Diffusion System Research Articles

Sliding mode control for uncertain fractional-order reaction–diffusion memristor neural networks with time delays

This paper investigates a sliding mode control method for a class of uncertain delayed fractional-order reaction–diffusion memristor neural networks. Different from most existing literature on sliding mode control for fractional-order reaction–diffusion systems, this study constructs a linear sliding mode switching function and designs the corresponding sliding mode control law. The sufficient theory for the globally asymptotic stability of the sliding mode dynamics are provided, and it is proven that the sliding mode surface is finite-time reachable under the proposed control law, with an estimate of the maximum reaching time. Finally, a numerical test is presented to validate the effectiveness of the theoretical analysis.

Investigating the Dynamic Behavior of Integer and Noninteger Order System of Predation with Holling’s Response

The paper’s primary objective is to examine the dynamic behavior of an integer and noninteger predator–prey system with a Holling type IV functional response in the Caputo sense. Our focus is on understanding how harvesting influences the stability, equilibria, bifurcations, and limit cycles within this system. We employ qualitative and quantitative analysis methods rooted in bifurcation theory, dynamical theory, and numerical simulation. We also delve into studying the boundedness of solutions and investigating the stability and existence of equilibrium points within the system. Leveraging Sotomayor’s theorem, we establish the presence of both the saddle-node and transcritical bifurcations. The analysis of the Hopf bifurcation is carried out using the normal form theorem. The model under consideration is extended to the fractional reaction–diffusion model which captures non-local and long-range effects more accurately than integer-order derivatives. This makes fractional reaction–diffusion systems suitable for modeling phenomena with anomalous diffusion or memory effects, improving the fidelity of simulations in turn. An adaptable numerical technique for solving this class of differential equations is also suggested. Through simulation results, we observe that one of the Lyapunov exponents has a negative value, indicating the potential for the emergence of a stable-limit cycle via bifurcation as well as chaotic and complex spatiotemporal distributions. We supplement our analytical investigations with numerical simulations to provide a comprehensive understanding of the system’s behavior. It was discovered that both the prey and predator populations will continue to coexist and be permanent, regardless of the choice of fractional parameter.

Pattern formation and delay-induced instability in a Leslie–Gower type prey–predator system with Smith growth function

In this article, we have investigated the spatiotemporal dynamics and delay-induced instability of a Leslie–Gower type prey–predator model under the influence of environmental toxicants with Smith growth function. This growth function is more realistic than logistic growth as it better describes the growth of the biological population. It has been used where the growth limitations are based on the proportion of available resources not utilized. A few works of Smith’s growth models are reported in the literature. Therefore, spatiotemporal dynamics and pattern formation with delay effect remain an exciting area of research, which motivates the present work. This work has studied two types of dynamical systems: (i) an ordinary differential temporal system with time delay and (ii) a reaction–diffusion system with time delay. The existence of equilibrium points and their stability conditions are discussed. Hopf bifurcation emerges in both proposed systems with respect to delay parameter. The stability and direction of Hopf bifurcation and delay–diffusion-driven instability have been investigated for the reaction–diffusion system. Numerical simulation is performed to support the analytical results and theorems. Moreover, the existence of Hopf and delay-induced instability are proved numerically. Interesting one-dimensional regular and irregular stripe patterns are obtained for increased values of the time delay parameter. Also, the presence of natural toxicants has a negative impact on the growth of prey–predator species.

Non-adaptive estimation for degenerate diffusion processes

We consider a degenerate system of stochastic differential equations. The first component of the system has a parameter θ 1 \theta _1 in a non-degenerate diffusion coefficient and a parameter θ 2 \theta _2 in the drift term. The second component has a drift term with a parameter θ 3 \theta _3 and no diffusion term. Parametric estimation of the degenerate diffusion system is discussed under a sampling scheme. We investigate the asymptotic behavior of the joint quasi-maximum likelihood estimator for ( θ 1 , θ 2 , θ 3 ) (\theta _1,\theta _2,\theta _3) . The estimation scheme is non-adaptive. The estimator incorporates information of the increments of both components, and under this construction, we show that the asymptotic variance of the estimator for θ 1 \theta _1 is smaller than the one for standard estimator based on the first component only, and that the convergence of the estimator for θ 3 \theta _3 is much faster than for the other parameters. By simulation studies, we compare the performance of the joint quasi-maximum likelihood estimator with the adaptive and one-step estimators investigated in Gloter and Yoshida [Electron. J. Statist 15 (2021), no. 1, 1424–1472].

