Diffusion System Research Articles

Bifurcation and stability analysis of a Leslie–Gower diffusion predator–prey model with prey refuge and Beddington–DeAngelis functional response

In this paper, we consider the spatiotemporal dynamics behaviors of a Leslie–Gower diffusion predator–prey system with prey refuge and Beddington–DeAnglis (B‐D) functional response. By using the Poincaré inequality and topological degree theory, we first investigate the Turing instability of the reaction–diffusion system and prove the existence of nonconstant positive steady‐state solutions. Then we discuss the steady‐state bifurcation and the direction to Hopf bifurcation of the PDE model by the local bifurcation theorem and center manifold theory. Finally, some numerical simulations are presented to supplement the analytic results in one dimension which indicates that changes in prey refuge and diffusion coefficient can increase the complexity of the system.

Dynamics of Hindmarsh–Rose diffusive system

Dynamics of Hindmarsh–Rose diffusive system

The Kinetics of Polymer Brush Growth in the Frame of the Reaction Diffusion Front Formalism

We studied the properties of a reaction front that forms in irreversible reaction–diffusion systems with concentration-dependent diffusivities during the synthesis of polymer brushes. A coarse-grained model of the polymerization process during the formation of polymer brushes was designed and investigated for this purpose. In this model, a certain amount of initiator was placed on an impenetrable surface, and the “grafted from” procedure of polymerization was carried out. The system consisted of monomer molecules and growing chains. The obtained brush consisted of linear chains embedded in nodes of a face-centered cubic lattice with excluded volume interactions only. The simulations were carried out for high rafting densities of 0.1, 0.3, and 0.6 and for reaction probabilities of 0.02, 0.002, and 0.0002. Simulations were performed by means of the Monte Carlo method while employing the Dynamic Lattice Liquid model. Some universal behavior was found, i.e., irrespective of reaction rate and grafting density, the width of the reaction front as well as the height of the front show for long times the same scaling with respect to time. During the formation of the polymer layer despite the observed difference in dispersion of chain lengths for different grafting densities and reaction rates at a given layer height, the quality of the polymer layer does not seem to depend on these parameters.

Diffusion Cascades and Mutually Coupled Diffusion Processes

In this paper, we define and investigate a system of coupled regular diffusion equations in which each concentration acts as a driving term in the next diffusion equation. Such systems can be understood as a kind of cascade process which appear in different fields of physics like diffusion and reaction processes or turbulence. As a solution, we apply the time-dependent self-similar Ansatz method, the obtained solutions can be expressed as the product of a Gaussian and a Kummer’s function. This model physically means that the first diffusion works as a catalyst in the second diffusion system. The coupling of these diffusion systems is only one way. In the second part of the study we investigate mutually coupled diffusion equations which also have the self-similar trial function. The derived solutions show some similarities to the former one. To make our investigation more complete, different kinds of couplings were examined like the linear, the power-law, and the Lorentzian. Finally, a special coupling was investigated which is capable of describing isomerization with temporal decay.

Normal form of Turing–Turing bifurcation for the diffusive Bazykin system with prey-taxis

In this paper, we introduce prey-taxis to the diffusive Bazykin system and study the codimension-two Turing–Turing bifurcation of this modified system. For the local system, i.e. without diffusion terms and the prey-taxis term, we investigate the stability of the unique positive equilibrium. Then, for the diffusive Bazykin system with prey-taxis, we examine the stability of the unique positive constant steady state, Turing instability, and Turing bifurcation. Moreover, by choosing the random diffusion coefficients as bifurcation parameters, we investigate the Turing–Turing bifurcation of the diffusive Bazykin system with prey-taxis. Furthermore, to demonstrate and classify the spatiotemporal dynamics near the Turing–Turing bifurcation point, we derive an algorithm for calculating the third-order truncated normal form of this bifurcation using the center manifold theorem and normal form theory. Finally, to verify the theoretical analysis results and demonstrate the effectiveness of the derived algorithm, we perform several numerical simulations using matlab software. We also investigate the influence of the prey-taxis coefficient on the Turing instability of the diffusive Bazykin system with prey-taxis.

Development of an Additively Manufactured Stationary Diffusion System for a Research Aeroengine Centrifugal Compressor

Abstract This paper introduces the next generation of the Centrifugal Stage for Aerodynamic Research (CSTAR) facility at the Purdue University Compressor Research Lab. The research centrifugal compressor is designed with an additively manufactured stationary diffusion system which comprises a vaned diffuser, a turn-to-axial bend, and deswirler vanes. The dimensional accuracy and surface finish of the diffusion system has allowed for rapid prototyping of several iterative diffusion system designs and has allowed for configuration of in situ pressure and temperature measurements to experimentally evaluate the aerodynamics and performance of centrifugal compressor diffusion systems. The diffusion system is manufactured from a commercially available stereolithography (SLA) resin, and the design of the parts is adapted to the constraints imposed by current technological and material limits of resin 3D printing. The implementation of additive manufacturing in the prototyping of the diffusion system has allowed for decreased cost and lead time while allowing rapid turnaround to generate experimental data. Performance data have verified the repeatability and temperature independence of the additively manufactured diffusion system through several design iterations. A survey of candidate materials for additively manufacturing these parts is also presented, and the tradeoffs of their material properties are discussed.

