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Discrete & Continuous Dynamical Systems - B

September 2020 , Volume 25 , Issue 9

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Geometric method for global stability of discrete population models
Zhanyuan Hou
2020, 25(9): 3305-3334 doi: 10.3934/dcdsb.2020063 +[Abstract](560) +[HTML](179) +[PDF](934.07KB)
Abstract:

A class of autonomous discrete dynamical systems as population models for competing species are considered when each nullcline surface is a hyperplane. Criteria are established for global attraction of an interior or a boundary fixed point by a geometric method utilising the relative position of these nullcline planes only, independent of the growth rate function. These criteria are universal for a broad class of systems, so they can be applied directly to some known models appearing in the literature including Ricker competition models, Leslie-Gower models, Atkinson-Allen models, and generalised Atkinson-Allen models. Then global asymptotic stability is obtained by finding the eigenvalues of the Jacobian matrix at the fixed point. An intriguing question is proposed: Can a globally attracting fixed point induce a homoclinic cycle?

Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms
Mihaela Negreanu
2020, 25(9): 3335-3356 doi: 10.3934/dcdsb.2020064 +[Abstract](593) +[HTML](170) +[PDF](472.78KB)
Abstract:

This paper is concerned with a general asymptotic stabilization of arbitrary global positive bounded solutions for the Lotka Volterra reaction diffusion systems, with an additional chemotactic influence and constant coefficients. We consider the dynamics of a mathematical model involving two biological species, both of which move according to random diffusion and are attracted/ repulsed by chemical stimulus produced by the other. The biological species present the ability to orientate their movement towards the concentration of the chemical secreted by the other species. The nonlinear system consists of two parabolic equations with Lotka-Volterra-type kinetic terms coupled with chemotactic cross-diffusion, along with two elliptic equations describing the behavior of the chemicals. We prove that the solution to the corresponding Neumann initial boundary value problem is global and bounded for regular and positive initial data. Moreover, for different ranges of parameters, we show that any positive and bounded solution converges to a spatially constant homogeneous state.

Spectral methods for two-dimensional space and time fractional Bloch-Torrey equations
Hong Lu, Ji Li and Mingji Zhang
2020, 25(9): 3357-3371 doi: 10.3934/dcdsb.2020065 +[Abstract](373) +[HTML](134) +[PDF](453.42KB)
Abstract:

In this paper, we consider the numerical approximation of the space and time fractional Bloch-Torrey equations. A fully discrete spectral scheme based on a finite difference method in the time direction and a Galerkin-Legendre spectral method in the space direction is developed. In order to reduce the amount of computation, an alternating direction implicit (ADI) spectral scheme is proposed. Then the stability and convergence analysis are rigorously established. Finally, numerical results are presented to support our theoretical analysis.

Dynamical behavior of a rotavirus disease model with two strains and homotypic protection
Kun Lu, Wendi Wang and Jianquan Li
2020, 25(9): 3373-3391 doi: 10.3934/dcdsb.2020066 +[Abstract](353) +[HTML](145) +[PDF](534.93KB)
Abstract:

A two-strain rotavirus model with vaccination and homotypic protection is proposed to study the survival of the two strains of rotavirus within the host. Corresponding to the different efficacy of monovalent vaccine against different strains, the vaccination reproduction numbers of the two strains and the reproduction numbers of their mutual invasion are found. Based on the existence and local stability of equilibria, our results suggest that the obtained reproduction numbers determine together the dynamics of the model, and that the two-strain rotavirus dies out as both the numbers is less than unity. The coexistence of two strains, one of which is dominant, is also related to the two reproduction numbers.

Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations
Manil T. Mohan
2020, 25(9): 3393-3436 doi: 10.3934/dcdsb.2020067 +[Abstract](511) +[HTML](178) +[PDF](747.2KB)
Abstract:

The dynamics of three dimensional Kelvin-Voigt-Brinkman- Forchheimer equations in bounded domains is considered in this work. The existence and uniqueness of strong solution to the system is obtained by exploiting the \begin{document}$ m $\end{document}-accretive quantization of the linear and nonlinear operators. The long-term behavior of solutions of such systems is also examined in this work. We first establish the existence of an absorbing ball in appropriate spaces for the semigroup associated with the solutions of the 3D Kelvin-Voigt-Brinkman-Forchheimer equations. Then, we prove that the semigroup is asymptotically compact, which implies the existence of a global attractor for the system. Next, we show the differentiability of the semigroup with respect to the initial data and then establish that the global attractor has finite Hausdorff and fractal dimensions. Furthermore, we establish the existence of an exponential attractor and discuss about its fractal dimensions for the associated semigroup of such systems. Finally, we discuss about the inviscid limit of the 3D Kelvin-Voigt-Brinkman-Forchheimer equations to the 3D Navier-Stokes-Voigt system and then to the simplified Bardina model.

Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization
Ling Liu, Jiashan Zheng and Gui Bao
2020, 25(9): 3437-3460 doi: 10.3934/dcdsb.2020068 +[Abstract](320) +[HTML](148) +[PDF](591.55KB)
Abstract:

We consider an initial-boundary value problem for the incompressible four-component Keller-Segel-Navier-Stokes system with rotational flux

in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^3 $\end{document} with smooth boundary, where \begin{document}$ \kappa\in \mathbb{R} $\end{document} is given constant, \begin{document}$ S $\end{document} is a matrix-valued sensitivity satisfying \begin{document}$ |S(x,n,c)|\leq C_S(1+n)^{-\alpha} $\end{document} with some \begin{document}$ C_S> 0 $\end{document} and \begin{document}$ \alpha\geq 0 $\end{document}. As the case \begin{document}$ \kappa = 0 $\end{document} (with \begin{document}$ \alpha\geq\frac{1}{3} $\end{document} or the initial data satisfy a certain smallness condition) has been considered in [18], based on new gradient-like functional inequality, it is shown in the present paper that the corresponding initial-boundary problem with \begin{document}$ \kappa \neq 0 $\end{document} admits at least one global weak solution if \begin{document}$ \alpha>0 $\end{document}. To the best of our knowledge, this is the first analytical work for the full three-dimensional four-component chemotaxis-Navier-Stokes system.

Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition
Xun Cao and Weihua Jiang
2020, 25(9): 3461-3489 doi: 10.3934/dcdsb.2020069 +[Abstract](348) +[HTML](148) +[PDF](1124.9KB)
Abstract:

A diffusive predator-prey model with nonlocal prey competition and the homogeneous Neumann boundary conditions is considered, to explore the effects of nonlocal reaction term. Firstly, conditions of the occurrence of Hopf, Turing, Turing-Turing and double zero bifurcations, are established. Then, several concise formulas of computing normal form at a double zero singularity for partial functional differential equations, are provided. Next, via analyzing normal form derived by utilizing these formulas, we find that diffusive predator-prey system admits interesting spatiotemporal dynamics near the double zero singularity, like tristable phenomenon that a stable spatially inhomogeneous periodic solution with the shape of \begin{document}$ \cos\omega_0 t\cos\frac{x}{l}- $\end{document}like which is unstable in model without nonlocal competition and also greatly different from these with the shape of \begin{document}$ \cos\omega_0 t+\cos\frac{x}{l}- $\end{document}like resulting from Turing-Hopf bifurcation, coexists with a pair of spatially inhomogeneous steady states with the shape of \begin{document}$ \cos\frac{x}{l}- $\end{document}like. At last, numerical simulations are shown to support theory analysis. These investigations indicate that nonlocal reaction term could stabilize spatially inhomogeneous periodic solutions with the shape of \begin{document}$ \cos\omega_0 t\cos\frac{kx}{l}- $\end{document}like for reaction-diffusion systems subject to the homogeneous Neumann boundary conditions.

