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Discrete & Continuous Dynamical Systems - B

May 2020 , Volume 25 , Issue 5

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Persistence properties for the generalized Camassa-Holm equation
Yongsheng Mi, Boling Guo and Chunlai Mu
2020, 25(5): 1623-1630 doi: 10.3934/dcdsb.2019243 +[Abstract](234) +[HTML](93) +[PDF](364.44KB)

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.

Stability and bifurcation analysis of Filippov food chain system with food chain control strategy
Soliman A. A. Hamdallah and Sanyi Tang
2020, 25(5): 1631-1647 doi: 10.3934/dcdsb.2019244 +[Abstract](276) +[HTML](153) +[PDF](987.57KB)

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.

A free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure
Siyu Liu, Haomin Huang and Mingxin Wang
2020, 25(5): 1649-1670 doi: 10.3934/dcdsb.2019245 +[Abstract](220) +[HTML](97) +[PDF](465.87KB)

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.

A nonlinear Stefan problem with variable exponent and different moving parameters
Huiling Li, Xiaoliu Wang and Xueyan Lu
2020, 25(5): 1671-1698 doi: 10.3934/dcdsb.2019246 +[Abstract](173) +[HTML](81) +[PDF](575.47KB)

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 \begin{document}$ m(x) $\end{document} satisfies that \begin{document}$ m(x)-1 $\end{document} can change its sign. Local existence and uniqueness of solution are established firstly, and then, some sufficient conditions are achieved for finite time blowup, and as well for global existence. Asymptotic behavior is further investigated for global solution, and existences of fast solution and slow solution are presented by making use of upper-sub solutions, energy and scaling arguments.

Influence of feedback controls on the global stability of a stochastic predator-prey model with Holling type Ⅱ response and infinite delays
Kexin Wang
2020, 25(5): 1699-1714 doi: 10.3934/dcdsb.2019247 +[Abstract](208) +[HTML](90) +[PDF](810.75KB)

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.

The effect of noise intensity on parabolic equations
Guangying Lv, Hongjun Gao, Jinlong Wei and Jiang-Lun Wu
2020, 25(5): 1715-1728 doi: 10.3934/dcdsb.2019248 +[Abstract](244) +[HTML](93) +[PDF](425.79KB)

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 \begin{document}$ u $\end{document} at time \begin{document}$ t $\end{document}. Furthermore, we improve existing results by introducing a simple method to verify the existing results in the literature.

A gradient-type algorithm for constrained optimization with application to microstructure optimization
Cristian Barbarosie, Anca-Maria Toader and Sérgio Lopes
2020, 25(5): 1729-1755 doi: 10.3934/dcdsb.2019249 +[Abstract](200) +[HTML](86) +[PDF](973.14KB)

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.

Stability analysis of traveling wave solutions for lattice reaction-diffusion equations
Cheng-Hsiung Hsu and Jian-Jhong Lin
2020, 25(5): 1757-1774 doi: 10.3934/dcdsb.2020001 +[Abstract](184) +[HTML](79) +[PDF](447.72KB)

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.

The two-grid and multigrid discretizations of the $ C^0 $IPG method for biharmonic eigenvalue problem
Hao Li, Hai Bi and Yidu Yang
2020, 25(5): 1775-1789 doi: 10.3934/dcdsb.2020002 +[Abstract](210) +[HTML](83) +[PDF](497.89KB)

In this paper, for the biharmonic eigenvalue problem with clamped boundary condition in \begin{document}$ \mathbb{R}^{2} $\end{document}, we study the two-grid discretization based on shifted-inverse iteration of \begin{document}$ C^0 $\end{document}IPG method. With our scheme, the solution of a biharmonic eigenvalue problem on a fine mesh \begin{document}$ \pi_h $\end{document} can be reduced to the solution of the eigenvalue problem on a coarser mesh \begin{document}$ \pi_H $\end{document} and the solution of a linear algebraic system on the fine mesh \begin{document}$ \pi_h $\end{document}. We prove that the resulting solution still maintains an asymptotically optimal accuracy when \begin{document}$ h\geq O(H^3) $\end{document}. In addition, we also discuss the multigrid discretization and the adaptive \begin{document}$ C^0 $\end{document}IPG algorithm based on Rayleigh quotient iteration. Numerical experiments are provided to validate the theoretical analysis.

