# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

January 2017 , Volume 22 , Issue 1

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2017, 22(1): i-ii doi: 10.3934/dcdsb.201701i +[Abstract](189) +[HTML](100) +[PDF](75.6KB)
Abstract:
2017, 22(1): 1-28 doi: 10.3934/dcdsb.2017001 +[Abstract](255) +[HTML](39) +[PDF](608.1KB)
Abstract:

We study the problem of inventory control, with simultaneous pricing optimization in continuous time. For the classical inventory control problem in continuous time, see [5], as a recent reference. We incorporate pricing decisions together with inventory decisions. We consider the situation without fixed cost for an infinite horizon. Without pricing, under very natural assumptions, the optimal ordering policy is given by a Base stock, which we review briefly. With pricing, the natural generalization is the so called "Base Stock list price" (BSLP) term coined by E. Porteus, see [36], and was shown in discrete time by A. Federgruen and A. Herching to be the optimal strategy, see [14]. We extend the concept to continuous time which not only complicates the dynamics of the problem, which has never been considered before.

2017, 22(1): 29-58 doi: 10.3934/dcdsb.2017002 +[Abstract](192) +[HTML](27) +[PDF](2269.1KB)
Abstract:

It has been hypothesized that the generation of new neural cells (neurogenesis) in the developing and adult brain is guided by the extracellular matrix. The extracellular matrix of the neurogenic niches features specialized structures termed fractones, which are scattered in between stem/progenitor cells and bind and activate growth factors at the surface of stem/progenitor cells to influence their proliferation. We present a mathematical control model that considers the role of fractones as captors and activators of growth factors, controlling the rate of proliferation and directing the location of the newly generated neuroepithelial cells in the forming brain. The model is a hybrid control system that incorporates both continuous and discrete dynamics. The continuous dynamics of the model features the diffusion of multiple growth factor concentrations through the mass of cells, with fractones acting as sinks that absorb and hold growth factor. When a sufficient amount has been captured, growth is assumed to occur instantaneously in the discrete dynamics of the model, causing an immediate rearrangement of cells, and potentially altering the dynamics of the diffusion. The fractones in the model are represented by controls that allow for their dynamic placement in and removal from the evolving cell mass.

2017, 22(1): 59-81 doi: 10.3934/dcdsb.2017003 +[Abstract](245) +[HTML](21) +[PDF](502.0KB)
Abstract:

This paper considers a new stochastic volatility model with regime switches and uncertain noise in discrete time and discusses its theoretical development for filtering and estimation. The model incorporates important features for asset price models, such as stochastic volatility, regime switches and parameter uncertainty in Gaussian noises for both the return and volatility processes. In particular, both drift and volatility uncertainties for the return and volatility processes are incorporated by introducing a family of real-world probability measures. Then, by modifying the reference probability approach to filtering, a sequence of conditional sub-linear expectations is used to provide a robust approach for describing the drift and volatility uncertainties in the Gaussian noises. Filtering theory, based on conditional sublinear expectations and the Viterbi algorithm are adopted to derive filters for the hidden Markov chain and filter-based estimates of the unknown parameters.

2017, 22(1): 83-99 doi: 10.3934/dcdsb.2017004 +[Abstract](168) +[HTML](20) +[PDF](1647.2KB)
Abstract:

The goal of this work is to introduce a mathematical model of multicellular developmental design based on morphological analysis in order to study the robustness of multi-cellular organism development.

In this model each cell is a controlled system and has the same information, an ordered list of cell type. Cells perceive their neighbours during the growth process and decide to divide in a direction given by the reading advancement of the virtual genetic material and depending on the complex interplay between genetic, epigenetic and environment.

Cells can perform distinct functions but in our simulator, two cell types just differ by there color and by permuting the segmentation direction according to the virtual genetic material and the epigenetic control. The switching on and switching off of genes depends on the environment of the cell.The multi-cellular organism has to reach a shape in a given environment to which it has to adapt.

We present in this paper an algorithm based model which is implemented in a virtual 3D-environment. Moreover, the algorithm follows the principle of inertia in that the cells progress through the reading of its virtual genetic material after a punctuated equilibrium or when its viability is at stake.

2017, 22(1): 101-113 doi: 10.3934/dcdsb.2017005 +[Abstract](184) +[HTML](30) +[PDF](407.0KB)
Abstract:

The Ait-Sahalia-Rho model is an important tool to study a number of financial problems, including the term structure of interest rate. However, since the functions of this model do not satisfy the linear growth condition, we cannot study the properties for the solution of this model by using the traditional techniques. In this paper we overcome the mathematical difficulties due to the nonlinear growth condition by using numerical simulation. Thus we first discuss analytical properties of the model and the convergence property of numerical solutions in probability for the Ait-Sahalia-Rho model. Finally, an example for option pricing is given to illustrate that the numerical solution is an effective method to estimate the expected payoffs.