Multiregion Light Control in Diffusive Media via Wavefront Shaping.

Wavefront shaping allows focusing light through or inside strongly scattering media, but the background intensity also increases which reduces the target's contrast. By combining transmission or deposition matrices for different regions, we construct joint operators to achieve spatially resolved control of light in diffusive systems. The eigenmode of a contrast operator can maximize the power contrast between a target and its surrounding. A difference operator enhances the power delivery to a target while avoiding the background increase. This work opens the door to coherent control of nonlocal effects in wave transport for practical applications.

A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: II stationary pattern formation

This paper is a continuation of the study conducted by Ko and Ryu (2024) [5], which introduces and analyzes a generalized predator-prey reaction-diffusion system incorporating (repulsive) prey-taxis and a hunting cooperation effect in predators, under homogeneous Neumann boundary conditions. In the study, the existence and uniqueness of global and classical solutions for the time- and space-dependent system are analytically examined. Furthermore, the study examines the local and global stability and convergence rate at the constant predator-extinction and coexistence states. In our paper, we analyze the stationary system corresponding to the system in [5], with a specific focus on examining the existence and nonexistence of positive and nonconstant solutions. The nonexistence occurs when the diffusion rate of prey is sufficiently high. On the other hand, the existence occurs when the prey-tactic rate is sufficiently high, indicating a strong repulsive prey-taxis, and the diffusion rate of prey is sufficiently low. For this investigation, we separately employ the energy method and the Leray-Schauder degree theory.

Patient experiences in high-grade gliomas: from symptoms to radiotherapy

ObjectivesLittle is known in the literature about felt experience and supportive care needs of patients with new diagnosis of adult-type diffuse gliomas Central Nervous System WHO grades 3 and 4....

Spatiotemporal Dynamics of an Intraguild Predation Model with Intraspecies Competition

Different species in the environment compete with one another for shared resources and suitable habitats. As a result, it becomes crucial to identify the essential components involved in preserving biodiversity. This paper describes the emergence of a wide range of temporal and spatiotemporal dynamics of an intraguild predation (IGP) model with intraspecies competition and Holling type-II response function between the IG-prey and IG-predator. To explore the dynamics of the temporal model, we study the local stability of all the biologically feasible steady states, global stability of the interior steady state, and Hopf bifurcation. Performing partial rank correlation coefficient (PRCC) sensitivity analysis, we picked more sensitive parameters and plotted the stability regions in two-parameter spaces. The dynamics exhibited by the system can demonstrate some important hallmarks noticed in the environments, such as the existence and nonexistence of oscillatory behavior in the IGP model. The analytical formulations for the stability and bifurcation of the coexistence of the steady-state are not easy to obtain, but we have verified numerically that the limit cycle bifurcates from the coexistence of the steady-state and there eventually emerges a chaotic behavior through period-doubling bifurcations. By plotting the largest Lyapunov exponent (LLE), we have verified that the temporal system experiences chaotic behavior. It is demonstrated that in the case of a diffusive system, diffusion may cause the coexistence of the steady state, which is otherwise stable, to become unstable. Criteria for the occurrence of Turing instability connected to the IGP model are derived. Our numerical illustrations demonstrate that diffusion-driven instability develops different stationary patterns based on the different initial conditions and diffusion parameters. The spatiotemporal patterns are observed in the Turing, Hopf–Turing, and Hopf regions.

Spatial patterns of the Brusselator model with asymmetric Lévy diffusion

Abstract The formation of spatial patterns is an important issue in reaction–diffusion systems. Previous studies have mainly focused on the spatial patterns in reaction–diffusion models equipped with symmetric diffusion (such as normal or fractional Laplace diffusion), namely, assuming that spatial environments of the systems are homogeneous. However, the complexity and heterogeneity of spatial environments of biochemical reactions in vivo can lead to asymmetric diffusion of reactants. Naturally, there arises an open question of how the asymmetric diffusion affects dynamical behaviors of biochemical reaction systems. To answer this, we build a general asymmetric Lévy diffusion model based on the theory of a continuous time random walk. In addition, we investigate the two-species Brusselator model with asymmetric Lévy diffusion, and obtain a general condition for the formation of Turing and wave patterns. More interestingly, we find that even though the Brusselator model with symmetric diffusion cannot produce steady spatial patterns for some parameters, the asymmetry of Lévy diffusion for this model can produce wave patterns. This is different from the previous result that wave instability requires at least a three-species model. In addition, the asymmetry of Lévy diffusion can significantly affect the amplitude and frequency of the spatial patterns. Our results enrich our knowledge of the mechanisms of pattern formation.