Bifurcations, ultimate boundedness and singular orbits in a unified hyperchaotic Lorenz-type system
Haijun Wang and Fumin Zhang
2020, 25(5): 1791-1820 doi: 10.3934/dcdsb.2020003 +[Abstract](198) +[HTML](93) +[PDF](7488.19KB)

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 \begin{document}$ b > 0 $\end{document}, i.e. conjugate hyperchaotic Lorenz-type attractors (CHCLTA) and nearby a short-duration transient of singularly degenerate heteroclinic cycles approaching infinity or singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci and stable node-foci, etc. In particular, by a linear scaling, a possibly new forming mechanism behind the creation of well-known hyperchaotic attractor with \begin{document}$ (a_{1}, a_{2}, c, d, e, f, b, p, q) = (-12, 12, 23,-1, -1, 1, 2.1, -6, -0.2) $\end{document}, consisting of occurrence of degenerate pitchfork bifurcation at \begin{document}$ S_{z} $\end{document}, the change in the stability index of the saddle at the origin as \begin{document}$ b $\end{document} crosses the null value, explosion of normally hyperbolic stable node-foci, collapse of singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci or saddle-nodes, and stable node-foci, is revealed. The findings and results of this paper may provide theoretical support in some future applications, since they improve and complement the known ones.

On the Abel differential equations of third kind
Regilene Oliveira and Cláudia Valls
2020, 25(5): 1821-1834 doi: 10.3934/dcdsb.2020004 +[Abstract](182) +[HTML](81) +[PDF](552.41KB)

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.

On the limit cycles of a class of discontinuous piecewise linear differential systems
Jaume Llibre and Lucyjane de A. S. Menezes
2020, 25(5): 1835-1858 doi: 10.3934/dcdsb.2020005 +[Abstract](191) +[HTML](83) +[PDF](2377.9KB)

In this paper we consider discontinuous piecewise linear differential systems whose discontinuity set is a straight line \begin{document}$ L $\end{document} which does not pass through the origin. These systems are formed by two linear differential systems of the form \begin{document}$ \dot{x} = Ax\pm b $\end{document}. We study the limit cycles of this class of discontinuous piecewise linear differential systems. We do this study by analyzing the fixed points of the return map of the system defined on the straight line \begin{document}$ L $\end{document}. This kind of differential systems appear in control theory.

Traveling waves for a reaction-diffusion model with a cyclic structure
Tianran Zhang
2020, 25(5): 1859-1870 doi: 10.3934/dcdsb.2020006 +[Abstract](183) +[HTML](81) +[PDF](390.17KB)

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 \begin{document}$ c^* $\end{document} of traveling wave solutions is given. The existence of traveling semi-fronts with \begin{document}$ c>c^* $\end{document} is proved by Schauder's fixed-point theorem. The traveling semi-fronts are shown to be bounded by rescaling method and comparison principle. The existence of traveling semi-front with \begin{document}$ c = c^* $\end{document} is obtained by limit arguments. Finally, the traveling semi-fronts are shown to connect to the positive equilibrium by a Lyapunov function.

Transition between monostability and bistability of a genetic toggle switch in Escherichia coli
Jie Li and Weinian Zhang
2020, 25(5): 1871-1894 doi: 10.3934/dcdsb.2020007 +[Abstract](183) +[HTML](72) +[PDF](940.56KB)

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 \begin{document}$ n\ge1 $\end{document}, in the case that repressors 1 and 2 multimerize with \begin{document}$ n $\end{document} and 1 subunits respectively and its equilibria are decided by a polynomial of degree \begin{document}$ n+1 $\end{document}. Since one hardly solves such a polynomial equation, a known result on bistability was given by omitting some small terms under the assumption that the promoters are strong and the expression ratio between the ON state and the OFF state is large. In this paper, determining distribution of zeros qualitatively for the polynomial of high degree, we analytically discuss on the system without the assumption and completely give qualitative properties for all equilibria, which corrects the known result of bistability. Furthermore, we prove that there may occur in the system a codimension 2 bifurcation, called cusp bifurcation, which is a collision of two saddle-node bifurcations and manifests the transition between bistability and monostability. We exhibit the global dynamics of repressors in various cases by analyzing equilibria at infinity and proving nonexistence of closed orbits.

