# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

All Issues

## Discrete & Continuous Dynamical Systems - B

Open Access Articles

2019, 24(5): 2293-2313 doi: 10.3934/dcdsb.2019096 +[Abstract](866) +[HTML](210) +[PDF](471.3KB)
Abstract:

We give answer to an open question by proving a sufficient optimality condition for state-linear optimal control problems with time delays in state and control variables. In the proof of our main result, we transform a delayed state-linear optimal control problem to an equivalent non-delayed problem. This allows us to use a well-known theorem that ensures a sufficient optimality condition for non-delayed state-linear optimal control problems. An example is given in order to illustrate the obtained result.

2019, 24(5): 2365-2381 doi: 10.3934/dcdsb.2019099 +[Abstract](883) +[HTML](204) +[PDF](440.63KB)
Abstract:

We consider a generational and continuous-time two-phase model of the cell cycle. The first model is given by a stochastic operator, and the second by a piecewise deterministic Markov process. In the second case we also introduce a stochastic semigroup which describes the evolution of densities of the process. We study long-time behaviour of these models. In particular we prove theorems on asymptotic stability and sweeping. We also show the relations between both models.

2018, 23(8): 3387-3413 doi: 10.3934/dcdsb.2018239 +[Abstract](1924) +[HTML](483) +[PDF](1087.46KB)
Abstract:

Tuberculosis (TB) is a leading cause of death from infectious disease. TB is caused mainly by a bacterium called Mycobacterium tuberculosis which often initiates in the respiratory tract. The interaction of macrophages and T cells plays an important role in the immune response during TB infection. Recent experimental results support that active TB infection may be induced by the dysfunction of Treg cell regulation that provides a balance between anti-TB T cell responses and pathology. To better understand the dynamics of TB infection and Treg cell regulation, we build a mathematical model using a system of differential equations that qualitatively and quantitatively characterizes the dynamics of macrophages, Th1 and Treg cells during TB infection. For sufficiently analyzing the interaction between immune response and bacterial infection, we separate our model into several simple subsystems for further steady state and stability studies. Using this system, we explore the conditions of parameters for three situations, recovery, latent disease and active disease, during TB infection. Our numerical simulations support that Th1 cells and Treg cells play critical roles in TB infection: Th1 cells inhibit the number of infected macrophages to reduce the chance of active disease; Treg cell regulation reduces the immune response to stabilize the dynamics of the system.

2018, 23(1): 459-472 doi: 10.3934/dcdsb.2018031 +[Abstract](2475) +[HTML](765) +[PDF](1511.3KB)
Abstract:

We consider a model of phenotypic evolution in populations with assortative mating of individuals. The model is given by a nonlinear operator acting on the space of probability measures and describes the relation between parental and offspring trait distributions. We study long-time behavior of trait distribution and show that it converges to a combination of Dirac measures. This result means that assortative mating can lead to a polymorphic population and sympatric speciation.

2018, 23(1): 377-385 doi: 10.3934/dcdsb.2018026 +[Abstract](2395) +[HTML](631) +[PDF](346.0KB)
Abstract:

A new theorem on asymptotic stability of stochastic semigroups is given. This theorem is applied to a stochastic semigroup corresponding to Stein's neuronal model. Asymptotic properties of models with and without the refractory period are compared.

2017, 22(1): i-ii doi: 10.3934/dcdsb.201701i +[Abstract](1292) +[HTML](730) +[PDF](75.6KB)
Abstract:
2017, 22(6): 2169-2206 doi: 10.3934/dcdsb.2017091 +[Abstract](3149) +[HTML](1307) +[PDF](5226.6KB)
Abstract:

The control of complex nonlinear dynamical networks is an ongoing challenge in diverse contexts ranging from biology to social sciences. To explore a practical framework for controlling nonlinear dynamical networks based on meaningful physical and experimental considerations, we propose a new concept of the domain control for nonlinear dynamical networks, i.e., the control of a nonlinear network in transition from the domain of attraction of an undesired state (attractor) to the domain of attraction of a desired state. We theoretically prove the existence of a domain control. In particular, we offer an approach for identifying the driver nodes that need to be controlled and design a general form of control functions for realizing domain controllability. In addition, we demonstrate the effectiveness of our theory and approaches in three realistic disease-related networks: the epithelial-mesenchymal transition (EMT) core network, the T helper (Th) differentiation cellular network and the cancer network. Moreover, we reveal certain genes that are critical to phenotype transitions of these systems. Therefore, the approach described here not only offers a practical control scheme for nonlinear dynamical networks but also helps the development of new strategies for the prevention and treatment of complex diseases.

