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Discrete and Continuous Dynamical Systems - B

February 2022 , Volume 27 , Issue 2

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Stability analysis and optimal control of production-limiting disease in farm with two vaccines
Yue Liu and Wing-Cheong Lo
2022, 27(2): 619-638 doi: 10.3934/dcdsb.2021058 +[Abstract](1205) +[HTML](474) +[PDF](741.95KB)

The transmission of production-limiting disease in farm, such as Neosporosis and Johne's disease, has brought a huge loss worldwide due to reproductive failure. This paper aims to provide a modeling framework for controlling the disease and investigating the spread dynamics of Neospora caninum-infected dairy as a case study. In particular, a dynamic model for production-limiting disease transmission in the farm is proposed. It incorporates the vertical and horizontal transmission routes and two vaccines. The threshold parameter, basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document}, is derived and qualitatively used to explore the stability of the equilibria. Global stability of the disease-free and endemic equilibria is investigated using the comparison theorem or geometric approach. On the case study of Neospora caninum-infected dairy in Switzerland, sensitivity analysis of all involved parameters with respect to the basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document} has been performed. Through Pontryagin's maximum principle, the optimal control problem is discussed to determine the optimal vaccination coverage rate while minimizing the number of infected individuals and control cost at the same time. Moreover, numerical simulations are performed to support the analytical findings. The present study provides useful information on the understanding of production-limiting disease prevention on a farm.

Qualitative analysis of integro-differential equations with variable retardation
Martin Bohner and Osman Tunç
2022, 27(2): 639-657 doi: 10.3934/dcdsb.2021059 +[Abstract](1233) +[HTML](625) +[PDF](520.89KB)

The paper is concerned with a class of nonlinear time-varying retarded integro-differential equations (RIDEs). By the Lyapunov–Krasovski$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ı} $ functional method, two new results with weaker conditions related to uniform stability (US), uniform asymptotic stability (UAS), integrability, boundedness, and boundedness at infinity of solutions of the RIDEs are given. For illustrative purposes, two examples are provided. The study of the results of this paper shows that the given theorems are not only applicable to time-varying linear RIDEs, but also applicable to time-varying nonlinear RIDEs.

Periodicity and stability analysis of impulsive neural network models with generalized piecewise constant delays
Kuo-Shou Chiu
2022, 27(2): 659-689 doi: 10.3934/dcdsb.2021060 +[Abstract](1227) +[HTML](455) +[PDF](2663.71KB)

In this paper, the global exponential stability and periodicity are investigated for impulsive neural network models with Lipschitz continuous activation functions and generalized piecewise constant delay. The sufficient conditions for the existence and uniqueness of periodic solutions of the model are established by applying fixed point theorem and the successive approximations method. By constructing suitable differential inequalities with generalized piecewise constant delay, some sufficient conditions for the global exponential stability of the model are obtained. The methods, which does not make use of Lyapunov functional, is simple and valid for the periodicity and stability analysis of impulsive neural network models with variable and/or deviating arguments. The results extend some previous results. Typical numerical examples with simulations are utilized to illustrate the validity and improvement in less conservatism of the theoretical results. This paper ends with a brief conclusion.

Bifurcation analysis in a delayed toxic-phytoplankton and zooplankton ecosystem with Monod-Haldane functional response
Zhichao Jiang, Zexian Zhang and Maoyan Jie
2022, 27(2): 691-715 doi: 10.3934/dcdsb.2021061 +[Abstract](1029) +[HTML](439) +[PDF](2833.03KB)

We structure a phytoplankton zooplankton interaction system by incorporating (i) Monod-Haldane type functional response function; (ii) two delays accounting, respectively, for the gestation delay \begin{document}$ \tau $\end{document} of the zooplankton and the time \begin{document}$ \tau_1 $\end{document} required for the maturity of TPP. Firstly, we give the existence of equilibrium and property of solutions. The global convergence to the boundary equilibrium is also derived under a certain criterion. Secondly, in the case without the maturity delay \begin{document}$ \tau_1 $\end{document}, the gestation delay \begin{document}$ \tau $\end{document} may lead to stability switches of the positive equilibrium. Then fixed \begin{document}$ \tau $\end{document} in stable interval, the effect of \begin{document}$ \tau_1 $\end{document} is investigated and find \begin{document}$ \tau_1 $\end{document} can also cause the oscillation of system. Specially, when \begin{document}$ \tau = \tau_1 $\end{document}, under certain conditions, the periodic solution will exist with the wide range as delay away from critical value. To deal with the local stability of the positive equilibrium under a general case with all delays being positive, we use the crossing curve methods, it can obtain the stable changes of positive equilibrium in \begin{document}$ (\tau, \tau_1) $\end{document} plane. When choosing \begin{document}$ \tau $\end{document} in the unstable interval, the system still can occur Hopf bifurcation, which extends the crossing curve methods to the system exponentially decayed delay-dependent coefficients. Some numerical simulations are given to indicate the correction of the theoretical analyses.