Methodology for the characterization of the electrical power demand curve, by means of fractal orbit diagrams on the complex plane of Mandelbrot set
Héctor A. Tabares-Ospina and Mauricio Osorio
2020, 25(5): 1895-1905 doi: 10.3934/dcdsb.2020008 +[Abstract](190) +[HTML](73) +[PDF](779.78KB)

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.

Dynamical analysis of chikungunya and dengue co-infection model
Salihu Sabiu Musa, Nafiu Hussaini, Shi Zhao and He Daihai
2020, 25(5): 1907-1933 doi: 10.3934/dcdsb.2020009 +[Abstract](484) +[HTML](93) +[PDF](1257.01KB)

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.

Asset price volatility and price extrema
Carey Caginalp and Gunduz Caginalp
2020, 25(5): 1935-1958 doi: 10.3934/dcdsb.2020010 +[Abstract](154) +[HTML](72) +[PDF](492.1KB)

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.

Traveling waves for nonlocal Lotka-Volterra competition systems
Bang-Sheng Han, Zhi-Cheng Wang and Zengji Du
2020, 25(5): 1959-1983 doi: 10.3934/dcdsb.2020011 +[Abstract](251) +[HTML](95) +[PDF](1885.45KB)

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 \begin{document}$ (0, 0) $\end{document} to some unknown positive steady state for wave speed \begin{document}$ c>c^* = \max\left\{2, 2\sqrt{dr}\right\} $\end{document} and there is no such traveling wave solutions for \begin{document}$ c<c^* $\end{document}, where \begin{document}$ d $\end{document} and \begin{document}$ r $\end{document} respectively corresponds to the diffusion coefficients and intrinsic rate of an competition species. Furthermore, we also demonstrate the unknown steady state just is the positive equilibrium of the system when the nonlocal delays only appears in the interspecific competition term, which implies that the nonlocal delay appearing in the interspecific competition terms does not affect the existence of traveling wave solutions. Finally, for a specific kernel function, some numerical simulations are given to show that the traveling wave solutions may connect the zero equilibrium to a periodic steady state.

On the threshold dynamics of the stochastic SIRS epidemic model using adequate stopping times
Adel Settati, Aadil Lahrouz, Mustapha El Jarroudi, Mohamed El Fatini and Kai Wang
2020, 25(5): 1985-1997 doi: 10.3934/dcdsb.2020012 +[Abstract](199) +[HTML](77) +[PDF](2611.8KB)

As it is well known, the dynamics of the stochastic SIRS epidemic model with mass action is governed by a threshold \begin{document}$ \mathcal{R_S} $\end{document}. If \begin{document}$ \mathcal{R_S}<1 $\end{document} the disease dies out from the population, while if \begin{document}$ \mathcal{R_S}> 1 $\end{document} the disease persists. However, when \begin{document}$ \mathcal{R_S} = 1 $\end{document}, classical techniques used to study the asymptotic behaviour do not work any more. In this paper, we give answer to this open problem by using a new approach involving some adequate stopping times. Our results show that if \begin{document}$ \mathcal{R_S} = 1 $\end{document} then, small noises promote extinction while the large one promote persistence. So, it is exactly the opposite role of the noises in case when \begin{document}$ \mathcal{R_S}\neq 1 $\end{document}.

Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment
Siyao Zhu and Jinliang Wang
2020, 25(5): 1999-2019 doi: 10.3934/dcdsb.2020013 +[Abstract](143) +[HTML](86) +[PDF](490.89KB)

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.

Corrigendum to "Singularity of controls in a simple model of acquired chemotherapy resistance"
Piotr Bajger, Mariusz Bodzioch and Urszula Foryś
2020, 25(5): 2021-2021 doi: 10.3934/dcdsb.2020105 +[Abstract](59) +[HTML](48) +[PDF](147.14KB)

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