Nonstandard Nearly Exact Analysis of the FitzHugh–Nagumo Model

The FitzHugh–Nagumo model has been used empirically to model certain types of neuronal activities. It is also a non-linear dynamical system applicable to chemical kinetics, population dynamics, epidemiology and pattern formation. In the literature, many approaches have been proposed to study its dynamics. In this paper, initially, we have employed cutting-edge tools from discrete dynamics for discretization and fixed points. It has been proven that an exact discrete scheme exists for this paradigm. This project also considers the phase space and integral surfaces of these evolutionary equations. In addition, it carries out a thorough symmetry analysis of this reaction diffusion system to find equivalent systems. Moreover, steady-state solutions are obtained using ansatzes for traveling wave solutions. The existence of infinite traveling wave solutions has also been proven. Yet again, this investigation establishes the potential of symmetry methods to unravel non-linearity. Finally, singular perturbation theory has been employed to obtain analytical approximations and to study stability in different parameter regimes.

Epidemiological profile of neuroendocrine tumors in adults in Brazil.

Neuroendocrine tumors (NETs) are a set of diseases that originate from neuroendocrine cells, which comprises a diffuse endocrine system present in various organs of the body. These tumors are more frequent in the gastrointestinal tract (70%) and the bronchopulmonary system (20%-30%). A NET incidence rate of 1-5 per 100,000 inhabitants has been estimated for several European countries and the USA employing 20 years of data. However, no comprehensive studies on this rare neoplasm are available in Brazil. In this context, the aim of this study was to characterize the epidemiological NET profile in the country. This is a retrospective descriptive observational study based on data from Hospital Cancer Records available at the Brazilian National Cancer Institute and the São Paulo Oncocentro Foundation. Demographic, clinical and treatmentrelated variables were analyzed from selected cases employing descriptive statistics. A total of 15,859 cases were identified, most occurring in males (53.4%) and in individuals under 65 years old (63.3%). Small cell carcinoma was the most frequent histological type (46.7%). Bronchopulmonary tumors were the most frequent NETs, followed by pancreatic tumors, with cases mostly concentrated in high complexity centers in the Brazilian Southeast and treated mainly with surgery and chemotherapy, with over half of the patients diagnosed in advanced stages of the disease.

Evolution of Cooperation in Spatio-Temporal Evolutionary Games with Public Goods Feedback.

In biology, evolutionary game-theoretical models often arise in which players' strategies impact the state of the environment, driving feedback between strategy and the surroundings. In this case, cooperative interactions can be applied to studying ecological systems, animal or microorganism populations, and cells producing or actively extracting a growth resource from their environment. We consider the framework of eco-evolutionary game theory with replicator dynamics and growth-limiting public goods extracted by population members from some external source. It is known that the two sub-populations of cooperators and defectors can develop spatio-temporal patterns that enable long-term coexistence in the shared environment. To investigate this phenomenon and unveil the mechanisms that sustain cooperation, we analyze two eco-evolutionary models: a well-mixed environment and a heterogeneous model with spatial diffusion. In the latter, we integrate spatial diffusion into replicator dynamics. Our findings reveal rich strategy dynamics, including bistability and bifurcations, in the temporal system and spatial stability, as well as Turing instability, Turing-Hopf bifurcations, and chaos in the diffusion system. The results indicate that effective mechanisms to promote cooperation include increasing the player density, decreasing the relative timescale, controlling the density of initial cooperators, improving the diffusion rate of the public goods, lowering the diffusion rate of the cooperators, and enhancing the payoffs to the cooperators. We provide the conditions for the existence, stability, and occurrence of bifurcations in both systems. Our analysis can be applied to dynamic phenomena in fields as diverse as human decision-making, microorganism growth factors secretion, and group hunting.

An innovative Vieta–Fibonacci wavelet collocation method for the numerical solution of three-component Brusselator reaction diffusion system of fractional order

An innovative Vieta–Fibonacci wavelet collocation method for the numerical solution of three-component Brusselator reaction diffusion system of fractional order

Strong stability preserving multiderivative time marching methods for stiff reaction–diffusion systems

The present study introduces a new class of unconditionally strong stability preserving (SSP) multi-derivative Runge–Kutta methods for the numerical simulation of reaction–diffusion systems in the stiff regime. The unconditional SSP property of the methods makes them highly efficient for the simulation of reaction–diffusion systems without any restrictive time-step requirements. These methods have been tested for accuracy using L∞ error analysis, and comparisons with existing literature have shown that they perform better even for larger time-steps. In addition, the robustness and efficiency of the derived method are also validated by numerical simulations of the Brusselator, Gray–Scott, and Schnakenberg models.

