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Discrete & Continuous Dynamical Systems - B

2017 , Volume 22 , Issue 10

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Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$
Linfang Liu, Xianlong Fu and Yuncheng You
2017, 22(10): 3629-3651 doi: 10.3934/dcdsb.2017143 +[Abstract](149) +[HTML](1) +[PDF](475.4KB)

In this paper we study the asymptotic dynamics of the weak solutions of nonautonomous stochastic reaction-diffusion equations driven by a time-dependent forcing term and the multiplicative noise. By conducting the uniform estimates we show that the cocycle generated by this SRDE has a pullback \begin{document}$(L^2, H^1)$\end{document} absorbing set and it is pullback asymptotically compact through the pullback flattening approach. The existence of a pullback \begin{document}$(L^2, H^1)$\end{document} random attractor for this random dynamical system in space \begin{document}$H^{1}(\mathbb{R}^{n})$\end{document} is proved.

A PDE model of intraguild predation with cross-diffusion
Robert Stephen Cantrell, Xinru Cao, King-Yeung Lam and Tian Xiang
2017, 22(10): 3653-3661 doi: 10.3934/dcdsb.2017145 +[Abstract](140) +[HTML](1) +[PDF](356.5KB)

This note concerns a quasilinear parabolic system modeling an intraguild predation community in a focal habitat in \begin{document}$\mathbb{R}^n$\end{document}, \begin{document}$n ≥ 2$\end{document}. In this system the intraguild prey employs a fitness-based dispersal strategy whereby the intraguild prey moves away from a locale when predation risk is high enough to render the locale undesirable for resource acquisition. The system modifies the model considered in Ryan and Cantrell (2015) by adding an element of mutual interference among predators to the functional response terms in the model, thereby switching from Holling Ⅱ forms to Beddington-DeAngelis forms. We show that the resulting system can be realized as a semi-dynamical system with a global attractor for any \begin{document}$n ≥ 2$\end{document}. In contrast, the orginal model was restricted to two dimensional spatial habitats. The permanance of the intraguild prey then follows as in Ryan and Cantrell by means of the Acyclicity Theorem of Persistence Theory.

Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity
Wei Wang, Yan Li and Hao Yu
2017, 22(10): 3663-3669 doi: 10.3934/dcdsb.2017147 +[Abstract](83) +[HTML](0) +[PDF](345.5KB)

In this paper we study the global boundedness of solutions to the fully parabolic chemotaxis system with singular sensitivity:\begin{document}$u_t=\Delta u-\chi\nabla·(\frac{u}{v}\nabla v)$\end{document}, \begin{document}$v_t=k\Delta v-v+u$\end{document}, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain \begin{document}$\Omega\subset\mathbb{R}^{n}$\end{document} (\begin{document}$n\ge 2$\end{document}), where \begin{document}$\chi, \, k>0$\end{document}. It is shown that the solution is globally bounded provided \begin{document}$0<\chi<\frac{-(k-1)+\sqrt{(k-1)^2+\frac{8k}{n}}}{2}$\end{document}. This result removes the additional restriction of \begin{document}$n \le 8 $\end{document} in Zhao, Zheng [15] for the global boundedness of solutions.

Area preserving geodesic curvature driven flow of closed curves on a surface
Miroslav KolÁŘ, Michal BeneŠ and Daniel ŠevČoviČ
2017, 22(10): 3671-3689 doi: 10.3934/dcdsb.2017148 +[Abstract](144) +[HTML](2) +[PDF](715.1KB)

We investigate a non-local geometric flow preserving surface area enclosed by a curve on a given surface evolved in the normal direction by the geodesic curvature and the external force. We show how such a flow of surface curves can be projected into a flow of planar curves with the non-local normal velocity. We prove that the surface area preserving flow decreases the length of the evolved surface curves. Local existence and continuation of classical smooth solutions to the governing system of partial differential equations is analysed as well. Furthermore, we propose a numerical method of flowing finite volume for spatial discretization in combination with the Runge-Kutta method for solving the resulting system. Several computational examples demonstrate variety of evolution of surface curves and the order of convergence.

