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Discrete & Continuous Dynamical Systems - B

Open Access Articles

Chaotic dynamics in a simple predator-prey model with discrete delay
Guihong Fan and Gail S. K. Wolkowicz
2020 doi: 10.3934/dcdsb.2020263 +[Abstract](143) +[HTML](81) +[PDF](5399.39KB)

A discrete delay is included to model the time between the capture of the prey and its conversion to viable biomass in the simplest classical Gause type predator-prey model that has equilibrium dynamics without delay. As the delay increases from zero, the coexistence equilibrium undergoes a supercritical Hopf bifurcation, two saddle-node bifurcations of limit cycles, and a cascade of period doublings, eventually leading to chaos. The resulting periodic orbits and the strange attractor resemble their counterparts for the Mackey-Glass equation. Due to the global stability of the system without delay, this complicated dynamics can be solely attributed to the introduction of the delay. Since many models include predator-prey like interactions as submodels, this study emphasizes the importance of understanding the implications of overlooking delay in such models on the reliability of the model-based predictions, especially since temperature is known to have an effect on the length of certain delays.

From approximate synchronization to identical synchronization in coupled systems
Chih-Wen Shih and Jui-Pin Tseng
2020, 25(9): 3677-3714 doi: 10.3934/dcdsb.2020086 +[Abstract](741) +[HTML](242) +[PDF](970.2KB)

We establish a framework to investigate approximate synchronization of coupled systems under general coupling schemes. The units comprising the coupled systems may be nonidentical and the coupling functions are nonlinear with delays. Both delay-dependent and delay-independent criteria for approximate synchronization are derived, based on an approach termed sequential contracting. It is explored and elucidated that the synchronization error, the distance between the asymptotic state and the synchronous set, decreases with decreasing difference between subsystems, difference between the row sums of connection matrix, and difference of coupling time delays between different units. This error vanishes when these factors decay to zero, and approximate synchronization becomes identical synchronization for the coupled system comprising identical subsystems and connection matrix with identical row sums, and with identical coupling delays. The application of the present theory to nonlinearly coupled heterogeneous FitzHugh-Nagumo neurons is illustrated. We extend the analysis to study approximate synchronization and asymptotic synchronization for coupled Lorenz systems and show that for some coupling schemes, the synchronization error decreases as the coupling strength increases, whereas in another case, the error remains at a substantial level for large coupling strength.

Local orthogonal rectification: Deriving natural coordinates to study flows relative to manifolds
Benjamin Letson and Jonathan E. Rubin
2020, 25(9): 3725-3747 doi: 10.3934/dcdsb.2020088 +[Abstract](697) +[HTML](240) +[PDF](1964.9KB)

We recently derived a method, local orthogonal rectification (LOR), that provides a natural and useful geometric frame for analyzing dynamics relative to a base curve in the phase plane for two-dimensional systems of ODEs (Letson and Rubin, SIAM J. Appl. Dyn. Syst., 2018). This work extends LOR to apply to any embedded base manifold in a system of ODEs of arbitrary dimension and establishes a corresponding system of LOR equations for analyzing dynamics within the LOR frame, which maps naturally back to the original phase space. The LOR equations encode geometric properties of the underlying flow and remain valid, in general, beyond a local neighborhood of the embedded manifold. In addition to developing a general theory for LOR that makes use of a given normal frame, we show how to construct a normal frame that conveniently simplifies the computations involved in LOR. Finally, we illustrate the utility of LOR by showing that a blow-up transformation on the LOR equations provides a useful decomposition for studying trajectories' behavior relative to the embedded base manifold and by using LOR to identify canard behavior near a fold of a critical manifold in a two-timescale system.

A sufficient optimality condition for delayed state-linear optimal control problems
Ana P. Lemos-Paião, Cristiana J. Silva and Delfim F. M. Torres
2019, 24(5): 2293-2313 doi: 10.3934/dcdsb.2019096 +[Abstract](2202) +[HTML](533) +[PDF](471.3KB)

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.

Applications of stochastic semigroups to cell cycle models
Katarzyna Pichór and Ryszard Rudnicki
2019, 24(5): 2365-2381 doi: 10.3934/dcdsb.2019099 +[Abstract](2106) +[HTML](531) +[PDF](440.63KB)

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.

Mathematical analysis of macrophage-bacteria interaction in tuberculosis infection
Danyun He, Qian Wang and Wing-Cheong Lo
2018, 23(8): 3387-3413 doi: 10.3934/dcdsb.2018239 +[Abstract](3565) +[HTML](838) +[PDF](1087.46KB)

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.

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](4054) +[HTML](1075) +[PDF](1511.3KB)

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](4066) +[HTML](943) +[PDF](346.0KB)

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.

Honglei Xu, Yi Zhang and Ka Fai Cedric Yiu
2017, 22(1): i-ii doi: 10.3934/dcdsb.201701i +[Abstract](1909) +[HTML](1036) +[PDF](75.6KB)
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](4283) +[HTML](1720) +[PDF](5226.6KB)

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](3162) +[HTML](1693) +[PDF](639.7KB)

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.

Chris Cosner, Yuan Lou, Shigui Ruan and Wenxian Shen
2017, 22(3): ⅰ-ⅱ doi: 10.3934/dcdsb.201703i +[Abstract](1695) +[HTML](879) +[PDF](77.4KB)
Tomás Caraballo, María J. Garrido-Atienza and Wilfried Grecksch
2016, 21(9): i-ii doi: 10.3934/dcdsb.201609i +[Abstract](1537) +[PDF](91.4KB)
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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Xiaoying Han and Qing Nie
2016, 21(7): i-ii doi: 10.3934/dcdsb.201607i +[Abstract](1468) +[PDF](85.1KB)
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](2767) +[PDF](4573.4KB)
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.
Jin Liang and Lihe Wang
2016, 21(5): i-ii doi: 10.3934/dcdsb.201605i +[Abstract](1659) +[PDF](173.9KB)
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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José M. Amigó and Karsten Keller
2015, 20(10): i-iii doi: 10.3934/dcdsb.2015.20.10i +[Abstract](1346) +[PDF](136.8KB)
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](3028) +[PDF](606.5KB)
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.
Robert Stephen Cantrell, Suzanne Lenhart, Yuan Lou and Shigui Ruan
2015, 20(6): i-iii doi: 10.3934/dcdsb.2015.20.6i +[Abstract](1928) +[PDF](127.7KB)
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](3588) +[PDF](3533.2KB)
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.

2019  Impact Factor: 1.27




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