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Discrete & Continuous Dynamical Systems - B

June 2019 , Volume 24 , Issue 6

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Confinement of a hot temperature patch in the modified SQG model
Roberto Garra
2019, 24(6): 2407-2416 doi: 10.3934/dcdsb.2018258 +[Abstract](823) +[HTML](445) +[PDF](340.11KB)

In this paper we study the time evolution of a temperature patch in $\mathbb{R}^2$ according to the modified Surface Quasi-Geostrophic (SQG) patch equation. In particular we give a temporal estimate on the growth of the support, providing a rigorous proof of the confinement of a hot patch of temperature in absence of external forcing, under the quasi-geostrophic approximation.

Oscillations and asymptotic convergence for a delay differential equation modeling platelet production
Loïs Boullu, Mostafa Adimy, Fabien Crauste and Laurent Pujo-Menjouet
2019, 24(6): 2417-2442 doi: 10.3934/dcdsb.2018259 +[Abstract](923) +[HTML](528) +[PDF](617.09KB)

We analyze the existence of oscillating solutions and the asymptotic convergence for a nonlinear delay differential equation arising from the modeling of platelet production. We consider four different cell compartments corresponding to different cell maturity levels: stem cells, megakaryocytic progenitors, megakaryocytes, and platelets compartments, and the quantity of circulating thrombopoietin (TPO), a platelet regulation cytokine.

Our initial model consists in a nonlinear age-structured partial differential equation system, where each equation describes the dynamics of a single compartment. This system is reduced to a single nonlinear delay differential equation describing the dynamics of the platelet population, in which the delay accounts for a differentiation time.

After introducing the model, we prove the existence of a unique steady state for the delay differential equation. Then we determine necessary and sufficient conditions for the existence of oscillating solutions. Next we set up conditions to get local asymptotic stability and asymptotic convergence of this steady state. Finally we present a short analysis of the influence of the conditions at t < 0 on the proof for asymptotic convergence.

Asymptotics of the Lebowitz-Rubinow-Rotenberg model of population development
Adam Gregosiewicz
2019, 24(6): 2443-2472 doi: 10.3934/dcdsb.2018260 +[Abstract](965) +[HTML](420) +[PDF](688.44KB)

We study a mathematical model of cell populations dynamics proposed by J. Lebowitz and S. Rubinow, and analysed by M. Rotenberg. Here, a cell is characterized by her maturity and speed of maturation. The growth of cell populations is described by a partial differential equation with a boundary condition. In the first part of the paper we exploit semigroup theory approach and apply Lord Kelvin's method of images in order to give a new proof that the model is well posed. A semi-explicit formula for the semigroup related to the model obtained by the method of images allows two types of new results. First of all, we give growth order estimates for the semigroup, applicable also in the case of decaying populations. Secondly, we study asymptotic behavior of the semigroup in the case of approximately constant population size. More specifically, we formulate conditions for the asymptotic stability of the semigroup in the case in which the average number of viable daughters per mitosis equals one. To this end we use methods developed by K. Pichór and R. Rudnicki.

Symmetries of nonlinear vibrations in tetrahedral molecular configurations
Irina Berezovik, Carlos García-Azpeitia and Wieslaw Krawcewicz
2019, 24(6): 2473-2491 doi: 10.3934/dcdsb.2018261 +[Abstract](823) +[HTML](447) +[PDF](278.49KB)

We study nonlinear vibrational modes of oscillations for tetrahedral configurations of particles. In the case of tetraphosphorus, the interaction of atoms is given by bond stretching and van der Waals forces. Using the equivariant gradient degree, we present a topological classification of the spatio-temporal symmetries of the periodic solutions with finite Weyl's group. This procedure describes all the symmetries of the nonlinear vibrations for general force fields.

Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations
Peng Gao
2019, 24(6): 2493-2510 doi: 10.3934/dcdsb.2018262 +[Abstract](762) +[HTML](418) +[PDF](381.7KB)

In this paper, we first establish two global Carleman estimates for linear stochastic nonclassical diffusion equations. Based on these estimates, we obtain two types of Unique Continuation Property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations.

