject to homogeneous Dirichlet boundary conditions is considered. By regarding the delay as the bifurcation parameter and analyzing in detail the spectrum of the associated linear operator, the stability of the positive steady state bifurcating from the zero solution is studied. In particular, it is shown that the system can undergo a forward Hopf bifurcation at the positive steady state solution when the delay take a sequence of critical values via the implicit function theorem. To verify the obtained theoretical results, some numerical simulations are also included.
Global stability of a five-dimensional model with immune responses and delay
Yincui Yan and  Wendi Wang
2012, 17(1): 401-416 doi: 10.3934/dcdsb.2012.17.401 +[Abstract](63) +[PDF](356.2KB)
In this article, we study a virus model with immune responses and an intracellular delay which is relatively large compared with virus life-cycle and is an indispensable factor in understanding virus infections, such as HIV and HBV infections. By constructing suitable Liapunov functionals, the global dynamics of the model is completely determined by the reproductive numbers for viral infection $R_0$, for CTL immune response $R_1$, for antibody immune response $R_2$, for CTL immune competition $R_3$ and for antibody immune competition $R_4$. The global stability of the model precludes the existence of periodic solution and other complex dynamical behaviors.
Traveling wave solutions in an integro-differential competition model
Liang Zhang and  Bingtuan Li
2012, 17(1): 417-428 doi: 10.3934/dcdsb.2012.17.417 +[Abstract](74) +[PDF](366.3KB)
We study the existence of traveling wave solutions for the two-species Lotka-Volterra competition model in the form of integro-differential equations. The model incorporates asymmetric dispersal kernels that describe long distance dispersal processes of competing species in space. Using lower and upper traveling wave solutions, we show that the model has traveling wave solutions that connect the origin and the coexistence equilibrium with speeds greater than the spreading speed of each species in the absence of its rival.
Blowup, global fast and slow solutions to a parabolic system with double fronts free boundary
Qunying Zhang and  Zhigui Lin
2012, 17(1): 429-444 doi: 10.3934/dcdsb.2012.17.429 +[Abstract](31) +[PDF](411.0KB)
The purpose of this paper is to investigate a reaction-diffusion model with double fronts free boundary. Our approach to the local existence and uniqueness of the solution is based on the contraction mapping theorem. Also we study the blowup property of the solution. The result shows that blowup occurs if the initial datum is large enough. Finally the long-time behavior of global solutions is presented. It is proved that the solution is global and fast, which decays uniformly at an exponential rate if the initial datum is small, while there is a global and slow solution provided that the initial value is suitably large.

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