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In this paper, we aim to investigate the influence of delay on the global attractivity of a tick population dynamics model incorporating two distinctive time-varying delays. By exploiting some differential inequality techniques and with the aid of the fluctuation lemma, we first prove the persistence and positiveness for all solutions of the addressed equation. Consequently, a delay-dependent criterion is derived to assure the global attractivity of the positive equilibrium point. And lastly, some numerical simulations are presented to verify that the obtained results improve and complement some existing ones.

This paper is concerned with the dynamic characteristics of a class of Nicholson’s blowflies system with patch structure and multiple pairs of distinct timevarying delays. We aim to find the influence of the distinct time-varying delays in the same reproductive function on its asymptotic behavior. First, we derive the global existence, positiveness and uniform persistence of solutions for the addressed system. Then, by employing the theory of functional differential equations, the fluctuation lemma and the technique of differential inequalities, we build up some new delay-dependent criteria for the global attractivity of the positive equilibrium point vector, which does not possess the same components. In addition, we exam the effectiveness and feasibility of the theoretical achievements by some numerical simulations.

In this paper, a prey-predator model with modified Leslie-Gower and simplified Holling-type Ⅳ functional responses is proposed to study the dynamic behaviors. For the deterministic system, we analyze the permanence of the system and the stability of the positive equilibrium point. For the stochastic system, we not only prove the existence and uniqueness of global positive solution, but also discuss the persistence in mean and extinction of the populations. In addition, we find that stochastic system has an ergodic stationary distribution under some parameter constraints. Finally, our theoretical results are verified by numerical simulations.

Time scales theory has been in use since the 1980s with many applications. Only very recently, it has been used to describe within-host and between-hosts dynamics of infectious diseases. In this study, we present explicit and implicit discrete epidemic models motivated by the time scales modeling approach. We use these models to formulate the basic reproduction number, which determines whether an outbreak occurs or the disease dies out. We discuss the stability of the disease-free and endemic equilibrium points using the linearization method and Lyapunov function. Furthermore, we apply our models to swine flu outbreak data to demonstrate that the discrete models can accurately describe the epidemic dynamics. Our comparison analysis shows that the implicit discrete model can best describe the data regardless of the data frequency. In addition, we perform the sensitivity analysis on the key parameters of the models to study how these parameters impact the basic reproduction number.

The problem of limit cycles for the Kolmogorov model is interesting and significant both in theory and applications. In this paper, we investigate the center-focus problems and limit cycles bifurcations for a class of cubic Kolmogorov model with three positive equilibrium points. The sufficient and necessary condition that each positive equilibrium point becomes a center is given. At the same time, we show that each one of point (1,2) and point (2,1) can bifurcate 1 small limit cycles under a certain condition, and 3 limit cycle can occur near (1,1) at the same step. Among the above limit cycles, 4 limit cycles can be stable. The limit cycles bifurcations problem for Kolmogorov model with several positive equilibrium points are hardly seen in published references. Our result is new and interesting.

This accessible book provides an introduction to the analysis and design of dynamic multiagent networks. Such networks are of great interest in a wide range of areas in science and engineering, including: mobile sensor networks, distributed robotics such as formation flying and swarming, quantum networks, networked economics, biological synchronization, and social networks. Focusing on graph theoretic methods for the analysis and synthesis of dynamic multiagent networks, the book presents a powerful new formalism and set of tools for networked systems. The book's three sections look at foundations, multiagent networks, and networks as systems. The authors give an overview of important ideas from graph theory, followed by a detailed account of the agreement protocol and its various extensions, including the behavior of the protocol over undirected, directed, switching, and random networks. They cover topics such as formation control, coverage, distributed estimation, social networks, and games over networks. And they explore intriguing aspects of viewing networks as systems, by making these networks amenable to control-theoretic analysis and automatic synthesis, by monitoring their dynamic evolution, and by examining higher-order interaction models in terms of simplicial complexes and their applications. The book will interest graduate students working in systems and control, as well as in computer science and robotics. It will be a standard reference for researchers seeking a self-contained account of system-theoretic aspects of multiagent networks and their wide-ranging applications. This book has been adopted as a textbook at the following universities: University of Stuttgart, GermanyRoyal Institute of Technology, SwedenJohannes Kepler University, AustriaGeorgia Tech, USAUniversity of Washington, USAOhio University, USA

