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Discrete & Continuous Dynamical Systems - B

March 2018 , Volume 23 , Issue 2

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A unifying approach to discrete single-species populations models
James Sandefur
2018, 23(2): 493-508 doi: 10.3934/dcdsb.2017194 +[Abstract](872) +[HTML](259) +[PDF](666.76KB)
Abstract:

In this article, we write the recruitment function \begin{document}$f$\end{document} for the discrete-time density-dependent population model

as \begin{document}$f(p) =p+r(p)p$\end{document} where \begin{document}$r$\end{document} is the per capita growth rate. Making reasonable assumptions about the intraspecies relationships for the population, we develop four conditions that the function \begin{document}$r$\end{document} should satisfy. We then analyze the implications of these conditions for the recruitment function \begin{document}$f$\end{document}. In particular, we compare our conditions to those of Cull [2007], finding that the Cull model, with two additional conditions, is equivalent to our model.

Studying the per capita growth rate when satisfying our four conditions gives insight into contest and scramble competition. In particular, depending on the properties of \begin{document}$r$\end{document} and \begin{document}$f$\end{document}, we have two different types of contest and scramble competitions, depending on the size of the population. We finally extend our approach to develop new models for discontinuous recruitment functions and for populations exhibiting Allee effects.

Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay
Rodrigo Samprogna and Tomás Caraballo
2018, 23(2): 509-523 doi: 10.3934/dcdsb.2017195 +[Abstract](833) +[HTML](172) +[PDF](388.44KB)
Abstract:

In this work we prove the existence of solution for a p-Laplacian non-autonomous problem with dynamic boundary and infinite delay. We ensure the existence of pullback attractor for the multivalued process associated to the non-autonomous problem we are concerned.

Asymptotic behaviour of the solutions to a virus dynamics model with diffusion
Toru Sasaki and Takashi Suzuki
2018, 23(2): 525-541 doi: 10.3934/dcdsb.2017206 +[Abstract](1012) +[HTML](333) +[PDF](690.66KB)
Abstract:

Asymptotic behaviour of the solutions to a basic virus dynamics model is discussed. We consider the population of uninfected cells, infected cells, and virus particles. Diffusion effect is incorporated there. First, the Lyapunov function effective to the spatially homogeneous part (ODE model without diffusion) admits the \begin{document}$L^1$\end{document} boundedness of the orbit. Then the pre-compactness of this orbit in the space of continuous functions is derived by the semigroup estimates. Consequently, from the invariant principle, if the basic reproductive number \begin{document}$R_0$\end{document} is less than or equal to 1, each orbit converges to the disease free spatially homogeneous equilibrium, and if \begin{document}$R_0>1$\end{document}, each orbit converges to the infected spatially homogeneous equilibrium, which means that the simple diffusion does not affect the asymptotic behaviour of the solutions.

Restrictions to the use of time-delayed feedback control in symmetric settings
Edward Hooton, Pavel Kravetc and Dmitrii Rachinskii
2018, 23(2): 543-556 doi: 10.3934/dcdsb.2017207 +[Abstract](520) +[HTML](127) +[PDF](646.25KB)
Abstract:

We consider the problem of stabilization of unstable periodic solutions to autonomous systems by the non-invasive delayed feedback control known as Pyragas control method. The Odd Number Theorem imposes an important restriction upon the choice of the gain matrix by stating a necessary condition for stabilization. In this paper, the Odd Number Theorem is extended to equivariant systems. We assume that both the uncontrolled and controlled systems respect a group of symmetries. Two types of results are discussed. First, we consider rotationally symmetric systems for which the control stabilizes the whole orbit of relative periodic solutions that form an invariant two-dimensional torus in the phase space. Second, we consider a modification of the Pyragas control method that has been recently proposed for systems with a finite symmetry group. This control acts non-invasively on one selected periodic solution from the orbit and targets to stabilize this particular solution. Variants of the Odd Number Limitation Theorem are proposed for both above types of systems. The results are illustrated with examples that have been previously studied in the literature on Pyragas control including a system of two symmetrically coupled Stewart-Landau oscillators and a system of two coupled lasers.