Pullback attractors of FitzHugh-Nagumo system on the time-varying domains
Zhen Zhang, Jianhua Huang and Xueke Pu
2017, 22(10): 3691-3706 doi: 10.3934/dcdsb.2017150 +[Abstract](99) +[HTML](3) +[PDF](440.4KB)

The existence and uniqueness of solutions satisfying energy equality is proved for non-autonomous FitzHugh-Nagumo system on a special time-varying domain which is a (possibly non-smooth) domain expanding with time. By constructing a suitable penalty function for the two cases respectively, we establish the existence of a pullback attractor for non-autonomous FitzHugh-Nagumo system on a special time-varying domain.

Dynamical behaviors of a generalized Lorenz family
Fuchen Zhang, Xiaofeng Liao, Guangyun Zhang, Chunlai Mu, Min Xiao and Ping Zhou
2017, 22(10): 3707-3720 doi: 10.3934/dcdsb.2017184 +[Abstract](237) +[HTML](2) +[PDF](363.3KB)

In this paper, the ultimate bound set and globally exponentially attractive set of a generalized Lorenz system are studied according to Lyapunov stability theory and optimization theory. The method of constructing Lyapunov-like functions applied to the former Lorenz-type systems (see, e.g. Lorenz system, Rossler system, Chua system) isn't applicable to this generalized Lorenz system. We overcome this difficulty by adding a cross term to the Lyapunov-like functions that used for the Lorenz system to study this generalized Lorenz system. The authors in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, Journal of Mathematical Analysis and Applications 323 (2006) 844-853] obtained the ultimate bound set of this generalized Lorenz system but only for some cases with $0 ≤ α < \frac{1}{{29}}.$ The ultimate bound set and globally exponential attractive set of this generalized Lorenz system are still unknown for $\alpha \notin \left[ {0, \frac{1}{{29}}} \right).$ Comparing with the best results in the current literature [D. Li, J. Lu, X. Wu, G. Chen, Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, Journal of Mathematical Analysis and Applications 323 (2006) 844-853], our new results fill up the gap of the estimate for the case of $\frac{1}{{29}} ≤ α < \frac{{14}}{{173}}.$ Furthermore, the estimation derived here contains the results given in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, J. Math. Anal. Appl. 323 (2006) 844-853] as special case for the case of $0 ≤ α < \frac{1}{{29}}.$

Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission
Jinliang Wang, Jiying Lang and Yuming Chen
2017, 22(10): 3721-3747 doi: 10.3934/dcdsb.2017186 +[Abstract](135) +[HTML](3) +[PDF](816.6KB)

In this paper, we are concerned with an age-structured HIV infection model incorporating latency and cell-to-cell transmission. The model is a hybrid system consisting of coupled ordinary differential equations and partial differential equations. First, we address the relative compactness and persistence of the solution semi-flow, and the existence of a global attractor. Then, applying the approach of Lyapunov functionals, we establish the global stability of the infection-free equilibrium and the infection equilibrium, which is completely determined by the basic reproduction number.

Impacts of cluster on network topology structure and epidemic spreading
Shuping Li and Zhen Jin
2017, 22(10): 3749-3770 doi: 10.3934/dcdsb.2017187 +[Abstract](282) +[HTML](3) +[PDF](571.5KB)

Considering the infection heterogeneity of different types of edges (lines and edges in the triangle in a network), we formulate and analyze an novel SIS model with cluster based mean-field approach for a network. We mainly focus on how network clustering influences network structure and the disease spreading over the network. In networks with double poisson distributions, power law-poisson distribution, poisson-power law distributions and double power law distributions, we find that cluster is positive(the clustering coefficient is increasing on the expected number of triangles) when the average degree of lines is fixed and the moment of triangles is less than some threshold. Once the moment of triangles exceeds that threshold, cluster will become negative(the clustering coefficient is decreasing on the expected number of triangles). For the disease, clustering always increases the basic reproduction number of the disease in networks with whether positive cluster or negative cluster. It is different from existing results that cluster always promotes the disease spread in the homogeneous or heterogeneous network.