2017, 22(4): 1295-1327 doi: 10.3934/dcdsb.2017063 +[Abstract](2106) +[HTML](1354) +[PDF](639.7KB)
Abstract:

A continuum model for a population of self-propelled particles interacting through nematic alignment is derived from an individual-based model. The methodology consists of introducing a hydrodynamic scaling of the corresponding mean field kinetic equation. The resulting perturbation problem is solved thanks to the concept of generalized collision invariants. It yields a hyperbolic but non-conservative system of equations for the nematic mean direction of the flow and the densities of particles flowing parallel or anti-parallel to this mean direction. Diffusive terms are introduced under a weakly non-local interaction assumption and the diffusion coefficient is proven to be positive. An application to the modeling of myxobacteria is outlined.

2017, 22(3): ⅰ-ⅱ doi: 10.3934/dcdsb.201703i +[Abstract](1117) +[HTML](618) +[PDF](77.4KB)
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2016, 21(9): i-ii doi: 10.3934/dcdsb.201609i +[Abstract](949) +[PDF](91.4KB)
Abstract:
It is a great honor and pleasure to dedicate this special issue of the journal Discrete and Continuous Dynamical Systems, Series B, to our colleague and friend Björn Schmalfuß, on the occasion of his 60th birthday.

2016, 21(7): i-ii doi: 10.3934/dcdsb.201607i +[Abstract](1003) +[PDF](85.1KB)
Abstract:
Stochasticity, sometimes referred to as noise, is unavoidable in biological systems. Noise, which exists at all biological scales ranging from gene expressions to ecosystems, can be detrimental or sometimes beneficial by performing unexpected tasks to improve biological functions. Often, the complexity of biological systems is a consequence of dealing with uncertainty and noise, and thus, consideration of noise is necessary in mathematical models. Recent advancement of technology allows experimental measurement on stochastic effects, showing multifaceted and perplexed roles of noise. As interrogating internal or external noise becomes possible experimentally, new models and mathematical theory are needed. Over the past few decades, stochastic analysis and the theory of nonautonomous and random dynamical systems have started to show their strong promise and relevance in studying complex biological systems. This special issue represents a collection of recent advances in this emerging research area.

2016, 21(7): 2275-2291 doi: 10.3934/dcdsb.2016047 +[Abstract](1795) +[PDF](4573.4KB)
Abstract:
Epithelial-mesenchymal transition (EMT) is an instance of cellular plasticity that plays critical roles in development, regeneration and cancer progression. Recent studies indicate that the transition between epithelial and mesenchymal states is a multi-step and reversible process in which several intermediate phenotypes might coexist. These intermediate states correspond to various forms of stem-like cells in the EMT system, but the function of the multi-step transition or the multiple stem cell phenotypes is unclear. Here, we use mathematical models to show that multiple intermediate phenotypes in the EMT system help to attenuate the overall fluctuations of the cell population in terms of phenotypic compositions, thereby stabilizing a heterogeneous cell population in the EMT spectrum. We found that the ability of the system to attenuate noise on the intermediate states depends on the number of intermediate states, indicating the stem-cell population is more stable when it has more sub-states. Our study reveals a novel advantage of multiple intermediate EMT phenotypes in terms of systems design, and it sheds light on the general design principle of heterogeneous stem cell population.
2016, 21(5): i-ii doi: 10.3934/dcdsb.201605i +[Abstract](833) +[PDF](173.9KB)
Abstract:
We dedicate this volume of the Journal of Discrete and Continuous Dynamical Systems-B to Professor Lishang Jiang on his 80th birthday. Professor Lishang Jiang was born in Shanghai in 1935. His family had migrated there from Suzhou. He graduated from the Department of Mathematics, Peking University, in 1954. After teaching at Beijing Aviation College, in 1957 he returned to Peking University as a graduate student of partial differential equations under the supervision of Professor Yulin Zhou. Later, as a professor, a researcher and an administrator, he worked at Peking University, Suzhou University and Tongji University at different points of his career. From 1989 to 1996, Professor Jiang was the President of Suzhou University. From 2001 to 2005, he was the Chairman of the Shanghai Mathematical Society.

2015, 20(10): i-iii doi: 10.3934/dcdsb.2015.20.10i +[Abstract](867) +[PDF](136.8KB)
Abstract:
It is our pleasure to thank Prof. Peter E. Kloeden for having invited us to guest edit a special issue of Discrete and Continuous Dynamical Systems - Series B on Entropy, Entropy-like Quantities, and Applications. From its inception this special issue was meant to be a blend of research papers, showing the diversity of current research on entropy, and a few surveys, giving a more systematic view of lasting developments. Furthermore, a general review should set the framework first.