On a distributed control problem for a coupled chemotaxis-fluid model
M. Ángeles Rodríguez-Bellido, Diego A. Rueda-Gómez and Élder J. Villamizar-Roa
2018, 23(2): 557-571 doi: 10.3934/dcdsb.2017208 +[Abstract](732) +[HTML](203) +[PDF](450.59KB)
Abstract:

In this paper we analyze an optimal distributed control problem where the state equations are given by a stationary chemotaxis model coupled with the Navier-Stokes equations. We consider that the movement and the interaction of cells are occurring in a smooth bounded domain of \begin{document}$\mathbb{R}^n,n = 2,3,$\end{document} subject to homogeneous boundary conditions. We control the system through a distributed force and a coefficient of chemotactic sensitivity, leading the chemical concentration, the cell density, and the velocity field towards a given target concentration, density and velocity, respectively. In addition to the existence of optimal solution, we derive some optimality conditions.

High order Gauss-Seidel schemes for charged particle dynamics
Yuezheng Gong, Jiaquan Gao and Yushun Wang
2018, 23(2): 573-585 doi: 10.3934/dcdsb.2018034 +[Abstract](609) +[HTML](134) +[PDF](6511.01KB)
Abstract:

Gauss-Seidel projection methods are designed for achieving desirable long-term computational efficiency and reliability in micromagnetics simulations. While conventional Gauss-Seidel schemes are explicit, easy to use and furnish a better stability as compared to Euler's method, their order of accuracy is only one. This paper proposes an improved Gauss-Seidel methodology for particle simulations of magnetized plasmas. A novel new class of high order schemes are implemented via composition strategies. The new algorithms acquired are not only explicit and symmetric, but also volume-preserving together with their adjoint schemes. They are highly favorable for long-term computations. The new high order schemes are then utilized for simulating charged particle motions under the Lorentz force. Our experiments indicate a remarkable satisfaction of the energy preservation and angular momentum conservation of the numerical methods in multi-scale plasma dynamics computations.

Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal
Guo-Bao Zhang, Ruyun Ma and Xue-Shi Li
2018, 23(2): 587-608 doi: 10.3934/dcdsb.2018035 +[Abstract](739) +[HTML](204) +[PDF](472.19KB)
Abstract:

This paper is concerned with the traveling waves of a nonlocal dispersal Lotka-Volterra strong competition model with bistable nonlinearity. We first establish the asymptotic behavior of traveling waves at infinity. Then by applying the stronger comparison principle and the sliding method, we prove that the traveling waves with nonzero speed are strictly monotone. Moreover, the uniqueness of wave speeds is also obtained.

Stability of travelling waves in a Wolbachia invasion
Matthew H. Chan, Peter S. Kim and Robert Marangell
2018, 23(2): 609-628 doi: 10.3934/dcdsb.2018036 +[Abstract](547) +[HTML](200) +[PDF](2427.96KB)
Abstract:

Numerous studies have examined the growth dynamics of Wolbachia within populations and the resultant rate of spatial spread. This spread is typically characterised as a travelling wave with bistable local growth dynamics due to a strong Allee effect generated from cytoplasmic incompatibility. While this rate of spread has been calculated from numerical solutions of reaction-diffusion models, none have examined the spectral stability of such travelling wave solutions. In this study we analyse the stability of a travelling wave solution generated by the reaction-diffusion model of Chan & Kim [4] by computing the essential and point spectrum of the linearised operator arising in the model. The point spectrum is computed via an Evans function using the compound matrix method, whereby we find that it has no roots with positive real part. Moreover, the essential spectrum lies strictly in the left half plane. Thus, we find that the travelling wave solution found by Chan & Kim [4] corresponding to competition between Wolbachia-infected and -uninfected mosquitoes is linearly stable. We employ a dimension counting argument to suggest that, under realistic conditions, the wavespeed corresponding to such a solution is unique.

Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media
Brahim Amaziane, Mladen Jurak, Leonid Pankratov and Anja Vrbaški
2018, 23(2): 629-665 doi: 10.3934/dcdsb.2018037 +[Abstract](480) +[HTML](154) +[PDF](759.23KB)
Abstract:

This paper presents a study of immiscible incompressible two-phase flow through fractured porous media. The results obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikelić (1996) and L. M. Yeh (2006) are revisited. The main goal is to incorporate some of the most recent improvements in the convergence of the solutions in the homogenization of such models. The microscopic model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat law. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by \begin{document}${\varepsilon }^θ$\end{document}, where \begin{document}$\varepsilon$ \end{document} is the size of a typical porous block and \begin{document}$θ>0$\end{document} is a parameter. The model involves highly oscillatory characteristics and internal nonlinear interface conditions. Under some realistic assumptions on the data, the convergence of the solutions, and the macroscopic models corresponding to various range of contrast are constructed using the two-scale convergence method combined with the dilation technique. The results improve upon previously derived effective models to highly heterogeneous porous media with discontinuous capillary pressures.

