ISSN:

1531-3492

eISSN:

1553-524X

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## Discrete & Continuous Dynamical Systems - B

May 2020 , Volume 25 , Issue 5

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*+*[Abstract](949)

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**Abstract:**

In present paper, we study the Cauchy problem for a generalized Camassa-Holm equation, which was discovered by Novikov. Our purpose here is to establish persistence properties and some unique continuation properties of the solutions of this equation in weighted spaces.

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**Abstract:**

In the present work, we introduce a control model to describe three species food chain interaction model composed of prey, middle predator, and top predator. The middle predator preys on prey and the top predator preys on middle predator. The control techniques of the exploited natural resources are used to modulate the harvesting effort to avoid high risks of extinction of the middle predator and keep stability of the food chain, by prohibiting fishing when the population density drops below a prescribed threshold. The behavior of the system stability of the regular, virtual, pseudo-equilibrium and tangent points are discussed. The complicated non-smooth dynamic behaviors (sliding and crossing segment and their domains) are analyzed. The bifurcation set of pseudo-equilibrium and the sliding crossing bifurcation have been investigated. Our analytical findings are verified through numerical investigations.

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In this paper we consider a free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure. In our model, the individuals of a new or invasive predatory species are classified as belonging to either the immature or mature case. Firstly, the global existence, uniqueness, regularity of the solution are derived. And then when vanishing happens, we get uniform estimates and the long time behavior of the solution. At last, a sharp criterion governing spreading and vanishing for the free boundary problem is studied by the upper and lower solution method.

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**Abstract:**

In this paper, we consider a nonlinear diffusion problem with variable exponent, accompanied by double free boundaries possessing different moving parameters, where the variable exponent function

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**Abstract:**

In this work a stochastic Holling-Ⅱ type predator-prey model with infinite delays and feedback controls is investigated. By constructing a Lyapunov function, together with stochastic analysis approach, we obtain that the stochastic controlled predator-prey model admits a unique global positive solution. We then utilize graphical method and stability theorem of stochastic differential equations to investigate the globally asymptotical stability of a unique positive equilibrium for the stochastic controlled predator-prey system. If the stochastic predator-prey system is globally stable, then we show that using suitable feedback controls can alter the position of the unique positive equilibrium and retain the stable property. If the predator-prey system is destabilized by large intensities of white noises, then by choosing the appropriate values of feedback control variables, we can make the system reach a new stable state. Some examples are presented to verify our main results.

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**Abstract:**

In this paper, the effect of noise intensity on parabolic equations is considered. We focus on the effect of noise on the energy solutions of stochastic parabolic equations. By utilising Ito's formula and the energy estimate method, we obtain excitation indices of the solution

*+*[Abstract](771)

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**Abstract:**

We propose a method to optimize periodic microstructures for obtaining homogenized materials with negative Poisson ratio, using shape and/or topology variations in the model hole. The proposed approach employs worst case design in order to minimize the Poisson ratio of the (possibly anisotropic) homogenized elastic tensor in several prescribed directions. We use a minimization algorithm for inequality constraints based on an active set strategy and on a new algorithm for solving minimization problems with equality constraints, belonging to the class of null-space gradient methods. It uses first order derivatives of both the objective function and the constraints. The step is computed as a sum between a steepest descent step (minimizing the objective functional) and a correction step related to the Newton method (aiming to solve the equality constraints). The linear combination between these two steps involves coefficients similar to Lagrange multipliers which are computed in a natural way based on the Newton method. The algorithm uses no projection and thus the iterates are not feasible; the constraints are only satisfied in the limit (after convergence). A local convergence result is proven for a general nonlinear setting, where both the objective functional and the constraints are not necessarily convex functions.

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**Abstract:**

In this work, we establish a framework to study the stability of traveling wave solutions for some lattice reaction-diffusion equations. The systems arise from epidemic, biological and many other applied models. Applying different kinds of comparison theorems, we show that all solutions of the Cauchy problem for the lattice differential equations converge exponentially to the traveling wave solutions provided that the initial perturbations around the traveling wave solutions belonging to suitable spaces. Our results can be applied to various discrete reaction-diffusion systems, e.g., the discrete multi-species Lotka-Volterra cooperative model, discrete epidemic model, three-species Lotka-Volterra competitive model, etc.

