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Discrete & Continuous Dynamical Systems - B

2013 , Volume 18 , Issue 7

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Interaction of waves in a one dimensional stochastic PDE model of excitable media
Hasan Alzubaidi and  Tony Shardlow
2013, 18(7): 1735-1754 doi: 10.3934/dcdsb.2013.18.1735 +[Abstract](31) +[PDF](1245.4KB)
Abstract:
We study the Barkley PDE model of excitable media in a one dimensional periodic domain with additive space time noise. Regions of excitation and kinks (i.e., boundaries between regions of excitation) form due to the additive noise and propagate due to the underlying dynamics of the excitable media. We study the resulting distribution of excitation and kinks by developing a reduced model.
A Rikitake type system with one control
Tudor Bînzar and  Cristian Lăzureanu
2013, 18(7): 1755-1776 doi: 10.3934/dcdsb.2013.18.1755 +[Abstract](31) +[PDF](2805.9KB)
Abstract:
A Rikitake type system with one control is defined and some of this geometrical and dynamical properties are pointed out.
Derivation of SDES for a macroevolutionary process
Ummugul Bulut and  Edward J. Allen
2013, 18(7): 1777-1792 doi: 10.3934/dcdsb.2013.18.1777 +[Abstract](23) +[PDF](312.2KB)
Abstract:
Systems of ordinary stochastic differential equations (SDEs) are derived that describe the evolutionary dynamics of genera and species. Two different hypotheses are made in the model construction, specifically, the rate of change of the number of genera is either proportional to the number of genera in the family or is proportional to the number of species in the family. Asymptotic and exact mean numbers of species per genera are derived for both hypotheses. Computational results for the derived systems of SDEs agree well with the observed results for several families. Moreover, for each family, the SDE models yield estimates of variability in the processes which are difficult to obtain using classical methods to study the dynamics of species and genera formation.
Attractivity for neutral functional differential equations
Tomás Caraballo and  Gábor Kiss
2013, 18(7): 1793-1804 doi: 10.3934/dcdsb.2013.18.1793 +[Abstract](34) +[PDF](347.7KB)
Abstract:
We study the long term dynamics of non-autonomous functional differential equations. Namely, we establish existence results on pullback attractors for non-linear neutral functional differential equations with time varying delays. The two main results differ in smoothness properties of delay functions.
Effects of white noise in multistable dynamics
Xinfu Chen , Carey Caginalp , Jianghao Hao and  Yajing Zhang
2013, 18(7): 1805-1825 doi: 10.3934/dcdsb.2013.18.1805 +[Abstract](28) +[PDF](489.7KB)
Abstract:
We study the invariant measure of multistable dynamics under the influence of white noise. We show that the invariant measure exists and in the limit of vanishing white noise, the invariant measure approaches a Dirac type measure concentrated at the most stable equilibria if fluctuations are uniform; however, a lesser stable equilibrium may be selected by the fluctuation if its ability to fluctuate is sufficiently smaller than other stable equilibria. Certain related mathematical issues are also addressed.
Phase-locked trajectories for dynamical systems on graphs
Jeremias Epperlein and  Stefan Siegmund
2013, 18(7): 1827-1844 doi: 10.3934/dcdsb.2013.18.1827 +[Abstract](30) +[PDF](471.9KB)
Abstract:
We prove a general result on the existence of periodic trajectories of systems of difference equations with finite state space which are phase-locked on certain components which correspond to cycles in the coupling structure. A main tool is the new notion of order-induced graph which is similar in spirit to a Lyapunov function. To develop a coherent theory we introduce the notion of dynamical systems on finite graphs and show that various existing neural networks, threshold networks, reaction-diffusion automata and Boolean monomial dynamical systems can be unified in one parametrized class of dynamical systems on graphs which we call threshold networks with refraction. For an explicit threshold network with refraction and for explicit cyclic automata networks we apply our main result to show the existence of phase-locked solutions on cycles.
Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise
Georgios T. Kossioris and  Georgios E. Zouraris
2013, 18(7): 1845-1872 doi: 10.3934/dcdsb.2013.18.1845 +[Abstract](42) +[PDF](544.2KB)
Abstract:
We consider an initial- and Dirichlet boundary- value problem for a linear Cahn-Hilliard-Cook equation, in one space dimension, forced by the space derivative of a space-time white noise. First, we propose an approximate stochastic parabolic problem discretizing the noise using linear splines. Then we construct fully-discrete approximations to the solution of the approximate problem using, for the discretization in space, a Galerkin finite element method based on $H^2-$piecewise polynomials, and, for time-stepping, the Backward Euler method. We derive strong a priori estimates: for the error between the solution to the problem and the solution to the approximate problem, and for the numerical approximation error of the solution to the approximate problem.
Long-time behaviour of a perturbed SIR model by white noise
Yuguo Lin and  Daqing Jiang
2013, 18(7): 1873-1887 doi: 10.3934/dcdsb.2013.18.1873 +[Abstract](77) +[PDF](439.6KB)