Global Hopf bifurcations of neutral functional differential equations with state-dependent delay
Xiuli Sun, Rong Yuan and Yunfei Lv
2018, 23(2): 667-700 doi: 10.3934/dcdsb.2018038 +[Abstract](704) +[HTML](162) +[PDF](540.04KB)
Abstract:

A global Hopf bifurcation theory for a system of neutral functional differential equations (NFDEs) with state-dependent delay is investigated by applying the \begin{document}$S^{1}$\end{document}-equivariant degree theory. We use the information about the characteristic equation of the formal linearization with frozen delay to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system. The results are important in studying bifurcations of NFDEs with state-dependent delay.

Cascades of alternating smooth bifurcations and border collision bifurcations with singularity in a family of discontinuous linear-power maps
Laura Gardini, Roya Makrooni and Iryna Sushko
2018, 23(2): 701-729 doi: 10.3934/dcdsb.2018039 +[Abstract](709) +[HTML](174) +[PDF](870.99KB)
Abstract:

We investigate the dynamics of a family of one-dimensional linear-power maps. This family has been studied by many authors mainly in the continuous case, associated with Nordmark systems. In the discontinuous case, which is much less studied, the map has vertical and horizontal asymptotes giving rise to new kinds of border collision bifurcations. We explain a mechanism of the interplay between smooth bifurcations and border collision bifurcations with singularity, leading to peculiar sequences of attracting cycles of periods \begin{document}$n,2n$\end{document}, \begin{document}$4n-1$\end{document}, \begin{document}$2(4n-1)$\end{document}, ..., \begin{document}$n≥3$\end{document}. We show also that the transition from invertible to noninvertible map may lead abruptly to chaos, and the role of organizing center in the parameter space is played by a particular bifurcation point related to this transition and to a flip bifurcation. Robust unbounded chaotic attractors characteristic for certain parameter ranges are also described. We provide proofs of some properties of the considered map. However, the complete description of its rich bifurcation structure is still an open problem.

Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction
Nika Lazaryan and Hassan Sedaghat
2018, 23(2): 731-747 doi: 10.3934/dcdsb.2018040 +[Abstract](499) +[HTML](174) +[PDF](491.67KB)
Abstract:

We study the evolution in discrete time of certain age-structured populations, such as adults and juveniles, with a Ricker fitness function. We determine conditions for the convergence of orbits to the origin (extinction) in the presence of the Allee effect and time-dependent vital rates. We show that when stages interact, they may survive in the absence of interior fixed points, a surprising situation that is impossible without inter-stage interactions. We also examine the shift in the interior Allee equilibrium caused by the occurrence of interactions between stages and find that the extinction or Allee threshold does not extend to the new boundaries set by the shift in equilibrium, i.e. no interior equilibria are on the extinction threshold.

Long-time behavior of a class of nonlocal partial differential equations
Chang Zhang, Fang Li and Jinqiao Duan
2018, 23(2): 749-763 doi: 10.3934/dcdsb.2018041 +[Abstract](1023) +[HTML](317) +[PDF](420.27KB)
Abstract:

This work is devoted to investigate the well-posedness and long-time behavior of solutions for the following nonlocal nonlinear partial differential equations in a bounded domain

Firstly, due to the lack of an upper growth restriction of the nonlinearity $f$, we have to utilize a weak compactness approach in an Orlicz space to obtain the well-posedness of weak solutions for the equations. We then establish the existence of \begin{document}$(L^2_0(Ω), L^2_0(Ω))$\end{document}-absorbing sets and \begin{document}$(L^2_0(Ω), H^{σ/2}_0(Ω))$\end{document}-absorbing sets for the solution semigroup \begin{document}$\{S(t)\}_{t≥q 0}$\end{document}. Finally, we prove the existence of \begin{document}$(L^2_0(Ω), L^2_0(Ω))$\end{document}-global attractor and \begin{document}$(L^2_0(Ω), H^{σ/2}_0(Ω))$\end{document}-global attractor by a asymptotic compactness method.

Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion
Xiaofeng Xu and Junjie Wei
2018, 23(2): 765-783 doi: 10.3934/dcdsb.2018042 +[Abstract](822) +[HTML](175) +[PDF](1211.07KB)
Abstract:

In this paper, the dynamics of a class of modified Leslie-Gower model with diffusion is considered. The stability of positive equilibrium and the existence of Turing-Hopf bifurcation are shown by analyzing the distribution of eigenvalues. The normal form on the centre manifold near the Turing-Hopf singularity is derived by using the method of Song et al. Finally, some numerical simulations are carried out to illustrate the analytical results. For spruce budworm model, the dynamics in the neighbourhood of the bifurcation point can be divided into six categories, each of which is exactly demonstrated by the numerical simulations. Then according to this dynamical classification, a stable spatially inhomogeneous periodic solution has been found, which can be used to explain the phenomenon of periodic outbreaks of spruce budworm.