Dynamic analysis and optimal control of a three-age-class HIV/AIDS epidemic model in China
Hongyong Zhao, Peng Wu and Shigui Ruan
2020, 25(9): 3491-3521 doi: 10.3934/dcdsb.2020070 +[Abstract](411) +[HTML](158) +[PDF](2151.47KB)
Abstract:

Based on the fact that HIV/AIDS manifests different transmission characteristics and pathogenesis in different age groups, and the proportions of youth and elderly HIV infected cases in total are increasing in China, we classify the whole population into three age groups, youth (15-24), adult (25-49), and elderly (\begin{document}$ \geqslant $\end{document}50), and establish a three-age-class HIV/AIDS epidemic model to investigate the transmission dynamics of HIV/AIDS in China. We derive the explicit expression for the basic reproduction number via the next generation matrix approach. Qualitative analysis of the model including the local, global behavior and permanence is carried out. In particular, numerical simulations are presented to reinforce these analytical results and demonstrate HIV epidemiological discrepancy among different age groups. We also formulate an optimal control problem and solve it using Pontryagin's Maximum Principle and an efficient iterative numerical methods. Our numerical results of optimal controls for the elderly group indicate that increasing the condom use and decreasing the rate of the formerly HIV infected persons converted to AIDS patients are important measures to control HIV/AIDS epidemic among elderly population.

Dynamics of a stage-structured population model with a state-dependent delay
Shangzhi Li and Shangjiang Guo
2020, 25(9): 3523-3551 doi: 10.3934/dcdsb.2020071 +[Abstract](347) +[HTML](142) +[PDF](571.63KB)
Abstract:

This paper is devoted to the dynamics of a predator-prey model with stage structure for prey and state-dependent maturation delay. Firstly, positivity and boundedness of solutions are addressed to describe the population survival and the natural restriction of limited resources. Secondly, the existence, uniqueness, and local asymptotical stability of (boundary and coexisting) equilibria are investigated by means of degree theory and Routh-Hurwitz criteria. Thirdly, the explicit bounds for the eventual behaviors of the mature population are obtained. Finally, by means of comparison principle and two auxiliary systems, it observed that the local asymptotical stability of either of the positive interior equilibrium and the positive boundary equilibrium implies that it is also globally asymptotical stable if the derivative of the delay function around this equilibrium is small enough.

Dynamics of non-autonomous fractional Ginzburg-Landau equations driven by colored noise
Hong Lu and Mingji Zhang
2020, 25(9): 3553-3576 doi: 10.3934/dcdsb.2020072 +[Abstract](285) +[HTML](133) +[PDF](511.92KB)
Abstract:

In this work, the existence and uniqueness of random attractors of a class of non-autonomous non-local fractional stochastic Ginzburg-Landau equation driven by colored noise with a nonlinear diffusion term is established. We comment that compared to white noise, the colored noise is much easier to handle in examining the pathwise dynamics of stochastic systems. Additionally, we prove the attractors of the random fractional Ginzburg-Landau system driven by a linear multiplicative colored noise converge to those of the corresponding stochastic system driven by a linear multiplicative white noise.

Singular renormalization group approach to SIS problems
Ning Sun, Shaoyun Shi and Wenlei Li
2020, 25(9): 3577-3596 doi: 10.3934/dcdsb.2020073 +[Abstract](274) +[HTML](138) +[PDF](472.74KB)
Abstract:

In this paper, we consider the boundary value problems of a one-dimensional steady-state SIS epidemic reaction-diffusion-advection system in the following two cases: (ⅰ) the advection rate is relatively large comparing to the diffusion rates of infected and susceptible populations; (ⅱ) the diffusion rate of the susceptible population approaches zero. By introducing a singular parameter, the system can be viewed as a singularly perturbed problem. By the renormalization group method, we construct the first-order approximate solutions and obtain error estimates.

Assignability of dichotomy spectra for discrete time-varying linear control systems
Le Viet Cuong and Thai Son Doan
2020, 25(9): 3597-3607 doi: 10.3934/dcdsb.2020074 +[Abstract](328) +[HTML](135) +[PDF](351.69KB)
Abstract:

In this paper, we show that for discrete time-varying linear control systems uniform complete controllability implies arbitrary assignability of dichotomy spectrum of closed-loop systems. This result significantly strengthens the result in [5] about arbitrary assignability of Lyapunov spectrum of discrete time-varying linear control systems.