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In this paper, for the biharmonic eigenvalue problem with clamped boundary condition in

*+*[Abstract](708)

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In this note, by using the theory of bifurcation and Lyapunov function, one performs a qualitative analysis on a novel four-dimensional unified hyperchaotic Lorenz-type system (UHLTS), including stability, pitchfork bifurcation, Hopf bifurcation, singularly degenerate heteroclinic cycle, ultimate bound estimation, global exponential attractive set, heteroclinic orbit and so on. Numerical simulations not only are consistent with the results of theoretical analysis, but also illustrate singularly degenerate heteroclinic cycles with distinct geometrical structures and nearby hyperchaotic attractors in the case of small

*+*[Abstract](706)

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**Abstract:**

Abel equations of the first and second kind have been widely studied, but one question that never has been addressed for the Abel polynomial differential systems is to understand the behavior of its solutions (without knowing explicitly them), or in other words, to obtain its qualitative behavior. This is a very hard task that grows exponentially as the number of parameters in the equation increases. In this paper, using Poincaré compactification we classify the topological phase portraits of a special kind of quadratic differential system, the Abel quadratic equations of third kind. We also describe the maximal number of polynomial solutions that Abel polynomial differential equations can have.

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In this paper we consider discontinuous piecewise linear differential systems whose discontinuity set is a straight line

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In this paper, a reaction-diffusion model with a cyclic structure is studied, which includes the SIS disease-transmission model and the nutrient-phytoplankton model. The minimal wave speed

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In this paper, we investigate a genetic toggle switch in *Escherichia Coli*, which models an artificial double-negative feedback loop with two mutually repressors. This model is a planar differential system with three parameters, one of which is an integer power

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**Abstract:**

The present article proposes a new geometric space in the complex plane of the Mandelbrot set, framed in the diagram of orbits and attractors, to characterize the dynamics of the curves of the demand of daily electrical power, with the purpose of discovering other observations enabling the elevation of new theoretical approaches. The result shows a different method to evaluate the dynamics of the electric power demand curve, using fractal orbital diagrams. This method is a new contribution that extends universal knowledge about the dynamics of complex systems and fractal geometry. Finally, the reader is informed that the data series used in this article was used in a previous publication, but using a different fractal technique to describe its dynamics.

*+*[Abstract](1319)

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The aim of this paper is to design and analyze a nonlinear mechanistic model for chikungunya (CHIKV) and dengue (DENV) co-endemicity. The model can assess the epidemiological consequences of the spread of each disease on the co-infection transmission dynamics. Although the two diseases are different, they exhibit similar dynamical features which show that to combat/control CHIKV virus (or co-infection with DENV virus) we can employ DENV control strategies and vice versa. Our analytical results show that each sub-model and the full model have two disease-free equilibria (i.e., trivial disease-free equilibrium (TDFE) and non-trivial disease-free equilibrium (NTDFE)). Further, qualitative analyses reveal that each of the sub-models exhibits the phenomenon of backward bifurcation (where a stable NTDFE co-exits with a stable endemic equilibrium (EE)). Epidemiologically, this implies that, in each case (CHIKV or DENV), the basic requirement of making the associated reproduction number to be less-than unity is no longer sufficient for the disease eradication. We further highlight that the full model, consisting of twenty-six (26) mutually exclusive compartments representing the human and mosquito dynamics, also exhibits the phenomenon of backward bifurcation. We fit the full model and its sub-models using realistic data from India. Sensitivity analysis using the partial rank correlation coefficient (PRCC) is used for ranking the importance of each parameter-output. The results suggested that the mosquito removal rates, the transmission rates, and the mosquito maturation rate are the top control parameters for combating CHIKV, DENV and CHIKV-DENV co-infection outbreaks.

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**Abstract:**

The relationship between price volatility and expected price market extremum is examined using a fundamental economics model of supply and demand. By examining randomness through a microeconomic setting, we obtain the implications of randomness in the supply and demand, rather than assuming that price has randomness on an empirical basis. Within a general setting of changing fundamentals, the volatility is maximum when expected prices are changing most rapidly, with the maximum of volatility reached prior to the maximum of expected price. A key issue is that randomness arises from the supply and demand, and the variance in the stochastic differential equation governing the logarithm of price must reflect this. Analogous results are obtained by further assuming that the supply and demand are dependent on the deviation from fundamental value of the asset.

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In this paper, we study the traveling wave solutions of a Lotka-Volterra diffusion competition system with nonlocal terms. We prove that there exists traveling wave solutions of the system connecting equilibrium

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As it is well known, the dynamics of the stochastic SIRS epidemic model with mass action is governed by a threshold

*+*[Abstract](598)

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In this paper, we investigate a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment. We first prove that the solution of the model is bounded when the susceptible and infected individuals have same or distinct dispersal rates. The global stability of the constant endemic equilibrium is proved by constructing suitable Lyapunov functionals when all parameters are positive constants. We employ the topological degree argument to show the existence of positive steady state. Most importantly, we have also investigated the asymptotic profiles of the positive steady state as the dispersal rate of susceptible or infected individuals tends to zero or infinity. Our result reveals that a linear source and spontaneous infection can significantly enhance disease persistence no matter what dispersal rate of the susceptible or infected population is small or large, which leads to the situation that when total population number allows to vary, disease becomes more difficult to control.

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**Abstract:**

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