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Discrete & Continuous Dynamical Systems - B

Open Access Articles

Does assortative mating lead to a polymorphic population? A toy model justification
Ryszard Rudnicki and Radoslaw Wieczorek
2018, 23(1): 459-472 doi: 10.3934/dcdsb.2018031 +[Abstract](671) +[HTML](224) +[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.

Stability of stochastic semigroups and applications to Stein's neuronal model
Katarzyna PichÓr and Ryszard Rudnicki
2018, 23(1): 377-385 doi: 10.3934/dcdsb.2018026 +[Abstract](550) +[HTML](192) +[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.

Preface
Honglei Xu, Yi Zhang and Ka Fai Cedric Yiu
2017, 22(1): i-ii doi: 10.3934/dcdsb.201701i +[Abstract](358) +[HTML](193) +[PDF](75.6KB)
Abstract:
Domain control of nonlinear networked systems and applications to complex disease networks
Suoqin Jin, Fang-Xiang Wu and Xiufen Zou
2017, 22(6): 2169-2206 doi: 10.3934/dcdsb.2017091 +[Abstract](796) +[HTML](278) +[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.

A continuum model for nematic alignment of self-propelled particles
Pierre Degond, Angelika Manhart and Hui Yu
2017, 22(4): 1295-1327 doi: 10.3934/dcdsb.2017063 +[Abstract](548) +[HTML](720) +[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.

Preface
Chris Cosner, Yuan Lou, Shigui Ruan and Wenxian Shen
2017, 22(3): ⅰ-ⅱ doi: 10.3934/dcdsb.201703i +[Abstract](256) +[HTML](169) +[PDF](77.4KB)
Abstract:
Preface
Tomás Caraballo, María J. Garrido-Atienza and Wilfried Grecksch
2016, 21(9): i-ii doi: 10.3934/dcdsb.201609i +[Abstract](295) +[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.

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Preface
Xiaoying Han and Qing Nie
2016, 21(7): i-ii doi: 10.3934/dcdsb.201607i +[Abstract](274) +[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.

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Controlling stochasticity in epithelial-mesenchymal transition through multiple intermediate cellular states
Catherine Ha Ta, Qing Nie and Tian Hong
2016, 21(7): 2275-2291 doi: 10.3934/dcdsb.2016047 +[Abstract](647) +[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.
Preface
Jin Liang and Lihe Wang
2016, 21(5): i-ii doi: 10.3934/dcdsb.201605i +[Abstract](242) +[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.

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Preface
José M. Amigó and Karsten Keller
2015, 20(10): i-iii doi: 10.3934/dcdsb.2015.20.10i +[Abstract](258) +[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.

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Classical converse theorems in Lyapunov's second method
Christopher M. Kellett
2015, 20(8): 2333-2360 doi: 10.3934/dcdsb.2015.20.2333 +[Abstract](477) +[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.
Preface
Robert Stephen Cantrell, Suzanne Lenhart, Yuan Lou and Shigui Ruan
2015, 20(6): i-iii doi: 10.3934/dcdsb.2015.20.6i +[Abstract](375) +[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).

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Optimal control of integrodifference equations in a pest-pathogen system
Marco V. Martinez, Suzanne Lenhart and K. A. Jane White
2015, 20(6): 1759-1783 doi: 10.3934/dcdsb.2015.20.1759 +[Abstract](801) +[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.
Preface
Alexandre N. Carvalho, José A. Langa and James C. Robinson
2015, 20(3): i-ii doi: 10.3934/dcdsb.2015.20.3i +[Abstract](293) +[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.

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Preface
Urszula Ledzewicz, Marek Galewski, Andrzej Nowakowski, Andrzej Swierniak, Agnieszka Kalamajska and Ewa Schmeidel
2014, 19(8): i-ii doi: 10.3934/dcdsb.2014.19.8i +[Abstract](337) +[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.