Canard cycle, relaxation oscillation and cross-diffusion induced pattern formation in a slow–fast ecological system with weak Allee effect

In this paper, we consider a Holling type IV functional response that describes a situation in which the predator’s per capita rate of predation decreases at sufficiently high prey density. Meanwhile it can be transformed into a Holling type II functional response in the limiting sense where predator’s attack rate increases at a decreasing rate with prey density until it becomes satiated. To explore the different dynamics of these two functional responses, we consider the slow–fast dynamic behavior of predator–prey system with Holling type IV and II functional responses, respectively, and make some simple comparisons of the dynamics of the system with these two functional responses. Specifically speaking, in the nonspatial case, firstly, the system with Holling type II functional response does not undergo a higher-codimension Hopf bifurcation. Then, the system with Holling type IV is extensively proved the existence of canard cycles and relaxation oscillations by using a range of analytical methods such as geometric singular perturbation technique, normal form of the slow–fast system, and the way in–way out function. In the spatial case, the temporal systems are extended to reaction–diffusion predator–prey systems. For the reaction–diffusion system with Holling IV, the different types of traveling wave are observed. Moreover, for the reaction–diffusion predator–prey system with Holling II, it is demonstrated that Turing instability occurs, which induces spatial heterogeneity patterns. Finally, the comparisons of the above dynamics of Holling Type IV and II functional responses in the non-spatial and spatial cases, respectively, are presented. From these comparisons, different Holling functional responses may be adopted for species at different stages or states, which is more conducive to maintaining species diversity and coexistence.

Nonconcentration phenomenon for one‐dimensional reaction–diffusion systems with mass dissipation

AbstractReaction–diffusion systems with mass dissipation are known to possess blow‐up solutions in high dimensions when the nonlinearities have super quadratic growth rates. In dimension 1, it has been shown recently that one can have global existence of bounded solutions if nonlinearities are at most cubic. For the cubic intermediate sum condition, that is, nonlinearities might have arbitrarily high growth rates, an additional entropy inequality had to be imposed. In this paper, we remove this extra entropy assumption completely and obtain global boundedness for reaction–diffusion systems with cubic intermediate sum condition. The novel idea is to show a nonconcentration phenomenon for mass dissipating systems, that is the mass dissipation implies a dissipation in a Morrey space for some . As far as we are concerned, it is the first time such a bound is derived for mass dissipating reaction–diffusion systems. The results are then applied to obtain global existence and boundedness of solutions to an oscillatory Belousov–Zhabotinsky system, which satisfies cubic intermediate sum condition but does not fulfill the entropy assumption. Extensions include global existence mass controlled systems with slightly super cubic intermediate sum condition.

Periodicity alters topological states in thermal diffusion system

Topological edge states are a ubiquitous phenomenon across diverse wave systems, including electromagnetic, acoustic and quantum systems. In contrast to wave systems, thermal transport is a kind of diffusion system characterized by pure dissipation. The imposition of different boundary conditions can significantly influence the evolution of thermal system, leading to the appearance of different topological states, which haven't been systematically analyzed yet, especially in practical heat transfer structures. In this work, via a non-Hermitian thermal diffusion lattice model, we discuss the periodicity-dependent topological states. Through rigorous theoretical analysis and numerical simulations of the temperature field, the Hamiltonian is obtained whose eigenvalues are purely imaginary corresponding to the decay rate of the temperature field. Our work paves the way for the investigation of how periodicity alters topological states in thermal diffusion systems, potentially revolutionizing the design of thermal metamaterials for topological thermal protection in thermal management.

Click metamaterials: Fast acquisition of thermal conductivity and functionality diversities

In material science, the development of metamaterials is crucial for advancing various technological applications. However, most metamaterial designs are still case by case due to lacking a fundamental mechanism for achieving reconfigurable thermal conductivities, largely hindering design flexibility and functional diversity. Inspired by the principles of click chemistry, known for its modular and efficient approach to creating molecular diversity, here we propose a universal concept of click metamaterials for fast realizing various thermal conductivities and functionalities. Tunable hollow-filled unit cells are constructed as the modified building blocks to change the thermal conductivity locally. Different configurations of unit cells with variable fill fractions can generate convertible thermal conductivities from isotropy to anisotropy, allowing click metamaterials to exhibit environment-free and reconfigurable thermal functionalities. The straightforward structures enable full-parameter regulation and simplify engineering preparation, making click metamaterials a promising candidate for practical use in other diffusion and wave systems.