Influence of mutations in phenotypically-structured populations in time periodic environment
Cécile Carrère and Grégoire Nadin
2020, 25(9): 3609-3630 doi: 10.3934/dcdsb.2020075 +[Abstract](477) +[HTML](313) +[PDF](564.29KB)
Abstract:

We study a parabolic Lotka-Volterra equation, with an integral term representing competition, and time periodic growth rate. This model represents a trait structured population in a time periodic environment. After showing the convergence of the solution to the unique positive and periodic solution of the problem, we study the influence of different factors on the mean limit population. As this quantity is the opposite of a certain eigenvalue, we are able to investigate the influence of the diffusion rate, the period length and the time variance of the environment fluctuations. We also give biological interpretation of the results in the framework of cancer, if the model represents a cancerous cells population under the influence of a periodic treatment. In this framework, we show that the population might benefit from a intermediate rate of mutation.

Approximating exit times of continuous Markov processes
Thomas Kruse and Mikhail Urusov
2020, 25(9): 3631-3650 doi: 10.3934/dcdsb.2020076 +[Abstract](523) +[HTML](282) +[PDF](476.21KB)
Abstract:

The time at which a one-dimensional continuous strong Markov process attains a boundary point of its state space is a discontinuous path functional and it is, therefore, unclear whether the exit time can be approximated by hitting times of approximations of the process. We prove a functional limit theorem for approximating weakly both the paths of the Markov process and its exit times. In contrast to the functional limit theorem in [3] for approximating the paths, we impose a stronger assumption here. This is essential, as we present an example showing that the theorem extended with the convergence of the exit times does not hold under the assumption in [3]. However, the EMCEL scheme introduced in [3] satisfies the assumption of our theorem, and hence we have a scheme capable of approximating both the process and its exit times for every one-dimensional continuous strong Markov process, even with irregular behavior (e.g., a solution of an SDE with irregular coefficients or a Markov process with sticky features). Moreover, our main result can be used to check for some other schemes whether the exit times converge. As an application we verify that the weak Euler scheme is capable of approximating the absorption time of the CEV diffusion and that the scale-transformed weak Euler scheme for a squared Bessel process is capable of approximating the time when the squared Bessel process hits zero.

Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps
Leonid Shaikhet
2020, 25(9): 3651-3657 doi: 10.3934/dcdsb.2020077 +[Abstract](552) +[HTML](282) +[PDF](149.57KB)
Abstract:

Delay differential equation is considered under stochastic perturbations of the type of white noise and Poisson's jumps. It is shown that if stochastic perturbations fade on the infinity quickly enough then sufficient conditions for asymptotic stability of the zero solution of the deterministic differential equation with delay provide also asymptotic mean square stability of the zero solution of the stochastic differential equation. Stability conditions are obtained via the general method of Lyapunov functionals construction and the method of Linear Matrix Inequalities (LMIs). Investigation of the situation when stochastic perturbations do not fade on the infinity or fade not enough quickly is proposed as an unsolved problem.

A note on a neuron network model with diffusion
Philippe Michel and Suman Kumar Tumuluri
2020, 25(9): 3659-3676 doi: 10.3934/dcdsb.2020085 +[Abstract](351) +[HTML](206) +[PDF](407.97KB)
Abstract:

We study the dynamics of an inhomogeneous neuronal network parametrized by a real number \begin{document}$ \sigma $\end{document} and structured by the time elapsed since the last discharge. The dynamics are governed by the parabolic PDE which describes the probability density of neurons with elapsed time \begin{document}$ s $\end{document} after its last discharge. We prove existence and uniqueness of a solution to the model. Moreover, we show that under some conditions on the connectivity and the firing rate, the network exhibits total desynchronization.