Global eradication for spatially structured populations by regional control
Sebastian Aniţa, Vincenzo Capasso and Ana-Maria Moşneagu
2019, 24(6): 2511-2533 doi: 10.3934/dcdsb.2018263 +[Abstract](767) +[HTML](457) +[PDF](491.96KB)

This paper concerns problems for the eradication of apopulation by acting on a subregion ω. The dynamics isdescribed by a general reaction-diffusion system, including one ormore populations, subject to a vital dynamics with either locallogistic or nonlocal logistic terms. For the one populationcase, a necessary condition and a sufficient condition foreradicability (zero-stabilizability) are obtained, in terms of the sign of the principaleigenvalue of a suitable elliptic operator acting on the domain\begin{document}$Ω \setminus \overline{ω }$\end{document}. A feedbackharvesting-like control with a large constant harvesting raterealizes eradication of the population. The problem oferadication is then reformulated in a more convenient way, bytaking into account the total cost of the damages produced by apest population and the costs related to the choice of therelevant subregion, and approximated by a regional optimalcontrol problem with a finite horizon. A conceptual iterativealgorithm is formulated for the simulation of the proposedoptimal control problem. Numerical tests are given to illustratethe effectiveness of the results. Relevant regional controlproblems for two populations reaction-diffusion models, such asprey-predator system, and an SIR epidemic system with spatialstructure and local/nonlocal force of infection, have beenanalyzed too.

Stability and bifurcation in an age-structured model with stocking rate and time delays
Shengqin Xu, Chuncheng Wang and Dejun Fan
2019, 24(6): 2535-2549 doi: 10.3934/dcdsb.2018264 +[Abstract](1162) +[HTML](509) +[PDF](490.86KB)

In this paper, a predator-prey model with age structure, stocking rate and two delays is investigated. We show that Hopf bifurcation occurs when one of the time delay $τ$ crosses a sequence of critical values, by applyingHopf bifurcation theory for abstract Cauchy problems with non-dense domain. Numerical simulations are included to verify our results and a summary is also given.

On the long-time behaviour of age and trait structured population dynamics
Tristan Roget
2019, 24(6): 2551-2576 doi: 10.3934/dcdsb.2018265 +[Abstract](825) +[HTML](409) +[PDF](641.84KB)

We study the long-time behaviour of a population structured by age and a phenotypic trait under a selection-mutation dynamics. By analysing spectral properties of a family of positive operators on measure spaces, we show the existence of eventually singular stationary solutions. When the stationary measures are absolutely continuous with a continuous density, we show the convergence of the dynamics to the unique equilibrium.

Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy
Luis A. Fernández and Cecilia Pola
2019, 24(6): 2577-2612 doi: 10.3934/dcdsb.2018266 +[Abstract](972) +[HTML](470) +[PDF](628.62KB)

We study a collection of problems associated with the optimization of cancer chemotherapy treatments, under the assumptions of Gomperztian-type tumor growth and that the drug killing effect is proportional to the rate of growth for the untreated tumor (Norton-Simon hypothesis). Classical pharmacokinetics and different pharmacodynamics (Skipper and Emax) are considered, together with a toxicity limit or the penalization of the accumulated drug effect. Existence and uniqueness of the optimal control is proved in some cases, while in others the total amount of drug is the unique relevant aspect to take into account and the existence of an infinite number of optimal controls is shown. In all cases, explicit expressions for the solutions are derived in terms of the problem data. Finally, numerical results of illustrative examples and some conclusions are presented.

Global solution and decay rate for a reduced gravity two and a half layer model
Yongming Liu and Lei Yao
2019, 24(6): 2613-2638 doi: 10.3934/dcdsb.2018267 +[Abstract](723) +[HTML](483) +[PDF](534.21KB)

In this paper we investigate the reduced gravity two and a half model in oceanic fluid dynamics. In a finite domain (for the initial-boundary value problem), we obtain time-independent estimates, which allow us to show the existence and uniqueness of regular solutions as well as the decay rate estimates. A collection of the decay rate estimates for \begin{document}$h_i-\widetilde{h}_i$ \end{document} (with \begin{document}$\widetilde{h}_i$ \end{document} being the stationary layer thickness) and \begin{document}$u_i(i = 1,2)$ \end{document} in \begin{document}$L^2(Ω)$ \end{document}-norm as well as \begin{document}$H^1(Ω)$ \end{document}-norm as time \begin{document}$t \to \infty $ \end{document} are established.

Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study
Tomás Caraballo, Renato Colucci and Luca Guerrini
2019, 24(6): 2639-2655 doi: 10.3934/dcdsb.2018268 +[Abstract](1051) +[HTML](465) +[PDF](1051.98KB)

In this article we consider a model introduced by Ucar in order to simply describe chaotic behaviour with a one dimensional ODE containing a constant delay. We study the bifurcation problem of the equilibria and we obtain an approximation of the periodic orbits generated by the Hopf bifurcation. Moreover, we propose and analyse a more general model containing distributed time delay. Finally, we propose some ideas for further study. All the theoretical results are supported and illustrated by numerical simulations.

Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains
Chia-Yu Hsieh
2019, 24(6): 2657-2681 doi: 10.3934/dcdsb.2018269 +[Abstract](852) +[HTML](497) +[PDF](470.08KB)

In this paper, we consider radial solutions of the Poisson-Nernst-Planck (PNP) system with variable dielectric coefficients \begin{document}$\varepsilon g(x)$\end{document} in \begin{document}$N$\end{document}-dimensional annular domains, \begin{document}$N≥2$\end{document}. When the parameter \begin{document}$\varepsilon$\end{document} tends to zero, the PNP system admits a boundary layer solution as a steady state, which satisfies the charge conserving Poisson-Boltzmann (CCPB) equation. For the stability of the radial boundary layer solutions to the time-dependent radial PNP system, we estimate the radial solution of the perturbed problem with global electroneutrality. We generalize the argument of the one spatial dimension case (cf. [18]) and find a new way to transform the perturbed problem. By choosing a suitable weighted norm, we then derive the associated energy law which can be used to prove that the \begin{document}$H^{-1}_x$\end{document} norm of the solution of the perturbed problem decays exponentially in time with the exponent independent of \begin{document}$\varepsilon$\end{document} if the coefficient of the Robin boundary condition of electrostatic potential has a suitable positive lower bound.

Nondegenerate multistationarity in small reaction networks
Anne Shiu and Timo de Wolff
2019, 24(6): 2683-2700 doi: 10.3934/dcdsb.2018270 +[Abstract](704) +[HTML](457) +[PDF](433.29KB)

Much attention has been focused in recent years on the following algebraic problem arising from applications: which chemical reaction networks, when taken with mass-action kinetics, admit multiple positive steady states? The interest behind this question is in steady states that are stable. As a step toward this difficult question, here we address the question of multiple nondegenerate positive steady states. Mathematically, this asks whether certain families of parametrized, real, sparse polynomial systems ever admit multiple positive real roots that are simple. Our main results settle this problem for certain types of small networks, and our techniques may point the way forward for larger networks.

Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space
Ruyun Ma and Man Xu
2019, 24(6): 2701-2718 doi: 10.3934/dcdsb.2018271 +[Abstract](693) +[HTML](460) +[PDF](463.33KB)

In this paper we study global bifurcation phenomena for the Dirichlet problem associated with the prescribed mean curvature equation in Minkowski space

Here \begin{document} $Ω$ \end{document} is a bounded regular domain in \begin{document} $\mathbb{R}^N$ \end{document}, the function \begin{document} $f$ \end{document} satisfies the Carathéodory conditions, and \begin{document} $f$ \end{document} is either superlinear or sublinear in \begin{document} $u$ \end{document} at \begin{document} $0$ \end{document}. The proof of our main results are based upon bifurcation techniques.

Long time behavior of fractional impulsive stochastic differential equations with infinite delay
Jiaohui Xu and Tomás Caraballo
2019, 24(6): 2719-2743 doi: 10.3934/dcdsb.2018272 +[Abstract](1161) +[HTML](498) +[PDF](565.85KB)

This paper is first devoted to the local and global existence of mild solutions for a class of fractional impulsive stochastic differential equations with infinite delay driven by both \begin{document}$\mathbb{K}$\end{document}-valued Q-cylindrical Brownian motion and fractional Brownian motion with Hurst parameter \begin{document}$H∈(1/2,1)$\end{document}. A general framework which provides an effective way to prove the continuous dependence of mild solutions on initial value is established under some appropriate assumptions. Furthermore, it is also proved the exponential decay to zero of solutions to fractional stochastic impulsive differential equations with infinite delay.