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Two-species particle aggregation and stability of co-dimension one solutions
Alan Mackey, Theodore Kolokolnikov and Andrea L. Bertozzi
2014, 19(5): 1411-1436 doi: 10.3934/dcdsb.2014.19.1411 +[Abstract](398) +[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.
Cops on the dots in a mathematical model of urban crime and police response
Joseph R. Zipkin, Martin B. Short and Andrea L. Bertozzi
2014, 19(5): 1479-1506 doi: 10.3934/dcdsb.2014.19.1479 +[Abstract](454) +[PDF](696.8KB)
Abstract:
Hotspots of crime localized in space and time are well documented. Previous mathematical models of urban crime have exhibited these hotspots but considered a static or otherwise suboptimal police response to them. We introduce a program of police response to hotspots of crime in which the police adapt dynamically to changing crime patterns. In particular, they choose their deployment to solve an optimal control problem at every time. This gives rise to a free boundary problem for the police deployment's spatial support. We present an efficient algorithm for solving this problem numerically and show that police presence can prompt surprising interactions among adjacent hotspots.
A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness
Yun Kang and Carlos Castillo-Chávez
2014, 19(1): 89-130 doi: 10.3934/dcdsb.2014.19.89 +[Abstract](527) +[PDF](8850.3KB)
Abstract:
The focus here is on the study disease dynamics under the assumption that a critical mass of susceptible individuals is required to guarantee the population's survival. Specifically, the emphasis is on the study of the role of an Allee effect on a Susceptible-Infectious (SI) model where the possibility that susceptible and infected individuals reproduce, with the S-class being the best fit. It is further assumed that infected individuals loose some of their ability to compete for resources, the cost imposed by the disease. These features are set in motion as simple model as possible. They turn out to lead to a rich set of dynamical outcomes. This toy model supports the possibility of multi-stability (hysteresis), saddle node and Hopf bifurcations, and catastrophic events (disease-induced extinction). The analyses provide a full picture of the system under disease-free dynamics including disease-induced extinction and proceed to identify required conditions for disease persistence. We conclude that increases in (i) the maximum birth rate of a species, or (ii) in the relative reproductive ability of infected individuals, or (iii) in the competitive ability of a infected individuals at low density levels, or in (iv) the per-capita death rate (including disease-induced) of infected individuals, can stabilize the system (resulting in disease persistence). We further conclude that increases in (a) the Allee effect threshold, or (b) in disease transmission rates, or in (c) the competitive ability of infected individuals at high density levels, can destabilize the system, possibly leading to the eventual collapse of the population. The results obtained from the analyses of this toy model highlight the significant role that factors like an Allee effect may play on the survival and persistence of animal populations. Scientists involved in biological conservation and pest management or interested in finding sustainability solutions, may find these results of this study compelling enough to suggest additional focused research on the role of disease in the regulation and persistence of animal populations. The risk faced by endangered species may turn out to be a lot higher than initially thought.
Preface
Thomas Hillen and Michael Winkler
2013, 18(10): i-ii doi: 10.3934/dcdsb.2013.18.10i +[Abstract](291) +[PDF](94.8KB)
Abstract:
Self-organization of micro organisms through the oriented movement of individuals along chemical gradients has caught researchers' imagination and interest for a long time. In fact, the process of aggregation of cells is a first step in the transition from individuals to a collective. Chemotaxis has been identified to play an important role in areas as diverse as ecological species (e.g. slime molds) and bacteria (E. coli), embryogenesis, immune response, wound healing, and cancer development. The first mathematical model for chemotaxis was introduced by Patlak (1953) and became later known as the Keller and Segel (1970) system of equations. The model has become a hot topic not only for the description of biological phenomena, but also mathematically. Sophisticated mathematical analysis has developed and it is the purpose of this special issue of DCDS-B to showcase some of the interesting and challenging mathematical questions that currently arise in the analysis of chemotaxis models.

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