Nonnegative States Research Articles

Wigner non-negative states that verify the Wigner entropy conjecture

Wigner non-negative states that verify the Wigner entropy conjecture

Turing Instability and Spatial Pattern Formation in a Model of Urban Crime

A nonlinear crime model is generalized by introducing self- and cross-diffusion terms. The effect of diffusion on the stability of non-negative constant steady states is applied. In particular, the cross-diffusion-driven instability, called Turing instability, is analyzed by linear stability analysis, and several Turing patterns driven by the cross-diffusion are studied through numerical investigations. When the Turing–Hopf conditions are satisfied, the type of instability highlighted in the ODE model persists in the PDE system, still showing an oscillatory behavior.

The dynamics of an eco-epidemiological prey–predator model with infectious diseases in prey

In this paper, we first propose an eco-epidemiological prey–predator model with infectious diseases in prey. Then study the ODE model, and diffusive model with the homogeneous Neumann and Dirichlet boundary conditions, respectively. For the ODE model and the diffusive model with the homogeneous Neumann boundary conditions, we give a complete conclusion about the stabilities of nonnegative equilibrium states (nonnegative constant equilibrium solutions). The results show that these two problems has no periodic solutions, and the diffusive model with the homogeneous Neumann boundary conditions has no yet Turing patterns. For the diffusive model with the homogeneous Dirichlet boundary conditions, we first establish the necessary and sufficient conditions for the existence of positive equilibrium solutions, and prove that the positive equilibrium solution is unique when it exists. Then we study the global asymptotic stabilities of trivial and semi-trivial nonnegative equilibrium solutions.

Turing-Hopf Bifurcation Analysis in a Diffusive Ratio-Dependent Predator-Prey Model with Allee Effect and Predator Harvesting.

This paper investigates the complex dynamics of a ratio-dependent predator-prey model incorporating the Allee effect in prey and predator harvesting. To explore the joint effect of the harvesting effort and diffusion on the dynamics of the system, we perform the following analyses: (a) The stability of non-negative constant steady states; (b) The sufficient conditions for the occurrence of a Hopf bifurcation, Turing bifurcation, and Turing-Hopf bifurcation; (c) The derivation of the normal form near the Turing-Hopf singularity. Moreover, we provide numerical simulations to illustrate the theoretical results. The results demonstrate that the small change in harvesting effort and the ratio of the diffusion coefficients will destabilize the constant steady states and lead to the complex spatiotemporal behaviors, including homogeneous and inhomogeneous periodic solutions and nonconstant steady states. Moreover, the numerical simulations coincide with our theoretical results.

A prey-refuge harvesting model using intuitionistic fuzzy sets

This study considers the classical Lotka–Volterra model for a prey–predator system with harvesting effort and catchability coefficients for both species incorporating a new prey-refuge parameter. Trapezoidal intuitionistic fuzzy numbers are considered to describe the initial values of the prey and predator population. Generalized Hukuhara intuitionistic fuzzy derivatives of type 1 and type 2 are used to formulate the prey–predator model. Non-negative feasible steady states and stability criteria are analyzed along with strong and weak solutions for the model. Numerical simulations with the help of MATLAB are performed for suitable parameter values, and the results are depicted graphically.

Control of positive systems with an unknown state-dependent power law input delay and input saturation

This paper is motivated by a class of positive systems with an input that is subject to an unknown state-dependent power law delay as well as saturation. For example, biological networks have non-negative protein concentration states. Mass action kinetics in these systems result in power law behavior, while complex interactions cause signal propagation delays. Incomplete network characterization makes delay state-dependence unknown. Manipulating network activity via modulated protein concentrations to attain desired performance is restricted by upper-bounds on concentration actuator authority. Here, an innovative control law exploits system dynamics to compensate for control domain restrictions. A Lyapunov stability analysis establishes that the reference tracking error of the closed-loop system is uniformly ultimately bounded. Numerical simulations on a human coagulation model show controller efficacy and better performance compared to the relevant literature. This example application steps toward personalized, closed-loop treatments for trauma coagulopathy, which currently has 30% mortality with open-loop clinical approaches.