H2-stability of some second order fully discrete schemes for the Navier-Stokes equations
Yinnian He, Pengzhan Huang and Jian Li
2019, 24(6): 2745-2780 doi: 10.3934/dcdsb.2018273 +[Abstract](998) +[HTML](513) +[PDF](403.29KB)

This paper considers the $H^2$-stability results for the second order fully discrete schemes based on the mixed finite element method for the 2D time-dependent Navier-Stokes equations with the initial data $u_0∈ H^α, $ where $α = 0, ~1$ and 2. A mixed finite element method is used to the spatial discretization of the Navier-Stokes equations, and the temporal treatments of the spatial discrete Navier-Stokes equations are the second order semi-implicit, implicit/explict and explicit schemes. The $H^2$-stability results of the schemes are provided, where the second order semi-implicit and implicit/explicit schemes are almost unconditionally $H^2$-stable, the second order explicit scheme is conditionally $H^2$-stable in the case of $\alpha = 2$, and the semi-implicit, implicit/explicit and explicit schemes are conditionally $H^2$-stable in the case of $\alpha = 1, ~0$. Finally, some numerical tests are made to verify the above theoretical results.

Immunosuppressant treatment dynamics in renal transplant recipients: an iterative modeling approach
Neha Murad, H. T. Tran, H. T. Banks, R. A. Everett and Eric S. Rosenberg
2019, 24(6): 2781-2797 doi: 10.3934/dcdsb.2018274 +[Abstract](727) +[HTML](455) +[PDF](2043.63KB)

Finding the optimal balance between over-suppression and under-suppression of the immune response is difficult to achieve in renal transplant patients, all of whom require lifelong immunosuppression. Our ultimate goal is to apply control theory to adaptively predict the optimal amount of immunosuppression; however, we first need to formulate a biologically realistic model. The process of quantitively modeling biological processes is iterative and often leads to new insights with every iteration. We illustrate this iterative process of modeling for renal transplant recipients infected by BK virus. We analyze and improve on the current mathematical model by modifying it to be more biologically realistic and amenable for designing an adaptive treatment strategy.

Distribution profiles in gene transcription activated by the cross-talking pathway
Feng Jiao, Qiwen Sun, Genghong Lin and Jianshe Yu
2019, 24(6): 2799-2810 doi: 10.3934/dcdsb.2018275 +[Abstract](705) +[HTML](392) +[PDF](782.85KB)

Gene transcription is a stochastic process, manifested by the heterogeneous mRNA distribution in an isogenic cell population. Bimodal distribution has been observed in the transcription of stress responsive genes which have evolved to be easily turned on and easily turned off. This is against the conclusion in the classical two-state model that bimodality occurs only when the gene is hardly turned on and hardly turned off. In this paper, we extend the gene activation process in the two-state model by introducing the cross-talking pathway that involves the random selection between a spontaneous weak basal pathway and a stress-induced strong signaling pathway. By deriving exact forms of mRNA distribution at steady-state, we find that the cross-talking pathway is much more likely to trigger the bimodal distribution. Our further analysis reveals an observed transition among the decaying, bimodal and unimodal mRNA distribution for stress gene upon enhanced stimulations. Especially, the bimodality occurs when the stress-induced signalling pathway is more frequently selected, reinforcing the assertion that bimodal transcription is a general feature of stress genes in response to environmental change.

Uniqueness and traveling waves in a cell motility model
Matthew S. Mizuhara and Peng Zhang
2019, 24(6): 2811-2835 doi: 10.3934/dcdsb.2018315 +[Abstract](500) +[HTML](310) +[PDF](955.79KB)

We study a non-linear and non-local evolution equation for curves obtained as the sharp interface limit of a phase-field model for crawling motion of eukaryotic cells on a substrate. We establish uniqueness of solutions to the sharp interface limit equation in the subcritical parameter regime. The proof relies on a Grönwall estimate for a specially chosen weighted \begin{document}$L^2$\end{document} norm.

As persistent motion of crawling cells is of central interest to biologists, we next study the existence of traveling wave solutions. We prove that traveling wave solutions exist in the supercritical parameter regime provided the non-linearity of the sharp interface limit equation possesses certain asymmetry (related, e.g., to myosin contractility).

Finally, we numerically investigate traveling wave solutions and simulate their dynamics. Due to non-uniqueness of solutions of the sharp interface limit equation we numerically solve a related, singularly perturbed PDE system which is uniquely solvable. Our simulations predict instability of traveling wave solutions and capture both bipedal wandering cell motion as well as rotating cell motion; these behaviors qualitatively agree with recent experimental and theoretical findings.