From approximate synchronization to identical synchronization in coupled systems
Chih-Wen Shih and Jui-Pin Tseng
2020, 25(9): 3677-3714 doi: 10.3934/dcdsb.2020086 +[Abstract](344) +[HTML](204) +[PDF](970.2KB)
Abstract:

We establish a framework to investigate approximate synchronization of coupled systems under general coupling schemes. The units comprising the coupled systems may be nonidentical and the coupling functions are nonlinear with delays. Both delay-dependent and delay-independent criteria for approximate synchronization are derived, based on an approach termed sequential contracting. It is explored and elucidated that the synchronization error, the distance between the asymptotic state and the synchronous set, decreases with decreasing difference between subsystems, difference between the row sums of connection matrix, and difference of coupling time delays between different units. This error vanishes when these factors decay to zero, and approximate synchronization becomes identical synchronization for the coupled system comprising identical subsystems and connection matrix with identical row sums, and with identical coupling delays. The application of the present theory to nonlinearly coupled heterogeneous FitzHugh-Nagumo neurons is illustrated. We extend the analysis to study approximate synchronization and asymptotic synchronization for coupled Lorenz systems and show that for some coupling schemes, the synchronization error decreases as the coupling strength increases, whereas in another case, the error remains at a substantial level for large coupling strength.

Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D
Mohamad Darwich
2020, 25(9): 3715-3724 doi: 10.3934/dcdsb.2020087 +[Abstract](578) +[HTML](195) +[PDF](383.85KB)
Abstract:

In this paper, we consider the Zakharov-Kuznetsov equation in 3D, with a dissipative term of order \begin{document}$ 0 < \alpha \leq 2 $\end{document} in the \begin{document}$ x $\end{document} direction. We prove that the problem is locally well-posed in \begin{document}$ H^{s}( { I\!\!R}^3) $\end{document}, for \begin{document}$ s > 1-\frac{\alpha}{2} $\end{document}, and by an a priori energy estimate, we prove that the problem is globally well-posed in \begin{document}$ H^{1}( { I\!\!R}^3) $\end{document}.

Local orthogonal rectification: Deriving natural coordinates to study flows relative to manifolds
Benjamin Letson and Jonathan E. Rubin
2020, 25(9): 3725-3747 doi: 10.3934/dcdsb.2020088 +[Abstract](320) +[HTML](188) +[PDF](1964.9KB)
Abstract:

We recently derived a method, local orthogonal rectification (LOR), that provides a natural and useful geometric frame for analyzing dynamics relative to a base curve in the phase plane for two-dimensional systems of ODEs (Letson and Rubin, SIAM J. Appl. Dyn. Syst., 2018). This work extends LOR to apply to any embedded base manifold in a system of ODEs of arbitrary dimension and establishes a corresponding system of LOR equations for analyzing dynamics within the LOR frame, which maps naturally back to the original phase space. The LOR equations encode geometric properties of the underlying flow and remain valid, in general, beyond a local neighborhood of the embedded manifold. In addition to developing a general theory for LOR that makes use of a given normal frame, we show how to construct a normal frame that conveniently simplifies the computations involved in LOR. Finally, we illustrate the utility of LOR by showing that a blow-up transformation on the LOR equations provides a useful decomposition for studying trajectories' behavior relative to the embedded base manifold and by using LOR to identify canard behavior near a fold of a critical manifold in a two-timescale system.

A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation
Zeyu Xia and Xiaofeng Yang
2020, 25(9): 3749-3768 doi: 10.3934/dcdsb.2020089 +[Abstract](371) +[HTML](179) +[PDF](437.55KB)
Abstract:

The nonlinear stability and convergence of a numerical scheme for the "Good" Boussinesq equation is provided in this article, with second order temporal accuracy and Fourier pseudo-spectral approximation in space. Instead of introducing an intermediate variable \begin{document}$ \psi $\end{document} to approximate the first order temporal derivative, we apply a direct approximation to the second order temporal derivative, which in turn leads to a reduction of the intermediate numerical variable and improvement in computational efficiency. A careful analysis reveals an unconditional stability and convergence for such a temporal discretization. In addition, by making use of the techniques of aliasing error control, we obtain an \begin{document}$ \ell^\infty (0,T^*; H^2) $\end{document} convergence for \begin{document}$ u $\end{document} and \begin{document}$ \ell^\infty (0,T^*; \ell^2) $\end{document} convergence for the discrete time-derivative of the solution in this paper, in comparison with the \begin{document}$ \ell^\infty (0,T^*; \ell^2) $\end{document} convergence for \begin{document}$ u $\end{document} and the \begin{document}$ \ell^\infty (0,T^*; H^{-2}) $\end{document} convergence for the time-derivative, given in [19].

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