Key Transdermal Patch Using Cannabidiol-Loaded Nanocarriers with Better Pharmacokinetics in vivo.

Cannabidiol (CBD) is a promising therapeutic drug with low addictive potential and a favorable safety profile. However, CBD did face certain challenges, including poor solubility in water and low oral bioavailability. To harness the potential of CBD by combining it with a transdermal drug delivery system (TDDS). This innovative approach sought to develop a transdermal patch dosage form with micellar vesicular nanocarriers to enhance the bioavailability of CBD, leading to improved therapeutic outcomes. A skin-penetrating micellar vesicular nanocarriers, prepared using nano emulsion method, cannabidiol loaded transdermal nanocarriers-12 (CTD-12) was presented with a small particle size, high encapsulation efficiency, and a drug-loaded ratio for CBD. The skin permeation ability used Strat-M™ membrane with a transdermal diffusion system to evaluate the CTD and patch of CTD-12 (PCTD-12) within 24 hrs. PCTD-12 was used in a preliminary pharmacokinetic study in rats to demonstrate the potential of the developed transdermal nanocarrier drug patch for future applications. In the transdermal application of CTD-12, the relative bioavailability of the formulation was 3.68 ± 0.17-fold greater than in the free CBD application. Moreover, PCTD-12 indicated 2.46 ± 0.18-fold higher relative bioavailability comparing with free CBD patch in the ex vivo evaluation. Most importantly, in the pharmacokinetics of PCTD-12, the relative bioavailability of PCTD-12 was 9.47 ± 0.88-fold higher than in the oral application. CTD-12, a transdermal nanocarrier, represents a promising approach for CBD delivery, suggesting its potential as an effective transdermal dosage form.

Equivariant Hopf Bifurcation in a Class of Partial Functional Differential Equations on a Circular Domain

Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points.

A toxicant-taxis reaction–diffusion model for the interaction between two species and a toxicant in an aquatic ecosystem

In this paper, we develop a toxicant-taxis reaction–diffusion model that describes the interaction dynamics between two species and a toxicant in a polluted aquatic ecosystem. We analyze the existence and stability of the equilibria of the corresponding space-absent ordinary differential equation system. We then perform linear stability analysis for our reaction–diffusion system to obtain sufficient conditions on model parameters under which the stable spatially homogeneous steady states become unstable. Numerical simulations are performed to illustrate that spatially inhomogeneous patterns and spatiotemporal patterns between the two species and the toxicant will typically emerge. We also numerically analyze the effects of spatially varying toxicant input and the intensity of toxicant-taxis on the population persistence and species diversity. It turns out that heterogeneous toxicant inputs and substantial toxicant taxis can boost the coexistence of two species, thereby promoting species diversity in ecosystems. By comparing the impact of a toxicant on a single species versus its impact on two species, we find that the collective sharing of the toxic burden among multiple species proves advantageous for biodiversity.

Asymptotic Behavior of Global Solution of an Anomalous Gray Scott Model

The object of this paper is to prove that existence global and asymptotic behavior of solutions for anomalous coupled reaction diffusion system (Gray Scott model) with homogeneous Neumann boundary conditions. The existence and uniqueness of the local solution are given by the Banach fixed point theorem. However, the asymptotic behavior is investiged by technique semi group estimates and the Sobolev embedding theorem

On wavefront patterns in a fractional reaction–diffusion model for predator–prey system with anti-predator behavior

AbstractIn recent studies, much attention has been paid to reaction–diffusion systems with anomalous diffusion. In this paper, we investigate the formation of wavefront patterns in a predator–prey model with anti-predator behavior under the influence of anomalous subdiffusion. We use methods of traveling wave analysis and numerical integration to establish the existence of traveling wavefront solutions. Further, obtained traveling wavefront solutions are validated through direct computer simulations of time-dependent solutions for fractional partial differential equation system. It is found that wavefronts exist in a range of system parameters, which travel faster in the subdiffusive system than in the normal diffusive one.