Tumor growth dynamics with nutrient limitation and cell proliferation time delay
Ahuod Alsheri, Ebraheem O. Alzahrani, Asim Asiri, Mohamed M. El-Dessoky and Yang Kuang
2017, 22(10): 3771-3782 doi: 10.3934/dcdsb.2017189 +[Abstract](66) +[HTML](1) +[PDF](391.0KB)

It is known that avascular spherical solid tumors grow monotonically, often tends to a limiting final size. This is repeatedly confirmed by various mathematical models consisting of mostly ordinary differential equations. However, cell growth is limited by nutrient and its proliferation incurs a time delay. In this paper, we formulate a nutrient limited compartmental model of avascular spherical solid tumor growth with cell proliferation time delay and study its limiting dynamics. The nutrient is assumed to enter the tumor proportional to its surface area. This model is a modification of a recent model which is built on a two-compartment model of cancer cell growth with transitions between proliferating and quiescent cells. Due to the limitation of resources, it is imperative that the population values or densities of a population model be nonnegative and bounded without any technical conditions. We confirm that our model meets this basic requirement. From an explicit expression of the tumor final size we show that the ratio of proliferating cells to the total tumor cells tends to zero as the death rate of quiescent cells tends to zero. We also study the stability of the tumor at steady states even though there is no Jacobian at the trivial steady state. The characteristic equation at the positive steady state is complicated so we made an initial effort to study some special cases in details. We find that delay may not destabilize the positive steady state in a very extreme situation. However, in a more general case, we show that sufficiently long cell proliferation delay can produce oscillatory solutions.

Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions
Wei Liu, Pavel Krejčí and Guoju Ye
2017, 22(10): 3783-3795 doi: 10.3934/dcdsb.2017190 +[Abstract](111) +[HTML](1) +[PDF](810.3KB)

It is well known that the Prandtl-Ishlinskii hysteresis operator is locally Lipschitz continuous in the space of continuous functions provided its primary response curve is convex or concave. This property can easily be extended to any absolutely continuous primary response curve with derivative of locally bounded variation. Under the same condition, the Prandtl-Ishlinskii operator in the Kurzweil integral setting is locally Lipschitz continuous also in the space of regulated functions. This paper shows that the Prandtl-Ishlinskii operator is still continuous if the primary response curve is only monotone and continuous, and that it may not even be locally Hölder continuous for continuously differentiable primary response curves.

Threshold dynamics of a reaction-diffusion epidemic model with stage structure
Liang Zhang and Zhi-Cheng Wang
2017, 22(10): 3797-3820 doi: 10.3934/dcdsb.2017191 +[Abstract](98) +[HTML](2) +[PDF](492.4KB)

A time-delayed reaction-diffusion epidemic model with stage structure and spatial heterogeneity is investigated, which describes the dynamics of disease spread only proceeding in the adult population. We establish the basic reproduction number \begin{document} $\mathcal{R}_0$ \end{document} for the model system, which gives the threshold dynamics in the sense that the disease will die out if \begin{document} $\mathcal{R}_0<1$ \end{document} and the disease will be uniformly persistent if \begin{document} $\mathcal{R}_0>1.$ \end{document} Furthermore, it is shown that there is at least one positive steady state when \begin{document} $\mathcal{R}_0>1.$ \end{document} Finally, in terms of general birth function for adult individuals, through introducing two numbers \begin{document} $\check{\mathcal{R}}_0$ \end{document} and \begin{document} $\hat{\mathcal{R}}_0$ \end{document}, we establish sufficient conditions for the persistence and global extinction of the disease, respectively.