Global dynamics of a nonlocal non-uniformly parabolic equation arising from the curvature flow* *The first author is supported by NSF of China (No. 12271437 and No. 12071203). The second author is supported by NSF of China (No. 11971498) and NSF of Guangdong Province (No. 2019A1515011339). The third author is supported by NSF of China (No. 11871148).

This paper studies a type of non-uniformly parabolic problem with nonlocal term ut=up(uxx+u−u¯)0<t<Tmax,0<x<a,ux(t,0)=ux(t,a)=00<t<Tmax,u(0,x)=u0(x)0<x<a, where p > 1, a > 0. First the classification of the finite-time blow-up/global existence phenomena based on the associated energy functional and explicit expression of all nonnegative steady states are demonstrated. More importantly, we derive that any bounded solution converges to some steady state as t → +∞. The difficulties in proving this convergence result lie in the existence of a continuum of steady states and the lack of the comparison principle due to the introduction of nonlocal term. To conquer these difficulties, we combine the applications of Lojasiewicz–Simon inequality and energy estimates.

Density matrix reconstruction using non-negative matrix product states

Quantum state tomography is a key technique for quantum information processing but is challenging due to the exponential growth of its complexity with the system size. In this work we propose an algorithm which iteratively finds the best non-negative matrix product state approximation based on a set of measurement outcomes whose size does not necessarily grow exponentially. Compared to the tomography method based on neural network states, our scheme utilizes a so-called tensor train representation that allows straightforward recovery of the unknown density matrix in the matrix product operator form. As applications, the effectiveness of our algorithm is numerically demonstrated to reconstruct the ground state of the XXZ spin chain under depolarizing noise.

On dynamics and stationary pattern formations of a diffusive predator-prey system with hunting cooperation

&lt;p style='text-indent:20px;'&gt;The current paper provides a qualitative study of a diffusive predator-prey system with a hunting-cooperation functional response under homogeneous Neumann boundary conditions. The considered functional response explains a behavioral mechanism in which predators search for and hunt their prey cooperatively. We investigate the global attractor for nonnegative time-dependent solutions to the system and the nonpersistence leading to the extinction of the predator species in the system. Moreover, we study the local stability at all feasible nonnegative constant steady states and the occurrence of Hopf bifurcation (i.e., the existence of time-periodic orbits) in the system. Finally, we study the existence and nonexistence of nonconstant positive steady states in the system, and the limiting behavior of the positive steady states according to the diffusion rate. Through this study, we observe interesting results, such as bistability, extinction of the predator species induced by a large consumption rate, and stationary patterns in the system with hunting cooperation in predators.&lt;/p&gt;

Consequences of refuge and diffusion in a spatiotemporal predator–prey model

In this investigation, we offer and examine a predator–prey interacting model with prey refuge in proportion to both the species and Beddington–DeAngelis functional response. We first prove the well-posedness of the temporal and spatiotemporal models which are restricted in a positive invariant region. Then for the temporal model, we analyse its temporal dynamics including uniform boundedness, permanence, stability of all feasible non-negative equilibria and show that refugia can induce periodic oscillation via Hopf bifurcation around the unique positive equilibrium; for the spatiotemporal model, we not only investigate its permanence, stability of non-negative constant steady states and Turing instability but also study the existence and non-existence of non-constant positive steady states by Leray–Schauder degree theory. The key observation is that the coefficient of refuge cooperates a significant part in modifying the dynamics of the current system and mediates the population permanence, stability of coexisting equilibrium and even the Turing instability parameter space. Finally, general numerical simulation consequences are given to illustrate the validity of the theoretical results. Through numerical simulations, one observes that the model dynamics shows prey refugia and self-diffusion control spatiotemporal pattern growth to spots, stripe–spot mixtures and stripes reproduction. The outcomes assign that the dynamics of the model with prey refuge is not simple, but rich and complex. Additionally, numerical simulations show that the other model parameters have an important effect on species’ spatially inhomogeneous distribution, which results in the formation of spots pattern, mixture of spots and stripes pattern, mixture of spots, stripes and rings pattern and anti-spot pattern. This may improve the model dynamics of the prey refuge on the reaction–diffusion predator–prey system.