Evaluating vaccination effectiveness of group-specific fractional-dose strategies
Zhimin Chen, Kaihui Liu and Xiuxiang Liu
2022, 27(2): 717-748 doi: 10.3934/dcdsb.2021062 +[Abstract](963) +[HTML](431) +[PDF](731.42KB)

In this paper, we formulate a multi-group SIR epidemic model with the consideration of proportionate mixing patterns between groups and group-specific fractional-dose vaccination to evaluate the effects of fractionated dosing strategies on disease control and prevention in a heterogeneously mixing population. The basic reproduction number \begin{document}$ \mathscr{R}_0 $\end{document}, the final size of the epidemic, and the infection attack rate are used as three measures of population-level implications of fractionated dosing programs. Theoretically, we identify the basic reproduction number, \begin{document}$ \mathscr{R}_0 $\end{document}, establish the existence and uniqueness of the final size and the final size relation with \begin{document}$ \mathscr{R}_0 $\end{document}, and obtain explicit calculation expressions of the infection attack rate for each group and the whole population. Furthermore, the simulation results suggest that dose fractionation policies take positive effects in lowering the \begin{document}$ \mathscr{R}_0 $\end{document}, decreasing the final size and reducing the infection attack rate only when the fractional-dose influenza vaccine efficacy is high enough rather than just similar to standard-dose. We find evidences that fractional-dose vaccination in response to influenza vaccine shortages take negative community-level effects. Our results indicate that the role of fractional dose vaccines should not be overestimated even though fractional dosing strategies could extend the vaccine coverage.

Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces
Lu Zhang, Aihong Zou, Tao Yan and Ji Shu
2022, 27(2): 749-768 doi: 10.3934/dcdsb.2021063 +[Abstract](992) +[HTML](403) +[PDF](446.31KB)

In this paper we discuss the weak pullback mean random attractors for stochastic Ginzburg-Landau equations defined in Bochner spaces. We prove the existence and uniqueness of weak pullback mean random attractors for the stochastic Ginzburg-Landau equations with nonlinear diffusion terms. We also establish the existence and uniqueness of such attractors for the deterministic Ginzburg-Landau equations with random initial data. In this case, the periodicity of the weak pullback mean random attractors is also proved whenever the external forcing terms are periodic in time.

On the Lorenz '96 model and some generalizations
John Kerin and Hans Engler
2022, 27(2): 769-797 doi: 10.3934/dcdsb.2021064 +[Abstract](1084) +[HTML](432) +[PDF](1431.87KB)

In 1996, Edward Lorenz introduced a system of ordinary differential equations that describes a scalar quantity evolving on a circular array of sites, undergoing forcing, dissipation, and rotation invariant advection. Lorenz constructed the system as a test problem for numerical weather prediction. Since then, the system has also found use as a test case in data assimilation. Mathematically, this is a dynamical system with a single bifurcation parameter (rescaled forcing) that undergoes multiple bifurcations and exhibits chaotic behavior for large forcing. In this paper, the main characteristics of the advection term in the model are identified and used to describe and classify possible generalizations of the system. A graphical method to study the bifurcation behavior of constant solutions is introduced, and it is shown how to use the rotation invariance to compute normal forms of the system analytically. Problems with site-dependent forcing, dissipation, or advection are considered and basic existence and stability results are proved for these extensions. We address some related topics in the appendices, wherein the Lorenz '96 system in Fourier space is considered, explicit solutions for some advection-only systems are found, and it is demonstrated how to use advection-only systems to assess numerical schemes.

Single-target networks
Gheorghe Craciun, Jiaxin Jin and Polly Y. Yu
2022, 27(2): 799-819 doi: 10.3934/dcdsb.2021065 +[Abstract](958) +[HTML](371) +[PDF](448.01KB)

Reaction networks can be regarded as finite oriented graphs embedded in Euclidean space. Single-target networks are reaction networks with an arbitrarily set of source vertices, but only one sink vertex. We completely characterize the dynamics of all mass-action systems generated by single-target networks, as follows: either (i) the system is globally stable for all choice of rate constants (in fact, is dynamically equivalent to a detailed-balanced system with a single linkage class), or (ii) the system has no positive steady states for any choice of rate constants and all trajectories must converge to the boundary of the positive orthant or to infinity. Moreover, we show that global stability occurs if and only if the target vertex of the network is in the relative interior of the convex hull of the source vertices.