Microbial-explicit soil organic carbon (SOC) cycling models are increasingly being recognized for their advantages over linear models in describing SOC dynamics. These models are known to exhibit oscillations, but it is not clear when they yield stable vs. unstable equilibrium points (EPs) – i.e., EPs that exist analytically but are not stable in relation to small perturbations and cannot be reached by transient simulations. The occurrence of such unstable EPs can lead to unexpected model behavior in transient simulations or unrealistic predictions of steady-state soil organic carbon (SOC) stocks. Here, we ask when and why unstable EPs can occur in an archetypal microbial-explicit model (representing SOC, dissolved OC (DOC), microbial biomass, and extracellular enzymes) and some simplified versions of it. Further, if a model formulation allows for physically meaningful but unstable EPs, can we find constraints in the model parameters (i.e., environmental conditions and microbial traits) that ensure stability of the EPs? We use analytical, numerical, and descriptive tools to answer these questions. We found that instability can occur when the resupply of a growth substrate (DOC) is (via a positive feedback loop) dependent on its abundance. We identified a conservative, sufficient condition in terms of model parameters to ensure the stability of EPs. Principally, three distinct strategies can avoid instability: (1) neglecting explicit DOC dynamics, (2) biomass-independent uptake rate, or (3) correlation between parameter values to obey the stability criterion. While the first two approaches simplify some mechanistic processes, the third approach points to the interactive effects of environmental conditions and parameters describing microbial physiology, highlighting the relevance of basic ecological principles for the avoidance of unrealistic (i.e., unstable) simulation outcomes. These insights can help to improve the applicability of microbial-explicit models, aid our understanding of the dynamics of these models, and highlight the relation between mathematical requirements and (in silico) microbial ecology.

The theory of alternate stable states provides an explanation for rapid ecosystem degradation, yielding important implications for ecosystem conservation and restoration. However, utilizing this theory to initiate transitions from degraded to desired ecosystem states remains a significant challenge. Applications of the alternative stable states framework may currently be impeded by a mismatch between local-scale driving processes and landscape-scale emergent system transitions. We show how nucleation theory provides an elegant bridge between local-scale positive feedback mechanisms and landscape-scale transitions between alternate stable ecosystem states. Geometrical principles can be used to derive a critical patch radius: a spatially explicit, local description of an unstable equilibrium point. This insight can be used to derive an optimal patch size that minimizes the cost of restoration, and to provide a framework to measure the resilience of desired ecosystem states to the synergistic effects of disturbance and environmental change.

This book places thermodynamics on a system-theoretic foundation so as to harmonize it with classical mechanics. Using the highest standards of exposition and rigor, the authors develop a novel formulation of thermodynamics that can be viewed as a moderate-sized system theory as compared to statistical thermodynamics. This middle-ground theory involves deterministic large-scale dynamical system models that bridge the gap between classical and statistical thermodynamics. The authors' theory is motivated by the fact that a discipline as cardinal as thermodynamics--entrusted with some of the most perplexing secrets of our universe--demands far more than physical mathematics as its underpinning. Even though many great physicists, such as Archimedes, Newton, and Lagrange, have humbled us with their mathematically seamless eurekas over the centuries, this book suggests that a great many physicists and engineers who have developed the theory of thermodynamics seem to have forgotten that mathematics, when used rigorously, is the irrefutable pathway to truth. This book uses system theoretic ideas to bring coherence, clarity, and precision to an extremely important and poorly understood classical area of science.

This paper investigates the existence of positive equilibrium as well as the stability of positive equilibrium and zero equilibrium in a nonlinear size-structured hierarchical population model. Under the condition that larger individuals are more competitive advantages than smaller ones, a non-zero fixed point theorem is used to show that there is at lest one positive equilibrium in the system. Moreover, we obtain the stability results of positive equilibrium and zero equilibrium by deriving characteristic equations and establishing Liapunov function. Finally, some numerical experiments are presented.