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Discrete & Continuous Dynamical Systems - B

July 2020 , Volume 25 , Issue 7

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Asymptotic properties of a stochastic chemostat model with two distributed delays and nonlinear perturbation
Xingwang Yu and Sanling Yuan
2020, 25(7): 2373-2390 doi: 10.3934/dcdsb.2020014 +[Abstract](479) +[HTML](103) +[PDF](968.69KB)

In this paper, a stochastic chemostat model with two distributed delays and nonlinear perturbation is proposed. We first transform the stochastic model into an equivalent high-dimensional system. Then we prove the existence and uniqueness of global positive solution of the model. Based on Khasminskii's theory, we study the existence of a stationary distribution of the model by constructing a suitable stochastic Lyapunov function. Then we also establish sufficient conditions for the extinction of the plankton. Finally, numerical simulations are carried out to illustrate the theoretical results and to conclude our study, which shows that environmental noise experienced by limiting nutrient completely determines the persistence and extinction of the plankton.

Complicated dynamics of tumor-immune system interaction model with distributed time delay
Min Yu, Gang Huang, Yueping Dong and Yasuhiro Takeuchi
2020, 25(7): 2391-2406 doi: 10.3934/dcdsb.2020015 +[Abstract](350) +[HTML](94) +[PDF](658.03KB)

In this paper, we propose a distributed delay model to investigate the dynamics of the interactions between tumor and immune system. And we choose a special form of delay kernel which combines two delay kernels: a monotonic delay kernel representing a fading memory and a nonmonotonic delay kernel describing a peaking memory. Then, we discuss the effect of such delay kernel on system dynamics. The results show that the introduction of nonmonotonic delay kernel does not change the stability of tumor-free equilibrium, but it can induce stability switches of tumor-presence equilibrium and cause a rich pattern of dynamical behaviors including stabilization. Moreover, our numerical simulation results reveal that the nonmonotonic delay kernel has more complicated effects on the stability compared with the monotonic delay kernel.

Stability of stochastic heroin model with two distributed delays
Miljana Jovanović and Vuk Vujović
2020, 25(7): 2407-2432 doi: 10.3934/dcdsb.2020016 +[Abstract](338) +[HTML](86) +[PDF](3495.21KB)

In this paper a stability of stochastic heroin model with two distributed delays is studied. Precisely, the deterministic model for dynamics of heroin users is extended by random perturbation that briefly describe how a environmental fluctuations lead an individual to become a heroin user. By using a suitable Lyapunov function stability conditions for heroin use free equilibrium are obtained. Furthermore, asymptotic behavior around the heroin spread equilibrium of the deterministic model is investigated by using appropriate Lyapunov functional. Theoretical studies, based on real data, are applied on modeling of number of heroin users in the USA from \begin{document}$ 01.01.2014. $\end{document}

Dynamical analysis of a diffusive SIRS model with general incidence rate
Yu Yang, Lan Zou, Tonghua Zhang and Yancong Xu
2020, 25(7): 2433-2451 doi: 10.3934/dcdsb.2020017 +[Abstract](523) +[HTML](123) +[PDF](635.25KB)

In this paper, we propose a diffusive SIRS model with general incidence rate and spatial heterogeneity. The formula of the basic reproduction number \begin{document}$ \mathcal R_0 $\end{document} is given. Then the threshold dynamics, including globally attractive of the disease-free equilibrium and uniform persistence, are established in terms of \begin{document}$ \mathcal{R}_0 $\end{document}. Special cases and numerical simulations are presented to support our main results.

Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply
Junde Wu and Shihe Xu
2020, 25(7): 2453-2460 doi: 10.3934/dcdsb.2020018 +[Abstract](289) +[HTML](67) +[PDF](312.68KB)

In this paper we study a nonlinear free boundary problem for the growth of radially symmetric tumor with a necrotic core. The proliferation of tumor cells depends on the concentration of nutrient which satisfies a diffusion equation within tumor and is periodically supplied by external tissues. The tumor outer surface and the inner interface of the necrotic core are both free boundaries. We give a sufficient and necessary condition for the existence and uniqueness of positive periodic solution, and show it is globally asymptotically stable under radial perturbations. Our analysis implies that tumor growth may finally synchronize the periodic external nutrient supply.

Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise
Renhai Wang and Bixiang Wang
2020, 25(7): 2461-2493 doi: 10.3934/dcdsb.2020019 +[Abstract](170) +[HTML](69) +[PDF](650.66KB)

This paper is concerned with the global existence and random dynamics of non-autonomous stochastic second-order lattice systems driven by infinite-dimensional nonlinear noise defined on higher-dimensional integer sets. We first show the existence and uniqueness of mean square solutions to the equations when the nonlinear drift term has a polynomial growth of arbitrary order and the diffusion term is locally Lipschitz continuous. We then prove that the mean random dynamical system associated with the solution operator possesses a unique tempered weak pullback mean random attractor in a Bochner space under certain conditions. We finally establish the existence of invariant measures for the stochastic systems in \begin{document}$ \ell^2\times\ell^2 $\end{document} by showing the tightness of a family of distribution laws of solutions via the idea of uniform tail-estimates on the solutions.

Long term behavior of random Navier-Stokes equations driven by colored noise
Anhui Gu, Boling Guo and Bixiang Wang
2020, 25(7): 2495-2532 doi: 10.3934/dcdsb.2020020 +[Abstract](180) +[HTML](61) +[PDF](634.88KB)

This paper is devoted to the study of long term behavior of the two-dimensional random Navier-Stokes equations driven by colored noise defined in bounded and unbounded domains. We prove the existence and uniqueness of pullback random attractors for the equations with Lipschitz diffusion terms. In the case of additive noise, we show the upper semi-continuity of these attractors when the correlation time of the colored noise approaches zero. When the equations are defined on unbounded domains, we establish the pullback asymptotic compactness of the solutions by Ball's idea of energy equations in order to overcome the difficulty introduced by the noncompactness of Sobolev embeddings.

Boundary switch on/off control approach to simultaneous identification of diffusion coefficient and initial state for one-dimensional heat equation
Zhi-Xue Zhao, Mapundi K. Banda and Bao-Zhu Guo
2020, 25(7): 2539-2554 doi: 10.3934/dcdsb.2020021 +[Abstract](170) +[HTML](75) +[PDF](556.47KB)

In this paper, we consider simultaneous reconstruction of diffusion coefficient and initial state for a one-dimensional heat equation through boundary control and measurement. The boundary measurement is proposed to make the system approximately observable, and both the coefficient and initial state are shown to be identifiable by this measurement under a boundary switch on/off control. By a Dirichlet series representation for the observation, we can transform the problem into an inverse process of reconstruction of the spectrum and coefficients for the Dirichlet series in terms of observation. This happens to be the reconstruction of spectral data for an exponential sequence with measurement error, and it enables us to develop an algorithm based on the matrix pencil method in signal analysis. A theoretical error analysis for the algorithm concerning the coefficient reconstruction is carried out for the proposed method. The numerical simulations are presented to verify the proposed algorithm.

Dynamics of positive steady-state solutions of a nonlocal dispersal logistic model with nonlocal terms
Li Ma and Youquan Luo
2020, 25(7): 2555-2582 doi: 10.3934/dcdsb.2020022 +[Abstract](185) +[HTML](99) +[PDF](531.08KB)

In this paper, we investigate a class of nonlocal dispersal logistic equations with nonlocal terms

where \begin{document}$ \Omega\subset \mathbb{R}^N(N\geq1) $\end{document} is a bounded domain, \begin{document}$ \lambda\in \mathbb{R} $\end{document}, \begin{document}$ 0<q\leq1 $\end{document}, \begin{document}$ p>0 $\end{document}, \begin{document}$ a,b\in C(\overline{\Omega}) $\end{document}, \begin{document}$ b\geq0 $\end{document}, \begin{document}$ b\neq0 $\end{document} and \begin{document}$ a $\end{document} verifies either \begin{document}$ a>0 $\end{document} or \begin{document}$ a<0 $\end{document}. \begin{document}$ Du = \int_\Omega J(x-y)u(y,t){\rm{d}}y-u(x,t) $\end{document} represents the nonlocal dispersal operators, which is continuous and nonpositive. Under some suitable assumptions we establish the existence, uniqueness or multiplicity and stability of positive stationary solution with nonlocal reaction term by using sub-supersolution methods, Lerray-Schauder degree theory and Lyapunov-Schmidt reduction and so on.

Point vortices for inviscid generalized surface quasi-geostrophic models
Carina Geldhauser and Marco Romito
2020, 25(7): 2583-2606 doi: 10.3934/dcdsb.2020023 +[Abstract](175) +[HTML](78) +[PDF](700.26KB)

We give a rigorous proof of the validity of the point vortex description for a class of inviscid generalized surface quasi-geostrophic models on the whole plane.