2015, 20(8): 2333-2360 doi: 10.3934/dcdsb.2015.20.2333 +[Abstract](1752) +[PDF](606.5KB)
Abstract:
Lyapunov's second or direct method is one of the most widely used techniques for investigating stability properties of dynamical systems. This technique makes use of an auxiliary function, called a Lyapunov function, to ascertain stability properties for a specific system without the need to generate system solutions. An important question is the converse or reversability of Lyapunov's second method; i.e., given a specific stability property does there exist an appropriate Lyapunov function? We survey some of the available answers to this question.
2015, 20(6): i-iii doi: 10.3934/dcdsb.2015.20.6i +[Abstract](1359) +[PDF](127.7KB)
Abstract:
The movement and dispersal of organisms have long been recognized as key components of ecological interactions and as such, they have figured prominently in mathematical models in ecology. More recently, dispersal has been recognized as an equally important consideration in epidemiology and in environmental science. Recognizing the increasing utility of employing mathematics to understand the role of movement and dispersal in ecology, epidemiology and environmental science, The University of Miami in December 2012 held a workshop entitled Everything Disperses to Miami: The Role of Movement and Dispersal in Ecology, Epidemiology and Environmental Science" (EDM).

2015, 20(6): 1759-1783 doi: 10.3934/dcdsb.2015.20.1759 +[Abstract](2370) +[PDF](3533.2KB)
Abstract:
We develop the theory of optimal control for a system of integrodifference equations modelling a pest-pathogen system. Integrodifference equations incorporate continuous space into a system of discrete time equations. We design an objective functional to minimize the damaged cost generated by an invasive species and the cost of controlling the population with a pathogen. Existence, characterization, and uniqueness results for the optimal control and corresponding states have been completed. We use a forward-backward sweep numerical method to implement our optimization which produces spatio-temporal control strategies for the gypsy moth case study.
2015, 20(3): i-ii doi: 10.3934/dcdsb.2015.20.3i +[Abstract](901) +[PDF](161.4KB)
Abstract:
We were very pleased to be given the opportunity by Prof. Peter Kloeden to edit this special issue of Discrete and Continuous Dynamical Systems - Series B on the asymptotic dynamics of non-autonomous systems.

2014, 19(8): i-ii doi: 10.3934/dcdsb.2014.19.8i +[Abstract](1145) +[PDF](447.1KB)
Abstract:
Most mathematicians who in their professional career deal with differential equations, PDEs, dynamical systems, stochastic equations and a variety of their applications, particularly to biomedicine, have come across the research contributions of Avner Friedman to these fields. However, not many of them know that his family background is actually Polish. His father was born in the small town of Włodawa on the border with Belarus and lived in another Polish town, Łomza, before he emigrated to Israel in the early 1920's (when it was still the British Mandate, Palestine). His mother came from the even smaller Polish town Knyszyn near Białystok and left for Israel a few years earlier. In May 2013, Avner finally had the opportunity to visit his father's hometown for the first time accompanied by two Polish friends, co-editors of this volume. His visit in Poland became an occasion to interact with Polish mathematicians. Poland has a long tradition of research in various fields related to differential equations and more recently there is a growing interest in biomedical applications. Avner visited two research centers, the Schauder Center in Torun and the Department of Mathematics of the Technical University of Lodz where he gave a plenary talk at a one-day conference on Dynamical Systems and Applications which was held on this occasion. In spite of its short length, the conference attracted mathematicians from the most prominent research centers in Poland including the University of Warsaw, the Polish Academy of Sciences and others, and even some mathematicians from other countries in Europe. Avner had a chance to get familiar with the main results in dynamical systems and applications presented by the participants and give his input in the scientific discussions. This volume contains some of the papers related to this meeting and to the overall research interactions it generated. The papers were written by mathematicians, mostly Polish, who wanted to pay tribute to Avner Friedman on the occasion of his visit to Poland.

2014, 19(5): 1411-1436 doi: 10.3934/dcdsb.2014.19.1411 +[Abstract](1449) +[PDF](2066.3KB)
Abstract:
Systems of pairwise-interacting particles model a cornucopia of physical systems, from insect swarms and bacterial colonies to nanoparticle self-assembly. We study a continuum model with densities supported on co-dimension one curves for two-species particle interaction in $\mathbb{R}^2$, and apply linear stability analysis of concentric ring steady states to characterize the steady state patterns and instabilities which form. Conditions for linear well-posedness are determined and these results are compared to simulations of the discrete particle dynamics, showing predictive power of the linear theory. Some intriguing steady state patterns are shown through numerical examples.

2018  Impact Factor: 1.008