Mckendrick–Tornquist Age-Dependent Population Model

The paper considers an age-dependent population model where the mortality function is given by the Tornquist distribution. As a result of the investigation, we obtain conditions of the existence of nonnegative stationary states in this model.

Stability Analysis of Delayed Tumor-Antigen-ActivatedImmune Response in Combined BCG and IL-2Immunotherapy of Bladder Cancer

We use a system biology approach to translate the interaction of Bacillus Calmette-Gurin (BCG) + interleukin 2 (IL-2) for the treatment of bladder cancer into a mathematical model. The main goal of this research is to predict the outcome of BCG + IL-2 treatment combinations. We examined whether the delay effect caused by the proliferation of tumor antigen-specific effector cells after the immune system destroys BCG-infected urothelium cells after BCG and IL-2 immunotherapy influences success in bladder cancer treatment. To do this, we introduce a system of differential equations where the variables are the main participants in the immune response after BCG installations to fight cancer: the number of tumor cells, BCG cells, immune cells, and cytokines involved in the tumor-immune response. The relevant parameters describing the dynamics of the system are taken from a variety of biological, clinical literature and estimated using the mathematical models. We examine the local stability analysis of non-negative equilibrium states of the model. In theory, treatment could improve system stability, and we analyze the stability of all equilibria using the method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). Our results prove that the period for the proliferation of tumor antigen-specific effector cells does not influence to the success of the non-responsive patients after an intensified combined BCG + IL-2 treatment.

H∞ Control for Observer-Based Non-Negative Edge Consensus of Discrete-Time Networked Systems.

This article is concerned with observer-based non-negative edge-consensus (OBNNEC) problems of networked discrete-time systems with or without actuator saturation. An algorithm which only uses actual outputs of neighboring edges is proposed by means of an H∞ control method and modified algebraic Riccati equation (MARE)-based technique. The observer matrix and feedback matrix are constructed by solving the MARE and linear matrix inequality (LMI), respectively. Then, sufficient conditions for guaranteeing bounded inputs and non-negative edge states are derived. In addition, the low-gain characteristic of the MARE-based method is instrumental in guaranteeing non-negative edge states and deriving the feasible observer matrix and feedback matrix. Finally, two examples are shown to demonstrate the obtained theoretical results.

Nonnegative controllability for a class of nonlinear degenerate parabolic equations with application to climate science

We consider a nonlinear degenerate reaction-diffusion equation. First we prove that if the initial state is nonnegative, then the solution remains nonnegative for all time. Then we prove the approximate controllability between nonnegative states via multiplicative controls, this is done using the reaction coefficient as control.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/59/abstr.html

H ∞ Control for Observer-Based Non-Negative Scaled Edge-Consensus of Networked Systems

This article investigates the observer-based non-negative scaled edge-consensus problems of a continues-time networked system with or without input saturation, in which the proposed algorithm only uses the neighboring edges’ outputs. First, the original networked model and preliminary algorithm are simplified by using the line graph theory and constructing a new variable. Then the specific mathematical formulations of feedback matrix and observer matrix are both derived by combining <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> control theory and low-gain feedback technique. Next, sufficient conditions which can guarantee the non-negative edge states and bounded control inputs are obtained. Moreover, the advantages of low-gain feedback technique for guaranteeing the non-negative edge states and designing the algorithm are developed. Finally, three cases are used to verify the validity of the theorems.

Robust Observer Design for Two-Dimensional Discrete Positive Switched Systems With Delays

The design problem of the robust positive observer for a class of two-dimensional (2-D) positive discrete switched systems in the presence of delays and uncertainties is considered in this brief. The delays and uncertainties in this brief are assumed to be time-varying and polytopic type, respectively. Firstly, the multiple Lyapunov functional (MLF) approach has been used to formulate some sufficient asymptotical stability conditions for 2-D switched systems with uncertainties, by choosing a suitable co-positive Lyapunov-Krasovskii functional (CPLKF). Secondly, under the assumption that the system states are not measurable, a robust observer design scheme is formulated that helps in the approximation of the corresponding non-negative states. It also ensures the asymptotic convergence of the associated error system. Finally, an appropriate example is discussed to show the usefulness of the proposed results.