Input-to-state stability of infinite-dimensional stochastic nonlinear systems
Pengfei Wang, Mengyi Zhang and Huan Su
2022, 27(2): 821-836 doi: 10.3934/dcdsb.2021066 +[Abstract](1294) +[HTML](414) +[PDF](326.54KB)

In this paper, the input-to-state stability (ISS), stochastic-ISS (SISS) and integral-ISS (iISS) for mild solutions of infinite-dimensional stochastic nonlinear systems (IDSNS) are investigated, respectively. By constructing a class of Yosida strong solution approximating systems for IDSNS and using the infinite-dimensional version Itô's formula, Lyapunov-based sufficient criteria are derived for ensuring ISS-type properties of IDSNS, which extend the existing corresponding results of infinite-dimensional deterministic systems. Moreover, two examples are presented to demonstrate the main results.

A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment
Bo Duan and Zhengce Zhang
2022, 27(2): 837-861 doi: 10.3934/dcdsb.2021067 +[Abstract](1202) +[HTML](390) +[PDF](396.17KB)

In this paper, we investigate a reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. The primary aim is to study the impact of small advection terms and heterogeneous environment, which is on two species' dynamics via a free boundary. The function \begin{document}$ m(x) $\end{document} represents heterogeneous environment, and it can satisfy positive everywhere condition or changeable sign condition. Firstly, on one hand, we provide long time behaviors of the solution in vanishing case when \begin{document}$ m(x) $\end{document} satisfies both conditions above; on the other hand, long time behaviors of the solution in spreading case are got when \begin{document}$ m(x) $\end{document} satisfies positive everywhere condition. Secondly, a spreading-vanishing dichotomy and several sufficient conditions through the initial data and the moving parameters are obtained to determine whether spreading or vanishing of two species happens when \begin{document}$ m(x) $\end{document} satisfies both conditions above. Furthermore, we derive estimates of spreading speed of the free boundary when \begin{document}$ m(x) $\end{document} satisfies positive everywhere condition and two species spreading occurs.

Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations
Liangliang Ma
2022, 27(2): 863-882 doi: 10.3934/dcdsb.2021068 +[Abstract](1238) +[HTML](395) +[PDF](359.34KB)

Stability problem on perturbations near the hydrostatic balance is one of the important issues for Boussinesq equations. This paper focuses on the asymptotic stability and large-time behavior problem of perturbations of the 2D fractional Boussinesq equations with only fractional velocity dissipation or fractional thermal diffusivity. Since the linear portion of the Boussinesq equations plays a crucial role in the stability properties, we firstly study the linearized fractional Boussinesq equations with only fractional velocity dissipation or fractional thermal diffusivity and complete the following work: 1) assessing the stability and obtaining the precise large-time asymptotic behavior for solutions to the linearized system satisfied the perturbation; 2) understanding the spectral property of the linearization; 3) showing the \begin{document}$ H^2 $\end{document}-stability for the linearized system, and prove that the \begin{document}$ L^2 $\end{document}-norm of \begin{document}$ \nabla{u} $\end{document} and \begin{document}$ \Delta{u} $\end{document} (or \begin{document}$ \nabla\theta $\end{document} and \begin{document}$ \Delta\theta $\end{document}), the \begin{document}$ L^\varrho $\end{document}-norm \begin{document}$ (2<\varrho<\infty) $\end{document} of \begin{document}$ u $\end{document} and \begin{document}$ \nabla{u} $\end{document} (or \begin{document}$ \theta $\end{document} and \begin{document}$ \nabla\theta $\end{document}) are all approaching to zero as \begin{document}$ t\rightarrow\infty $\end{document} when \begin{document}$ \alpha = 1 $\end{document} and \begin{document}$ \eta = 0 $\end{document} (or \begin{document}$ \nu = 0 $\end{document} and \begin{document}$ \beta = 1 $\end{document}). Secondly, we obtain the \begin{document}$ H^1 $\end{document}-stability for the full nonlinear system and prove the \begin{document}$ L^\varrho $\end{document}-norm \begin{document}$ (2<\varrho<\infty) $\end{document} of \begin{document}$ \theta $\end{document} and the \begin{document}$ L^2 $\end{document}-norm of \begin{document}$ \nabla\theta $\end{document} approaching to zero as \begin{document}$ t\rightarrow\infty $\end{document}.

Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework
Wensheng Yin, Jinde Cao and Guoqiang Zheng
2022, 27(2): 883-901 doi: 10.3934/dcdsb.2021072 +[Abstract](1056) +[HTML](386) +[PDF](350.98KB)

In this paper we establish a comparison approach to study stabilization of stochastic differential equations driven by \begin{document}$ G $\end{document}-Brownian motion with delayed (\begin{document}$ G $\end{document}-SDDEs for short) feedback control. This theory also extends to a general range of moment order and brings more choices of \begin{document}$ p $\end{document}. Finally, a simple example is proposed to demonstrate the applications of our theory.

A fast high order method for time fractional diffusion equation with non-smooth data
Haili Qiao and Aijie Cheng
2022, 27(2): 903-920 doi: 10.3934/dcdsb.2021073 +[Abstract](1664) +[HTML](581) +[PDF](632.0KB)

In this paper, we consider the time fractional diffusion equation with Caputo fractional derivative. Due to the singularity of the solution at the initial moment, it is difficult to achieve an ideal convergence order on uniform meshes. Therefore, in order to improve the convergence order, we discrete the Caputo time fractional derivative by a new \begin{document}$ L1-2 $\end{document} format on graded meshes, while the spatial derivative term is approximated by the classical central difference scheme on uniform meshes. We analyze the approximation about the time fractional derivative, and obtain the time truncation error, but the stability analysis remains an open problem. On the other hand, considering that the computational cost is extremely large, we present a reduced-order finite difference extrapolation algorithm for the time-fraction diffusion equation by means of proper orthogonal decomposition (POD) technique, which effectively reduces the computational cost. Finally, several numerical examples are given to verify the convergence of the scheme and the effectiveness of the reduced order extrapolation algorithm.

Quasi-periodic travelling waves for beam equations with damping on 3-dimensional rectangular tori
Bochao Chen and Yixian Gao
2022, 27(2): 921-944 doi: 10.3934/dcdsb.2021075 +[Abstract](958) +[HTML](372) +[PDF](431.07KB)

This paper concerns the mathematical analysis of quasi-periodic travelling wave solutions for beam equations with damping on 3-dimensional rectangular tori. Provided that the generators of the rectangular torus satisfy certain relationships, by excluding some values of two model parameters, we establish the existence of small amplitude quasi-periodic travelling wave solutions with three frequencies. Moreover, it can be shown that such solutions are either continuations of rotating wave solutions, or continuations of quasi-periodic travelling wave solutions with two frequencies, and that the set of two model parameters is dense in the positive quadrant.

Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents
Florian Dorsch and Hermann Schulz-Baldes
2022, 27(2): 945-976 doi: 10.3934/dcdsb.2021076 +[Abstract](1065) +[HTML](353) +[PDF](1274.28KB)

Randomly drawn \begin{document}$ 2\times 2 $\end{document} matrices induce a random dynamics on the Riemann sphere via the Möbius transformation. Considering a situation where this dynamics is restricted to the unit disc and given by a random rotation perturbed by further random terms depending on two competing small parameters, the invariant (Furstenberg) measure of the random dynamical system is determined. The results have applications to the perturbation theory of Lyapunov exponents which are of relevance for one-dimensional discrete random Schrödinger operators.

Dynamics of consumer-resource systems with consumer's dispersal between patches
Kun Hu and Yuanshi Wang
2022, 27(2): 977-1000 doi: 10.3934/dcdsb.2021077 +[Abstract](887) +[HTML](427) +[PDF](805.05KB)

This paper considers consumer-resource systems with Holling II functional response. In the system, the consumer can move between a source and a sink patch. By applying dynamical systems theory, we give a rigorous analysis on persistence of the system. Then we show local/global stability of equilibria and prove Hopf bifurcation by the Kuznetsov Theorem. It is shown that dispersal in the system could lead to results reversing those without dispersal. Varying a dispersal rate can change species' interaction outcomes from coexistence in periodic oscillation, to persistence at a steady state, to extinction of the predator, and even to extinction of both species. By explicit expressions of stable equilibria, we prove that dispersal can make the consumer reach overall abundance larger than if non-dispersing, and there exists an optimal dispersal rate that maximizes the abundance. Asymmetry in dispersal can also lead to those results. It is proven that the overall abundance is a ridge-like function (surface) of dispersal rates, which extends both previous theory and experimental observation. These results are biologically important in protecting endangered species.

Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain
Guodong Wang and Bijun Zuo
2022, 27(2): 1001-1027 doi: 10.3934/dcdsb.2021078 +[Abstract](1069) +[HTML](390) +[PDF](464.53KB)

In this paper, we study the energy equality for weak solutions to the 3D homogeneous incompressible magnetohydrodynamic equations with viscosity and magnetic diffusion in a bounded domain. Two types of regularity conditions are imposed on weak solutions to ensure the energy equality. For the first type, some global integrability condition for the velocity \begin{document}$ \mathbf u $\end{document} is required, while for the magnetic field \begin{document}$ \mathbf b $\end{document} and the magnetic pressure \begin{document}$ \pi $\end{document}, some suitable integrability conditions near the boundary are sufficient. In contrast with the first type, the second type claims that if some additional interior integrability is imposed on \begin{document}$ \mathbf b $\end{document}, then the regularity on \begin{document}$ \mathbf u $\end{document} can be relaxed.

Normal deviation of synchronization of stochastic coupled systems
Jicheng Liu and Meiling Zhao
2022, 27(2): 1029-1054 doi: 10.3934/dcdsb.2021079 +[Abstract](950) +[HTML](424) +[PDF](464.08KB)

This paper will prove the normal deviation of the synchronization of stochastic coupled system. According to the relationship between the stationary solution and the general solution, the martingale method is used to prove the normal deviation of the fixed initial value of the multi-scale system, thereby obtaining the normal deviation of the stationary solution. At the same time, with the relationship between the synchronized system and the multi-scale system, the normal deviation of the synchronization is obtained.

Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two
Dingheng Pi
2022, 27(2): 1055-1073 doi: 10.3934/dcdsb.2021080 +[Abstract](937) +[HTML](384) +[PDF](379.57KB)

In this paper we consider an \begin{document}$ n $\end{document} dimensional piecewise smooth dynamical system. This system has a co-dimension 2 switching manifold \begin{document}$ \Sigma $\end{document} which is an intersection of two hyperplanes \begin{document}$ \Sigma_1 $\end{document} and \begin{document}$ \Sigma_2 $\end{document}. We investigate the relation between periodic orbit of PWS system and periodic orbit of its double regularized system. If this PWS system has an asymptotically stable sliding periodic orbit(including type Ⅰ and type Ⅱ), we establish conditions to ensure that also a double regularization of the given system has a unique, asymptotically stable, periodic orbit in a neighbourhood of \begin{document}$ \gamma $\end{document}, converging to \begin{document}$ \gamma $\end{document} as both of the two regularization parameters go to \begin{document}$ 0 $\end{document} by applying implicit function theorem and geometric singular perturbation theory.

Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control
Mohamed Ouzahra
2022, 27(2): 1075-1090 doi: 10.3934/dcdsb.2021081 +[Abstract](1007) +[HTML](378) +[PDF](343.15KB)

In this paper we are concerned with the approximate controllability of a multidimensional semilinear reaction-diffusion equation governed by a multiplicative control, which is locally distributed in the reaction term. For a given initial state we provide sufficient conditions on the desirable state to be approximately reached within an arbitrarily small time interval. Our approaches are based on linear semigroup theory and some results on uniform approximation with smooth functions.

Discrete-time dynamics of structured populations via Feller kernels
Horst R. Thieme
2022, 27(2): 1091-1119 doi: 10.3934/dcdsb.2021082 +[Abstract](883) +[HTML](371) +[PDF](395.65KB)

Feller kernels are a concise means to formalize individual structural transitions in a structured discrete-time population model. An iteroparous populations (in which generations overlap) is considered where different kernels model the structural transitions for neonates and for older individuals. Other Feller kernels are used to model competition between individuals. The spectral radius of a suitable Feller kernel is established as basic turnover number that acts as threshold between population extinction and population persistence. If the basic turnover number exceeds one, the population shows various degrees of persistence that depend on the irreducibility and other properties of the transition kernels.

Positive solutions of singular multiparameter p-Laplacian elliptic systems
Meiqiang Feng and Yichen Zhang
2022, 27(2): 1121-1147 doi: 10.3934/dcdsb.2021083 +[Abstract](1190) +[HTML](359) +[PDF](402.18KB)

In this paper, by using the eigenvalue theory, the sub-supersolution method and the fixed point theory, we prove the existence, multiplicity, uniqueness, asymptotic behavior and approximation of positive solutions for singular multiparameter p-Laplacian elliptic systems on nonlinearities with separate variables or without separate variables. Various nonexistence results of positive solutions are also studied.