Linear programming formulations of deterministic infinite horizon optimal control problems in discrete time
Vladimir Gaitsgory, Alex Parkinson and Ilya Shvartsman
2017, 22(10): 3821-3838 doi: 10.3934/dcdsb.2017192 +[Abstract](126) +[HTML](1) +[PDF](427.6KB)

This paper is devoted to a study of infinite horizon optimal control problems with time discounting and time averaging criteria in discrete time. We establish that these problems are related to certain infinite-dimensional linear programming (IDLP) problems. We also establish asymptotic relationships between the optimal values of problems with time discounting and long-run average criteria.

Asymptotic dynamics in a two-species chemotaxis model with non-local terms
Tahir Bachar Issa and Rachidi Bolaji Salako
2017, 22(10): 3839-3874 doi: 10.3934/dcdsb.2017193 +[Abstract](98) +[HTML](1) +[PDF](607.9KB)

In this study, we consider an extended attraction two species chemotaxis system of parabolic-parabolic-elliptic type with nonlocal terms under homogeneous Neuman boundary conditions in a bounded domain \begin{document} $Ω \subset \mathbb{R}^n(n≥1)$ \end{document} with smooth boundary. We first prove the global existence of non-negative classical solutions for various explicit parameter regions. Next, under some further explicit conditions on the coefficients and on the chemotaxis sensitivities, we show that the system has a unique positive constant steady state solution which is globally asymptotically stable. Finally, we also find some explicit conditions on the coefficient and on the chemotaxis sensitivities for which the phenomenon of competitive exclusion occurs in the sense that as time goes to infinity, one of the species dies out and the other reaches its carrying capacity. The method of eventual comparison is used to study the asymptotic behavior.

Robustness of exponentially κ-dissipative dynamical systems with perturbations
Jin Zhang, Yonghai Wang and Chengkui Zhong
2017, 22(10): 3875-3890 doi: 10.3934/dcdsb.2017198 +[Abstract](73) +[HTML](1) +[PDF](455.3KB)

We study the robustness of exponentially \begin{document}$κ$\end{document}-dissipative dynamical systems with perturbed parameters \begin{document}$\varepsilon∈ E(\subset\mathbb{R})$\end{document}. In particular, under some proper assumptions, we will construct a family of compact sets \begin{document}$\{\mathcal A_\varepsilon\}_{\varepsilon∈ E}$\end{document}, which is positive invariant, uniformly exponentially attracting and equi-continuous. At last, an application to a Kirchhoff wave model is given.

A reaction-diffusion system arising in game theory: existence of solutions and spatial dominance
Hideo Deguchi
2017, 22(10): 3891-3901 doi: 10.3934/dcdsb.2017200 +[Abstract](80) +[HTML](1) +[PDF](371.0KB)

The initial value problem for a reaction-diffusion system with discontinuous nonlinearities proposed by Hofbauer in 1999 as an equilibrium selection model in game theory is studied from the viewpoint of the existence and stability of solutions. An equilibrium selection result using the stability of a constant stationary solution is obtained for finite symmetric 2 person games with a 1/2-dominant equilibrium.

On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity
Zilai Li and Zhenhua Guo
2017, 22(10): 3903-3919 doi: 10.3934/dcdsb.2017201 +[Abstract](109) +[HTML](3) +[PDF](404.8KB)

We obtain the existence of global strong solution to the free boundary problem in 1D compressible Navier-Stokes system for the viscous and heat conducting ideal polytropic gas flow, when the heat conductivity depends on temperature in power law of Chapman-Enskog and the viscosity coefficient be a positive constant.

Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model
Stefano Villa, Paola Goatin and Christophe Chalons
2017, 22(10): 3921-3952 doi: 10.3934/dcdsb.2017202 +[Abstract](58) +[HTML](1) +[PDF](914.1KB)

We introduce a second order model for traffic flow with moving bottlenecks. The model consists of the × 2$ Aw-Rascle-Zhang system with a point-wise flow constraint whose trajectory is governed by an ordinary differential equation. We define two Riemann solvers, characterize the corresponding invariant domains and propose numerical strategies, which are effective in capturing the non-classical shocks due to the constraint activation.

Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems
Xingwu Chen, Jaume Llibre and Weinian Zhang
2017, 22(10): 3953-3965 doi: 10.3934/dcdsb.2017203 +[Abstract](119) +[HTML](4) +[PDF](354.6KB)

It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in \begin{document}$\mathbb{R}^2$\end{document} is \begin{document}$\end{document}. In contrast here we consider discontinuous differential systems in \begin{document}$\mathbb{R}^2$\end{document} defined in two half-planes separated by a straight line. In one half plane we have a general linear center at the origin of \begin{document}$\mathbb{R}^2$\end{document}, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of \begin{document}$\mathbb{R}^2$\end{document}. Using averaging theory, we prove that the cyclicity of a Hopf bifurcation for such discontinuous differential systems is at least 5. Our computations show that only one of the averaged functions of fifth order can produce 5 limit cycles and there are no more limit cycles up to sixth order averaged function.

Wild oscillations in a nonlinear neuron model with resets: (Ⅰ) Bursting, spike-adding and chaos
Jonathan E. Rubin, Justyna Signerska-Rynkowska, Jonathan D. Touboul and Alexandre Vidal
2017, 22(10): 3967-4002 doi: 10.3934/dcdsb.2017204 +[Abstract](281) +[HTML](1) +[PDF](5697.5KB)

In a series of two papers, we investigate the mechanisms by which complex oscillations are generated in a class of nonlinear dynamical systems with resets modeling the voltage and adaptation of neurons. This first paper presents mathematical analysis showing that the system can support bursts of any period as a function of model parameters, and that these are organized in a period-incrementing structure. In continuous dynamical systems with resets, such structures are complex to analyze. In the present context, we use the fact that bursting patterns correspond to periodic orbits of the adaptation map that governs the sequence of values of the adaptation variable at the resets. Using a slow-fast approach, we show that this map converges towards a piecewise linear discontinuous map whose orbits are exactly characterized. That map shows a period-incrementing structure with instantaneous transitions. We show that the period-incrementing structure persists for the full system with non-constant adaptation, but the transitions are more complex. We investigate the presence of chaos at the transitions.

Wild oscillations in a nonlinear neuron model with resets: (Ⅱ) Mixed-mode oscillations
Jonathan E. Rubin, Justyna Signerska-Rynkowska, Jonathan D. Touboul and Alexandre Vidal
2017, 22(10): 4003-4039 doi: 10.3934/dcdsb.2017205 +[Abstract](147) +[HTML](2) +[PDF](2428.8KB)

This work continues the analysis of complex dynamics in a class of bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal voltage dynamics with adaptation and spike emission. We show that these models can generically display a form of mixed-mode oscillations (MMOs), which are trajectories featuring an alternation of small oscillations with spikes or bursts (multiple consecutive spikes). The mechanism by which these are generated relies fundamentally on the hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts, contrasting with classical MMO mechanisms in ordinary differential equations involving more than three dimensions and generally relying on a timescale separation. The decomposition of mechanisms reveals the geometrical origin of MMOs, allowing a relatively simple classification of points on the reset manifold associated to specific numbers of small oscillations. We show that the MMO pattern can be described through the study of orbits of a discrete adaptation map, which is singular as it features discrete discontinuities with unbounded left-and right-derivatives. We study orbits of the map via rotation theory for discontinuous circle maps and elucidate in detail complex behaviors arising in the case where MMOs display at most one small oscillation between each consecutive pair of spikes.

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