Stationary Pattern of a Reaction-Diffusion Mussel-Algae Model.

In this paper, we consider a reaction-diffusion mussel-algae model with state-dependent mussel mortality. This mortality involves a positive feedback term resulting from the reduction of dislodgment and predation and a negative feedback term resulting from the intraspecific competition for mussel. We first study the global stability of the nonnegative uniform steady states and then focus on the existence and nonexistence of nonconstant positive steady states. The global bifurcation of constant positive steady state is also considered. Our results suggest that the regular patterning in mussel beds may be caused by the high mobility of algae or the low diffusion of mussels.

Local controllability of reaction-diffusion systems around nonnegative stationary states

We consider an×nnonlinear reaction-diffusion system posed on a smooth bounded domain Ω of ℝN. This system models reversible chemical reactions. We act on the system throughmcontrols (1 ≤m&lt;n), localized in some arbitrary nonempty open subsetωof the domainΩ. We prove the local exact controllability to nonnegative (constant) stationary states in any timeT&gt; 0. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics that prevents controllability from happening in the whole spaceL∞(Ω)n. The proof relies on several ingredients. First, an adequate affine change of variables transforms the system into a cascade system with second order coupling terms. Secondly, we establish a new null-controllability result for the linearized system thanks to a spectral inequality for finite sums of eigenfunctions of the Neumann Laplacian operator, due to David Jerison, Gilles Lebeau and Luc Robbiano and precise observability inequalities for a family of finite dimensional systems. Thirdly, the source term method, introduced by Yuning Liu, Takéo Takahashi and Marius Tucsnak, is revisited in aL∞-context. Finally, an appropriate inverse mapping theorem in suitable spaces enables to go back to the nonlinear reaction-diffusion system.

Coexistence and extinction in a data-based ratio-dependent model of an insect community.

In theory, pure competition often leads to competitive exclusion of species. However, what we often see in nature is a large number of distinct predator or consumer species coexist in a community consisting a smaller number of prey or plant species. In an effort of dissecting how indirect competition and selective predation may have contributed to the coexistence of species in an insect community, according to the replicated cage experiments (two aphid species and a specialist parasitoid that attacks only one of the aphids) and proposed mathematical models, van Veen et. al. [5] conclude that the coexistence of the three species is due to a combination of density-mediated and trait-mediated indirect interactions. In this paper, we formulate an alternative model that observes the conventional law of mass conservation and provides a better fitting to their experimental data sets. Moreover, we present an initial attempt in studying the stabilities of the nonnegative steady states of this model.

Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator

In this paper, we study the spatiotemporal dynamics of a diffusive predator-prey model with generalist predator subject to homogeneous Neumann boundary condition. Some basic dynamics including the dissipation, persistence and non-persistence(i.e., one species goes extinct), the local and global stability of non-negative constant steady states of the model are investigated. The conditions of Turing instability due to diffusion at positive constant steady states are presented. A critical value \begin{document}$ \rho $\end{document} of the ratio \begin{document}$ \frac{d_2}{d_1} $\end{document} of diffusions of predator to prey is obtained, such that if \begin{document}$ \frac{d_2}{d_1}>\rho $\end{document} , then along with other suitable conditions Turing bifurcation will emerge at a positive steady state, in particular so it is with the large diffusion rate of predator or the small diffusion rate of prey; while if \begin{document}$ \frac{d_2}{d_1} , both the reaction-diffusion system and its corresponding ODE system are stable at the positive steady state. In addition, we provide some results on the existence and non-existence of positive non-constant steady states. These existence results indicate that the occurrence of Turing bifurcation, along with other suitable conditions, implies the existence of non-constant positive steady states bifurcating from the constant solution. At last, by numerical simulations, we demonstrate Turing pattern formation on the effect of the varied diffusive ratio \begin{document}$ \frac{d_2}{d_1} $\end{document} . As \begin{document}$ \frac{d_2}{d_1} $\end{document} increases, Turing patterns change from spots pattern, stripes pattern into spots-stripes pattern. It indicates that the pattern formation of the model is rich and complex.