On the stability and transition of the Cahn-Hilliard/Allen-Cahn system
Quan Wang and Dongming Yan
2020, 25(7): 2607-2620 doi: 10.3934/dcdsb.2020024 +[Abstract](200) +[HTML](88) +[PDF](415.6KB)

In this paper, the main objective is to study the stability and transition of the Cahn-Hilliard/Allen-Cahn system. By using the dynamic transition theory, combining with the spectral theorem for general linear completely continuous fields, we prove that the system undergoes a continuous transition and bifurcates from a trivial solution to an attractor as the control parameter crosses a certain critical value. In addition, for some special cases, i.e., the domain is \begin{document}$ n $\end{document}-dimensional box \begin{document}$ (n = 1,2,3) $\end{document}, we not only obtain the stability of the singular points of the attractors, the topological structure of the attractors is also illustrated.

The asymptotic behaviour of the $ p(x) $-Laplacian Steklov eigenvalue problem
Lujuan Yu
2020, 25(7): 2621-2637 doi: 10.3934/dcdsb.2020025 +[Abstract](165) +[HTML](65) +[PDF](494.82KB)

In this paper we study the asymptotic behaviour of the first eigenvalues \begin{document}$ \lambda^{1}_{p_{n}(\cdot)} $\end{document} and the corresponding eigenfunctions \begin{document}$ u_{n} $\end{document} of (1) as \begin{document}$ p_{n}(x)\rightarrow \infty $\end{document}. Under adequate hypotheses on the sequence \begin{document}$ p_{n} $\end{document}, we prove that \begin{document}$ \lambda^{1}_{p_{n}(\cdot)} $\end{document} converges to 1 and the positive first eigenfunctions \begin{document}$ u_{n} $\end{document}, normalized by \begin{document}$ |u_{n}|_{L^{p_{n}(x)}(\partial \Omega)} = 1 $\end{document}, converge, up to subsequences, to \begin{document}$ u_{\infty} $\end{document} uniformly in \begin{document}$ C^{\alpha}(\overline{\Omega}) $\end{document}, for some \begin{document}$ 0<\alpha<1 $\end{document}, where \begin{document}$ u_{\infty} $\end{document} is a nontrivial viscosity solution of a problem involving the \begin{document}$ \infty $\end{document}-Laplacian subject to appropriate boundary conditions.

Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy
Wenjie Li, Lihong Huang and Jinchen Ji
2020, 25(7): 2639-2664 doi: 10.3934/dcdsb.2020026 +[Abstract](222) +[HTML](83) +[PDF](754.37KB)

This paper studies the solution behaviour of a general delayed predator-prey model with discontinuous prey control strategy. The positiveness and boundeness of the solution of the system is firstly investigated using the comparison theorem. Then the sufficient conditions are derived for the existence of positive periodic solutions using the differential inclusion theory and the topological degree theory. Furthermore, the positive periodic solution is proved to be globally exponentially stable by employing the generalized Lyapunov approach. The global finite-time convergence is also discussed for the system state. Finally, the numerical simulations of four examples are given to validate the correctness of the theoretical results.

The existence and exponential behavior of solutions to time fractional stochastic delay evolution inclusions with nonlinear multiplicative noise and fractional noise
Yajing Li and Yejuan Wang
2020, 25(7): 2665-2697 doi: 10.3934/dcdsb.2020027 +[Abstract](191) +[HTML](71) +[PDF](623.12KB)

This article is devoted to study time fractional stochastic evolution inclusions with infinite delays driven by a nonlinear multiplicative noise and a fractional Brownian motion with Hurst parameter \begin{document}$ H\in(\frac{1}{2},1) $\end{document}. First of all, we investigate the local and global existence of mild solutions to such evolution inclusions by using the fractional resolvent operator theory and some new results on the measure of noncompactness for the stochastic integral term. Further, we prove that every mild solution decays exponentially to zero in the sense of mean-square topology.

Probabilistic continuity of a pullback random attractor in time-sample
Shulin Wang and Yangrong Li
2020, 25(7): 2699-2772 doi: 10.3934/dcdsb.2020028 +[Abstract](171) +[HTML](61) +[PDF](525.91KB)

Given a time-sample dependent attractor of a random dynamical system, we study its lower semi-continuity in probability along the time axis, and the criteria are established by using the local-sample asymptotically compactness for a triple-continuous system. The abstract results are applied to the non-autonomous stochastic p-Laplace equation on an unbounded domain with weakly dissipative nonlinearity. Without any additional hypotheses, we prove that the pullback random attractor is probabilistically continuous in both time and sample parameters.