Bronchial carcinoid. Stages of treatment

Carcinoids are tumors of the diffuse endocrine system that arise from hormone-producing cells of nervous origin (secretory neurocytes). Bronchial carcinoid was first described by R. Leannec [3] in 1831. Among all lung neoplasms, carcinoid tumors occur in 2%. CONCLUSION This method allowed the first stage to stop inflammatory processes in the lung tissue, which contributed to the effective implementation of the main stage of treatment. Lobectomy was performed without complications.

Universal stability of coherently diffusive one-dimensional systems with respect to decoherence

Universal stability of coherently diffusive one-dimensional systems with respect to decoherence

Higher-Order Topological In-Bulk Corner State in Pure Diffusion Systems.

Compared with conventional topological insulator that carries topological state at its boundaries, the higher-order topological insulator exhibits lower-dimensional gapless boundary states at its corners and hinges. Leveraging the form similarity between Schrödinger equation and diffusion equation, research on higher-order topological insulators has been extended from condensed matter physics to thermal diffusion. Unfortunately, all the corner states of thermal higher-order topological insulator reside within the band gap. Another kind of corner state, which is embedded in the bulk states, has not been realized in pure diffusion systems so far. Here, we construct higher-dimensional Su-Schrieffer-Heeger models based on sphere-rod structure to elucidate these corner states, which we term "in-bulk corner states." Because of the anti-Hermitian properties of diffusive Hamiltonian, we investigate the thermal behavior of these corner states through theoretical calculation, simulation, and experiment. Furthermore, we study the different thermal behaviors of in-bulk corner state and in-gap corner state. Our results would open a different gate for diffusive topological states and provide a distinct application for efficient heat dissipation.

Observation of parity-time symmetry in diffusive systems.

Phase modulation has scarcely been mentioned in diffusive physical systems because the diffusion process does not carry the momentum like waves. Recently, non-Hermitian physics provides a unique perspective for understanding diffusion and shows prospects in thermal phase regulation, exemplified by the discovery of anti-parity-time (APT) symmetry in diffusive systems. However, precise control of thermal phase remains elusive hitherto and can hardly be realized, due to the phase oscillations. Here we construct the PT-symmetric diffusive systems to achieve the complete suppression of thermal phase oscillation. The real coupling of diffusive fields is readily established through a strong convective background, and the decay-rate detuning is enabled by thermal metamaterial design. We observe the phase transition of PT symmetry breaking with the symmetry-determined amplitude and phase regulation of coupled temperature fields. Our work shows the existence of PT symmetry in dissipative energy exchanges and provides unique approaches for harnessing the mass transfer of particles, wave dynamics in strongly scattering systems, and thermal conduction.

Weak Pinning and Long-Range Anticorrelated Motion of Phase Boundaries in Driven Diffusive Systems.

We show that domain walls separating coexisting extremal current phases in driven diffusive systems exhibit complex stochastic dynamics with a subdiffusive temporal growth of position fluctuations due to long-range anticorrelated current fluctuations and a weak pinning at long times. This weak pinning manifests itself in a saturated width of the domain wall position fluctuations that increases sublinearly with the system size. As a function of time t and system size L, the width w(t,L) has a scaling behavior w(t,L)=L^{3/4}f(t/L^{9/4}), with f(u) constant for u≫1 and f(u)∼u^{1/3} for u≪1. An Orstein-Uhlenbeck process with long-range anticorrelated noise is shown to capture this scaling behavior. The exponent 9/4 is a new dynamical exponent for relaxation processes in driven diffusive systems.

Stabilization of coupled delayed nonlinear time fractional reaction diffusion systems using sampled‐in‐space sensing and actuation

AbstractThis paper is considered with the asymptotic stabilization of coupled delayed nonlinear time fractional reaction diffusion systems (FRDSs) governed by fractional parabolic partial differential equations (PDEs) with space‐dependent coefficients under sampled‐data in space control. It is assumed that state measurements can be averaged measurements (AMs) or point measurements (PMs), and a finite number of sensing and actuation devices are located in a spaced manner along the spatial domain of the interest. With the proposed sampled‐data in space controller, the closed‐loop stability is obtained. Tuning rules of system parameters and control parameters are derived using the fractional Halanay's inequality and the fractional Lyapunov method. Subsequently, the dual problem of observer design is formulated. Fractional examples are used to valid the theoretical result. Discussions on the extension of sampled‐data boundary feedback stabilization are provided finally.