How seasonal forcing influences the complexity of a predator-prey system
Xueping Li, Jingli Ren, Sue Ann Campbell, Gail S. K. Wolkowicz and Huaiping Zhu
2018, 23(2): 785-807 doi: 10.3934/dcdsb.2018043 +[Abstract](1338) +[HTML](348) +[PDF](4589.35KB)
Abstract:

Almost all population communities are strongly influenced by their seasonally varying living environments. We investigate the influence of seasons on populations via a periodically forced predator-prey system with a nonmonotonic functional response. We study four seasonality mechanisms via a continuation technique. When the natural death rate is periodically varied, we get six different bifurcation diagrams corresponding to different bifurcation cases of the unforced system. If the carrying capacity is periodic, two different bifurcation diagrams are obtained. Here we cannot get a "universal diagram" like the one in the periodically forced system with monotonic Holling type Ⅱ functional response; that is, the two elementary seasonality mechanisms have different effects on the population. When both the natural death rate and the carrying capacity are forced with two different seasonality mechanisms, the phenomena that arise are to some extent different. The bifurcation results also show that each seasonality mechanism can display complex dynamics such as multiple attractors including stable cycles of different periods, quasi-periodic solutions, chaos, switching between these attractors and catastrophic transitions. In addition, we give some orbits in phase space and corresponding Poincaré sections to illustrate different attractors.

Boundedness of positive solutions of a system of nonlinear delay differential equations
István Győri, Ferenc Hartung and Nahed A. Mohamady
2018, 23(2): 809-836 doi: 10.3934/dcdsb.2018044 +[Abstract](548) +[HTML](163) +[PDF](4742.82KB)
Abstract:

In this manuscript the system of nonlinear delay differential equations \begin{document}$\dot{x}_i(t) =\sum\limits_{j =1}^{n}\sum\limits_{\ell =1}^{n_0}α_{ij\ell} (t) h_{ij}(x_j(t-τ_{ij\ell}(t)))$\end{document}\begin{document}$-β_i(t)f_i(x_i(t))+ρ_i(t)$\end{document}, \begin{document}$t≥0$\end{document}, \begin{document}$1≤i ≤n$\end{document} is considered. Sufficient conditions are established for the uniform permanence of the positive solutions of the system. In several particular cases, explicit formulas are given for the estimates of the upper and lower limit of the solutions. In a special case, the asymptotic equivalence of the solutions is investigated.

Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method
Shuang Liu and Wenxue Li
2018, 23(2): 837-859 doi: 10.3934/dcdsb.2018045 +[Abstract](595) +[HTML](174) +[PDF](1325.86KB)
Abstract:

We consider the outer synchronization between drive-response systems on networks with time-varying delays, where we focus on the case when the underlying networks are not strongly connected. A hierarchical method is proposed to characterize large-scale networks without strong connectedness. The hierarchical algorithm can be implemented by some programs to overcome the difficulty resulting from the scale of networks. This method allows us to obtain two different kinds of sufficient outer synchronization criteria without the assumption of being strongly connected, by combining the theory of asymptotically autonomous systems with Lyapunov method and Kirchhoff's Matrix Tree Theorem in graph theory. The theory improves some existing results obtained by graph theory. As illustrations, the theoretic results are applied to delayed coupled oscillators and a numerical example is also given.

Hopf bifurcation of an age-structured virus infection model
Hossein Mohebbi, Azim Aminataei, Cameron J. Browne and Mohammad Reza Razvan
2018, 23(2): 861-885 doi: 10.3934/dcdsb.2018046 +[Abstract](976) +[HTML](343) +[PDF](794.93KB)
Abstract:

In this paper, we introduce and analyze a mathematical model of a viral infection with explicit age-since infection structure for infected cells. We extend previous age-structured within-host virus models by including logistic growth of target cells and allowing for absorption of multiple virus particles by infected cells. The persistence of the virus is shown to depend on the basic reproduction number \begin{document}$R_{0}$\end{document}. In particular, when \begin{document}$R_{0}≤1$\end{document}, the infection free equilibrium is globally asymptotically stable, and conversely if \begin{document}$R_{0}> 1$\end{document}, then the infection free equilibrium is unstable, the system is uniformly persistent and there exists a unique positive equilibrium. We show that our system undergoes a Hopf bifurcation through which the infection equilibrium loses the stability and periodic solutions appear.

Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis
Jaume Llibre, Y. Paulina Martínez and Claudio Vidal
2018, 23(2): 887-912 doi: 10.3934/dcdsb.2018047 +[Abstract](823) +[HTML](165) +[PDF](1304.14KB)
Abstract:

We provide the phase portraits in the Poincaré disk for all the linear type centers of polynomial Hamiltonian systems with nonlinearities of degree \begin{document}$4$\end{document} symmetric with respect to the \begin{document}$y$\end{document}-axis given by the Hamiltonian function \begin{document}$H(x,y) =1/2(x^2+y^2)+ax^4y+bx^2y^3+cy^5$\end{document} in function of its parameters.

Large deviation principle for the micropolar, magneto-micropolar fluid systems
Kazuo Yamazaki
2018, 23(2): 913-938 doi: 10.3934/dcdsb.2018048 +[Abstract](453) +[HTML](169) +[PDF](595.54KB)
Abstract:

Micropolar fluid and magneto-micropolar fluid systems are systems of equations with distinctive feature in its applicability and also mathematical difficulty. The purpose of this work is to follow the approach of [8] and show that another general class of systems of equations, that includes the two-dimensional micropolar and magneto-micropolar fluid systems, is well-posed and satisfies the Laplace principle, and consequently the large deviation principle, with the same rate function.

Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming
Sigurdur Hafstein, Skuli Gudmundsson, Peter Giesl and Enrico Scalas
2018, 23(2): 939-956 doi: 10.3934/dcdsb.2018049 +[Abstract](609) +[HTML](195) +[PDF](512.46KB)
Abstract:

We study the global asymptotic stability in probability of the zero solution of linear stochastic differential equations with constant coefficients. We develop a sum-of-squares program that verifies whether a parameterized candidate Lyapunov function is in fact a global Lyapunov function for such a system. Our class of candidate Lyapunov functions are naturally adapted to the problem. We consider functions of the form \begin{document} $V(\mathbf{x}) = \|\mathbf{x}\|_Q^p: = (\mathbf{x}^\top Q\mathbf{x})^{\frac{p}{2}}$ \end{document}, where the parameters are the positive definite matrix \begin{document} $Q$ \end{document} and the number \begin{document} $p>0$ \end{document}. We give several examples of our proposed method and show how it improves previous results.

Modeling the transmission of dengue fever with limited medical resources and self-protection
Min Zhu and Zhigui Lin
2018, 23(2): 957-974 doi: 10.3934/dcdsb.2018050 +[Abstract](724) +[HTML](126) +[PDF](757.32KB)
Abstract:

To capture the impacts of limited medical resources and self-protection on the transmission of dengue fever, we formulate an SIS v.s. SI dengue model with the nonlinear recovery rate and contact transmission rate. The spatial heterogeneity of environment is also taken into consideration. With the aid of the relevant eigenvalue problem, we explore some properties of the basic reproduction number, and show that it still plays its "traditional" role in determining the stability of equilibria, that is, the extinction and persistence of dengue fever. Moreover, we consider a special diffusive pattern in which there is only human diffusion, but no mosquitoes diffusion, then present the explicit expression of the basic reproduction number and exhibit the corresponding transmission dynamics. This paper ends up with some numerical simulations and epidemiological explanations, which confirm our analytical findings.

New approach of controlling cardiac alternans
Mounira Kesmia, Soraya Boughaba and Sabir Jacquir
2018, 23(2): 975-989 doi: 10.3934/dcdsb.2018051 +[Abstract](505) +[HTML](168) +[PDF](437.27KB)
Abstract:

The alternans of the cardiac action potential duration is a pathological rhythm. It is considered to be relating to the onset of ventricular fibrillation and sudden cardiac death. It is well known that, the predictive control is among the control methods that use the chaos to stabilize the unstable fixed point. Firstly, we show that alternans (or period-2 orbit) can be suppressed temporally by the predictive control of the periodic state of the system. Secondly, we determine an estimation of the size of a restricted attraction's basin of the unstable equilibrium point representing the unstable regular rhythm stabilized by the control. This result allows the application of predictive control after one beat of alternans. In particular, using predictive control of periodic dynamics, we can delay the onset of bifurcations and direct a trajectory to a desired target stationary state. Examples of the numerical results showing the stabilization of the unstable normal rhythm are given.

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