Global solutions and random dynamical systems for rough evolution equations
Robert Hesse and Alexandra Neamţu
2020, 25(7): 2723-2748 doi: 10.3934/dcdsb.2020029 +[Abstract](180) +[HTML](73) +[PDF](562.29KB)

We consider infinite-dimensional parabolic rough evolution equations. Using regularizing properties of analytic semigroups we prove global-in-time existence of solutions and investigate random dynamical systems for such equations.

No-oscillation theorem for the transient dynamics of the linear signal transduction pathway and beyond
Tiejun Li, Tongkai Li and Shaoying Lu
2020, 25(7): 2749-2774 doi: 10.3934/dcdsb.2020030 +[Abstract](139) +[HTML](71) +[PDF](503.55KB)

Understanding the connection between the topology of a biochemical reaction network and its dynamical behavior is an important topic in systems biology. We proved a no-oscillation theorem for the transient dynamics of the linear signal transduction pathway, that is, there are no dynamical oscillations for each species if the considered system is a simple linear transduction chain equipped with an initial stimulation. In the nonlinear case, we showed that the no-oscillation property still holds for the starting and ending species, but oscillations generally exist in the dynamics of intermediate species. We also discussed different generalizations on the system setup. The established theorem will provide insights on the understanding of network motifs and the choice of mathematical models when dealing with biological data.

Interior structural bifurcation of 2D symmetric incompressible flows
Deniz Bozkurt, Ali Deliceoğlu and Taylan Şengül
2020, 25(7): 2775-2791 doi: 10.3934/dcdsb.2020032 +[Abstract](161) +[HTML](79) +[PDF](1045.95KB)

The structural bifurcation of a 2D divergence free vector field \begin{document}$ \mathbf{u}(\cdot, t) $\end{document} when \begin{document}$ \mathbf{u}(\cdot, t_0) $\end{document} has an interior isolated singular point \begin{document}$ \mathbf{x}_0 $\end{document} of zero index has been studied by Ma and Wang [23]. Although in the class of divergence free fields which undergo a local bifurcation around a singular point, the ones with index zero singular points are generic, this class excludes some important families of symmetric flows. In particular, when \begin{document}$ \mathbf{u}(\cdot, t_0) $\end{document} is anti-symmetric with respect to \begin{document}$ \mathbf{x}_0 $\end{document}, or symmetric with respect to the axis located on \begin{document}$ \mathbf{x}_0 $\end{document} and normal to the unique eigendirection of the Jacobian \begin{document}$ D\mathbf{u}(\cdot, t_0) $\end{document}, the vector field must have index 1 or -1 at the singular point. Thus we study the structural bifurcation when \begin{document}$ \mathbf{u}(\cdot, t_0) $\end{document} has an interior isolated singular point \begin{document}$ \mathbf{x}_0 $\end{document} with index -1, 1. In particular, we show that if such a vector field with its acceleration at \begin{document}$ t_0 $\end{document} both satisfy the aforementioned symmetries then generically the flow will undergo a local bifurcation. Under these generic conditions, we rigorously prove the existence of flow patterns such as pairs of co-rotating vortices and double saddle connections. We also present numerical evidence of the Stokes flow in a rectangular cavity showing that the bifurcation scenarios we present are indeed realizable.

Random attractors for stochastic time-dependent damped wave equation with critical exponents
Qingquan Chang, Dandan Li and Chunyou Sun
2020, 25(7): 2793-2824 doi: 10.3934/dcdsb.2020033 +[Abstract](193) +[HTML](79) +[PDF](631.14KB)

We study the asymptotic behavior of solutions of a stochastic time-dependent damped wave equation. With the critical growth restrictions on the nonlinearity \begin{document}$ f $\end{document} and the time-dependent damped term, we prove the global existence of solutions and characterize their long-time behavior. We show the existence of random attractors with finite fractal dimension in \begin{document}$ H^1_0(U)\times L^2(U) $\end{document}. In particular, the periodicity of random attractors is also obtained with periodic force term and coefficient function. Furthermore, we construct the pullback random exponential attractors.

Global smooth solution for the Sipn-Polarized transport equation with Landau-Lifshitz-Bloch equation
Boling Guo and Fangfang Li
2020, 25(7): 2825-2840 doi: 10.3934/dcdsb.2020034 +[Abstract](144) +[HTML](72) +[PDF](361.79KB)

The Landau-Lifshitz-Bloch equation is often used to describe micromagnetic phenomenon under high temperature. In this paper, we establish the existence and uniqueness of global smooth solution for the initial problem of the spin-polarized transport equation with Landau-Lifshitz-Bloch equation in dimension two.

2019  Impact Factor: 1.27




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