A comparison of deterministic and stochastic predator-prey models with disease in the predator
Hongxiao Hu, Liguang Xu and Kai Wang
2019, 24(6): 2837-2863 doi: 10.3934/dcdsb.2018289 +[Abstract](819) +[HTML](529) +[PDF](2922.41KB)

In this paper, we study the dynamics of deterministic and stochastic models for a predator-prey, where the predator species is subject to an SIS form of parasitic infection. The deterministic model is a system of ordinary differential equations for a predator-prey model with disease in the predator only. The existence and local stability of the boundary equilibria and the uniform persistence for the ODE model are investigated. Based on these results, some threshold values for successful invasion of disease or prey species are obtained. A new stochastic model is derived in the form of continuous-time Markov chains. Branching process theory is applied to the continuous-time Markov chain models to estimate the probabilities for disease outbreak or prey species invasion. The deterministic and stochastic threshold theories are compared and some relationships between the deterministic and stochastic thresholds are derived. Finally, some numerical simulations are introduced to illustrate the main results and to highlight some of the differences between the deterministic and stochastic models.

A note on the convergence of the solution of the Novikov equation
Giuseppe Maria Coclite and Lorenzo di Ruvo
2019, 24(6): 2865-2899 doi: 10.3934/dcdsb.2018290 +[Abstract](857) +[HTML](437) +[PDF](499.03KB)

We consider the Novikov and Camass-Holm equations, which contain nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solution of the dispersive equation converges to the unique entropy solution of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the \begin{document}$L^p$\end{document} setting.

Dynamical behaviors of stochastic type K monotone Lotka-Volterra systems
Dejun Fan, Xiaoyu Yi, Ling Xia and Jingliang Lv
2019, 24(6): 2901-2922 doi: 10.3934/dcdsb.2018291 +[Abstract](633) +[HTML](408) +[PDF](325.33KB)

Two n-species stochastic type K monotone Lotka-Volterra systems are proposed and investigated. For non-autonomous system, we show that there is a unique positive solution to the model for any positive initial value. Moreover, sufficient conditions for stochastic permanence and global attractivity are established. For autonomous system, we prove that for each species, there is a constant which can be represented by the coefficients of the system. If the constant equals 1, then the corresponding species will be nonpersistent on average. To illustrate the theoretical results, the corresponding numerical simulations are also given.

Global analysis of a stochastic TB model with vaccination and treatment
Tao Feng and Zhipeng Qiu
2019, 24(6): 2923-2939 doi: 10.3934/dcdsb.2018292 +[Abstract](962) +[HTML](607) +[PDF](1003.42KB)

In this paper, a stochastic model is formulated to describe the transmission dynamics of tuberculosis. The model incorporates vaccination and treatment in the intervention strategies. Firstly, sufficient conditions for persistence in mean and extinction of tuberculosis are provided. In addition, sufficient conditions are obtained for the existence of stationary distribution and ergodicity. Moreover, numerical simulations are given to illustrate these analytical results. The theoretical and numerical results show that large environmental disturbances can suppress the spread of tuberculosis.

Polynomial maps with hidden complex dynamics
Xu Zhang and Guanrong Chen
2019, 24(6): 2941-2954 doi: 10.3934/dcdsb.2018293 +[Abstract](694) +[HTML](510) +[PDF](650.82KB)

The dynamics of a class of one-dimensional polynomial maps are studied, and interesting dynamics are observed under certain conditions: the existence of periodic points with even periods except for one fixed point; the coexistence of two attractors, an attracting fixed point and a hidden attractor; the existence of a double period-doubling bifurcation, which is different from the classical period-doubling bifurcation of the Logistic map; the existence of Li-Yorke chaos. Furthermore, based on this one-dimensional map, the corresponding generalized Hénon map is investigated, and some interesting dynamics are found for certain parameter values: the coexistence of an attracting fixed point and a hidden attractor; the existence of Smale horseshoe for a subshift of finite type and also Li-Yorke chaos.

Discontinuous phenomena in bioreactor system
Hany A. Hosham and Eman D Abou Elela
2019, 24(6): 2955-2969 doi: 10.3934/dcdsb.2018294 +[Abstract](877) +[HTML](569) +[PDF](1159.6KB)

This paper critically examines discontinuous bifurcation and stability issues in model of methane gas production from organic waste via decaying process in two cases, namely sliding and non-sliding flow. The presence of certain types of discontinuities in Monod curve lead to discontinuous system and therefore the criteria for the existence and stability of equilibrium points are established. The analysis highlights the presence of several types of border collision bifurcations depending upon the effect of the dilution factor, biomass concentration and solid-liquid-gas separator efficiency, like nonsmooth fold, persistence and grazing-sliding scenarios. In addition, numerical simulations are carried out to illustrate and validate the results.

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