2017, 22(1): 115-123 doi: 10.3934/dcdsb.2017006 +[Abstract](207) +[HTML](19) +[PDF](287.1KB)
Abstract:

Several computational models have been developed in the literature to describe the dynamics of the cell-cycle for the mammalian cell, in particular for cancer cells, using both traditional and new techniques and yielding some positive results. In this paper, we discuss how to optimise model parameters for these types of models and how this can serve to enhance numerical results. We pose the model parameter selection problem as an optimal parameter selection problem on both a normal cell and cancer cell cycle of growth. Various possible objectives are discussed and we illustrate the process with some numerical results.

2017, 22(1): 125-159 doi: 10.3934/dcdsb.2017007 +[Abstract](260) +[HTML](48) +[PDF](615.0KB)
Abstract:

Dynamic Stackelberg game models have been used to study sequential decision making in noncooperative games in various fields. In this paper we give relevant dynamic Stackelberg game models, and review their applications to operations management and marketing channels. A common feature of these applications is the specification of the game structure: a decentralized channel consists of a manufacturer and independent retailers, and a sequential decision process with a state dynamics. In operations management, Stackelberg games have been used to study inventory issues, such as wholesale and retail pricing strategies, outsourcing, and learning effects in dynamic environments. The underlying demand typically has a growing trend or seasonal variation. In marketing, dynamic Stackelberg games have been used to model cooperative advertising programs, store brand and national brand advertising strategies, shelf space allocation, and pricing and advertising decisions. The demand dynamics are usually extensions of the classic advertising capital models or sales-advertising response models. We begin each section by introducing the relevant dynamic Stackelberg game formulation along with the definition of the equilibrium used, and then review the models and results appearing in the literature.

2017, 22(1): 161-185 doi: 10.3934/dcdsb.2017008 +[Abstract](178) +[HTML](24) +[PDF](474.6KB)
Abstract:

It is observed empirically that mean-reverting processes are more realistic in modeling the inventory level of a company. In a typical mean-reverting process, the inventory level is assumed to be linearly dependent on the deviation of the inventory level from the long-term mean. However, when the deviation is large, it is reasonable to assume that the company might want to increase the intensity of interference to the inventory level significantly rather than in a linear manner. In this paper, we attempt to model inventory replenishment as a nonlinear continuous feedback process. We study both infinite horizon discounted cost and the long-run average cost, and derive the corresponding optimal (s, S) policy.

2017, 22(1): 187-198 doi: 10.3934/dcdsb.2017009 +[Abstract](200) +[HTML](27) +[PDF](396.0KB)
Abstract:

In this paper, we establish \begin{document} $Z$ \end{document}-eigenvalue inclusion theorems for general tensors, which reveal some crucial differences between \begin{document} $Z$ \end{document}-eigenvalues and \begin{document} $H$ \end{document}-eigenvalues. As an application, we obtain upper bounds for the largest \begin{document} $Z$ \end{document}-eigenvalue of a weakly symmetric nonnegative tensor, which are sharper than existing upper bounds.

2017, 22(1): 199-208 doi: 10.3934/dcdsb.2017010 +[Abstract](274) +[HTML](18) +[PDF](505.1KB)
Abstract:

In this paper, we study the exponential stability problem of discrete-time switched delay systems. Combining a multiple Lyapunov function method with a mode-dependent average dwell time technique, we develop novel sufficient conditions for exponential stability of the switched delay systems expressed by a set of numerically solvable linear matrix inequalities. Finally, numerical examples are presented to illustrate less conservativeness of the obtained results.

2017, 22(1): 209-226 doi: 10.3934/dcdsb.2017011 +[Abstract](246) +[HTML](37) +[PDF](486.4KB)
Abstract:

In this paper, we discuss the \begin{document} $p$ \end{document}th moment exponential stabilization of continuous-time hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. The hybrid stochastic functional differential equations are also known as stochastic functional differential equations with the Markovian switching. We follow Mao's paper to consider the auxiliary system whose control is based on continuous-time state observation. The lemma is provided that if the \begin{document} $p$ \end{document}th moment of the solution \begin{document} $y(t)$ \end{document} of the auxiliary system decays exponentially then the same with the \begin{document} $p$ \end{document}th moment of the functional \begin{document} $y_t$ \end{document}. With the help of this lemma, the criterion for \begin{document} $p$ \end{document}th moment exponential stability of the primary system is given, and the margin of the duration of the discrete-time state observation is presented. Then the special case like the linear system is considered and the discrete-time feedback control is designed.

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