Mixed Hegselmann-Krause dynamics
Hsin-Lun Li
2022, 27(2): 1149-1162 doi: 10.3934/dcdsb.2021084 +[Abstract](978) +[HTML](400) +[PDF](314.11KB)

The original Hegselmann-Krause (HK) model consists of a set of \begin{document}$ n $\end{document} agents that are characterized by their opinion, a number in \begin{document}$ [0, 1] $\end{document}. Each agent, say agent \begin{document}$ i $\end{document}, updates its opinion \begin{document}$ x_i $\end{document} by taking the average opinion of all its neighbors, the agents whose opinion differs from \begin{document}$ x_i $\end{document} by at most \begin{document}$ \epsilon $\end{document}. There are two types of HK models: the synchronous HK model and the asynchronous HK model. For the synchronous model, all the agents update their opinion simultaneously at each time step, whereas for the asynchronous HK model, only one agent chosen uniformly at random updates its opinion at each time step. This paper is concerned with a variant of the HK opinion dynamics, called the mixed HK model, where each agent can choose its degree of stubbornness and mix its opinion with the average opinion of its neighbors at each update. The degree of the stubbornness of agents can be different and/or vary over time. An agent is not stubborn or absolutely open-minded if its new opinion at each update is the average opinion of its neighbors, and absolutely stubborn if its opinion does not change at the time of the update. The particular case where, at each time step, all the agents are absolutely open-minded is the synchronous HK model. In contrast, the asynchronous model corresponds to the particular case where, at each time step, all the agents are absolutely stubborn except for one agent chosen uniformly at random who is absolutely open-minded. We first show that some of the common properties of the synchronous HK model, such as finite-time convergence, do not hold for the mixed model. We then investigate conditions under which the asymptotic stability holds, or a consensus can be achieved for the mixed model.

Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain
Weihua Jiang, Xun Cao and Chuncheng Wang
2022, 27(2): 1163-1178 doi: 10.3934/dcdsb.2021085 +[Abstract](2167) +[HTML](473) +[PDF](1848.72KB)

In this article, Turing instability and the formations of spatial patterns for a general two-component reaction-diffusion system defined on 2D bounded domain, are investigated. By analyzing characteristic equation at positive constant steady states and further selecting diffusion rate \begin{document}$ d $\end{document} and diffusion ratio \begin{document}$ \varepsilon $\end{document} as bifurcation parameters, sufficient and necessary conditions for the occurrence of Turing instability are established, which is called the first Turing bifurcation curve. Furthermore, parameter regions in which single-mode Turing patterns arise and multiple-mode (or superposition) Turing patterns coexist when bifurcations parameters are chosen, are described. Especially, the boundary of parameter region for the emergence of single-mode Turing patterns, consists of the first and the second Turing bifurcation curves which are given in explicit formulas. Finally, by taking diffusive Schnakenberg system as an example, parameter regions for the emergence of various kinds of spatially inhomogeneous patterns with different spatial frequencies and superposition Turing patterns, are estimated theoretically and shown numerically.

Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations
Jinye Shen and Xian-Ming Gu
2022, 27(2): 1179-1207 doi: 10.3934/dcdsb.2021086 +[Abstract](1174) +[HTML](379) +[PDF](481.92KB)

In this work, two fully novel finite difference schemes for two-dimensional time-fractional mixed diffusion and diffusion-wave equation (TFMDDWEs) are presented. Firstly, a Hermite and Newton quadratic interpolation polynomial have been used for time discretization and central quotient has used in spatial direction. The H2N2 finite difference is constructed. Secondly, in order to increase computational efficiency, the sum-of-exponential is used to approximate the kernel function in the fractional-order operator. The fast H2N2 finite difference is obtained. Thirdly, the stability and convergence of two schemes are studied by energy method. When the tolerance error \begin{document}$ \epsilon $\end{document} of fast algorithm is sufficiently small, it proves that both of difference schemes are of \begin{document}$ 3-\beta\; (1<\beta<2) $\end{document} order convergence in time and of second order convergence in space. Finally, numerical results demonstrate the theoretical convergence and effectiveness of the fast algorithm.

2021 Impact Factor: 1.497
5 Year Impact Factor: 1.527
2021 CiteScore: 2.3




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