Abstract:
In this paper, we consider a stochastic SIR model with perturbed disease transmission coefficient. We present sufficient conditions for the disease to extinct exponentially. In the case of persistence, we analyze long-time behaviour of densities of the distributions of the solution. We will prove that the densities of the solution can converge in $L^1$ to an invariant density under appropriate conditions. Also we find the support of the invariant density. Specially, when the intensity of white noise is relatively small, we find a new threshold for an epidemic to occur.
Optimal stochastic differential games with VaR constraints
Jingzhen Liu and  Ka-Fai Cedric Yiu
2013, 18(7): 1889-1907 doi: 10.3934/dcdsb.2013.18.1889 +[Abstract](32) +[PDF](472.9KB)
Abstract:
The nonlinear dynamic games between competing insurance companies are interesting and important problems because of the general practice of using re-insurance to reduce risks in the insurance industry. This problem becomes more complicated if a proper risk control is imposed on all the involving companies. In order to understand the dynamical properties, we consider the stochastic differential game between two insurance companies with risk constraints. The companies are allowed to purchase proportional reinsurance and invest their money into both risk free asset and risky (stock) asset. The competition between the two companies is formulated as a two player (zero-sum) stochastic differential game. One company chooses the optimal reinsurance and investment strategy in order to maximize the expected payoff, and the other one tries to minimize this value. For the purpose of risk management, the risk arising from the whole portfolio is constrained to some level. By the principle of dynamic programming, the problem is reduced to solving the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations for Nash equilibria. We derive the Nash equilibria explicitly and obtain closed form solutions to HJBI under different scenarios.
A mathematical model for control of vector borne diseases through media campaigns
A. K. Misra , Anupama Sharma and  Jia Li
2013, 18(7): 1909-1927 doi: 10.3934/dcdsb.2013.18.1909 +[Abstract](113) +[PDF](623.3KB)
Abstract:
Vector borne diseases spread rapidly in the population. Hence their control intervention must work quickly and target large area as well. A rational approach to combat these diseases is mobilizing people and making them aware through media campaigns. In the present paper, a non-linear mathematical model is proposed to assess the impact of creating awareness by the media on the spread of vector borne diseases. It is assumed that as a response to awareness, people will not only try to protect themselves but also take some potential steps to inhibit growth of vectors in the environment. The model is analyzed using stability theory of differential equations and numerical simulation. The equilibria and invasion threshold for infection i.e., basic reproduction number, has been obtained. It is found that the presence of awareness in the population makes the disease invasion difficult. Also, continuous efforts by the media along with the swift dissemination of awareness can completely eradicate the disease from the system.
Mean-field backward stochastic Volterra integral equations
Yufeng Shi , Tianxiao Wang and  Jiongmin Yong
2013, 18(7): 1929-1967 doi: 10.3934/dcdsb.2013.18.1929 +[Abstract](66) +[PDF](525.8KB)
Abstract:
Mean-field backward stochastic Volterra integral equations (MF-BSVIEs, for short) are introduced and studied. Well-posedness of MF-BSVIEs in the sense of introduced adapted M-solutions is established. Two duality principles between linear mean-field (forward) stochastic Volterra integral equations (MF-FSVIEs, for short) and MF-BSVIEs are obtained. A Pontryagin's type maximum principle is established for an optimal control of MF-FSVIEs.
Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model
Fei-Ying Yang , Yan Li , Wan-Tong Li and  Zhi-Cheng Wang
2013, 18(7): 1969-1993 doi: 10.3934/dcdsb.2013.18.1969 +[Abstract](36) +[PDF](444.7KB)
Abstract:
In this paper, we consider a Kermack-McKendrick epidemic model with nonlocal dispersal. We find that the existence and nonexistence of traveling wave solutions are determined by the reproduction number. To prove the existence of nontrivial traveling wave solutions, we construct an invariant cone in a bounded domain with initial functions being defined on, and apply Schauder's fixed point theorem as well as limiting argument. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Moreover, the nonexistence of traveling wave solutions is obtained by Laplace transform if the speed is less than the critical velocity.
An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system
Zaihong Wang , Jin Li and  Tiantian Ma
2013, 18(7): 1995-1997 doi: 10.3934/dcdsb.2013.18.1995 +[Abstract](28) +[PDF](252.5KB)
Abstract:
In this paper, we point out an error in the paper: Positive periodic solution for Brillouin electron beam focusing system, Discrete Contin. Dyn. Syst. Ser. B, 16(2011), 385-392. Meanwhile, it is pointed out that, for $0 < a < 1$, the conjecture that the Brillouin electron beam focusing system $x''+a(1+\cos 2t)x=1/x$ admits positive periodic solutions is still an open problem.

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