Multiple Time Scales and Normal Form of Hopf Bifurcation in a Delayed Reaction–Diffusion–Advection System

In the reaction–diffusion systems, the Laplacian is usually a self-adjoint operator. Incorporating advection terms generally destroys this and yields a quite complicated decomposition of phase space in the Center Manifold Reduction (CMR) method. Considering that the Multiple Time Scales (MTS) method can be directly applied to bifurcation analysis and normal form derivation based on algebraic operations, and the computational process is relatively universal, we apply the MTS method to analyze the Hopf bifurcation problem of reaction–diffusion–advection systems with time delay. First, we introduce the MTS method for the system with a specific boundary condition, which is used to derive the normal form of Hopf bifurcation. Calculating the normal form using the MTS method can be summarized as determining the bifurcation parameters, conducting Taylor expansion based on the MTS idea, and eliminating secular terms. Then, the key coefficients in the normal form are explicitly given, which are also compared with the results obtained by the CMR method, and both methods lead to the same bifurcation results. Finally, we use the MTS method to calculate the normal form of Hopf bifurcation for a predator–prey model with mixed boundary conditions. We find that the spatially nonhomogeneous periodic solutions appear near the equilibrium in the system when the time delay exceeds the critical value, and the theoretical results are illustrated by numerical simulations.

Accurate measurement of Soret coefficients in binary hydrocarbons mixtures based on composite methods

The Soret effect is a significant factor in various scenarios, with thermodiffusion in binary systems serving as a common method for the study. Most research focuses rarely on the distribution characteristics of components in diffusion systems; and Soret coefficients in the porous media could not be obtained by typical methods based on the thermodiffusion column, which are particularly important in the field of oil and gas development. Moreover, experiments on ground conditions have struggled to determine the Soret coefficient accurately due to the convective effect caused by gravity differentiation. The thermodiffusion behavior of n-pentane (C5) and n-heptane (C7) binary mixtures in both bulk and porous media conditions have been investigated, aiming to provide corrected coefficients that mitigate the influence of gravity using theoretical derivation. A new method was proposed to calculate the Soret coefficients in this work by establishing a model based on gas chromatography technology. Dynamic variation of component concentration along the path was studied, and the corresponding Soret coefficients were calculated and analyzed in parallel. The results indicate that the concentration and temperature exhibit a logarithmic relationship with the distance from the heat source. The Soret coefficient values obtained from measurements in porous media are closer to the theoretically corrected values, which do not account for gravity effects. Additionally, as the permeability decreases, the counteracting effect of porous media on convection becomes more pronounced. Therefore, it presents a novel method for accurately measuring the Soret coefficient that ignores convection to some extent.

Singular limit of a chemotaxis model with indirect signal production and phenotype switching

Abstract Convergence of solutions to a partially diffusive chemotaxis system with indirect signal production and phenotype switching is shown in a two-dimensional setting when the switching rate increases to infinity, thereby providing a rigorous justification of formal computations performed in the literature. The expected limit system being the classical parabolic–parabolic Keller–Segel system, the obtained convergence is restricted to a finite time interval for general initial conditions but valid for arbitrary bounded time intervals when the mass of the initial condition is appropriately small. Furthermore, if the solution to the limit system blows up in finite time, then neither of the two phenotypes in the partially diffusive system can be uniformly bounded with respect to the L 2-norm beyond that time.

Enhancing energy and nitrogen removal efficiency through automatic split injection and innovative aeration device: A study of low C/N ratio environments

A new aeration device based on Bernoulli's principle, Jetventrumixer (JVM), was introduced into an aeration tank in denitrification process, which involved an automatic split injection system (ASIS) into two denitrification tanks every 10 minutes. Real-time monitoring of influent water allowed the calculation of the C/N ratio, enhancing the utilization efficiency of internal carbon sources while reducing the need for external carbon. The comparison of the JVM with the conventional air diffuser for 100 days operation showed that the removal efficiency for NH4+-N in both systems was approximately 98 %, but the nitrification efficiencies were 84 % and 80 %, respectively. This indicates that the JVM achieves an high enough removal efficiency and nitrification efficiency compared with conventional air diffuser system with dramatic reduction in energy consumption by 52.1 %. When the influent wastewater was split and injected into duplicate denitrification tanks at ratios of 3:7, 5:5, and 7:3, the total nitrogen (TN) removal efficiencies were 77 %, 73 %, and 72 %, respectively. In contrast, with the implementation of the ASIS, the TN removal efficiency increased up to 82 %. The increase in TN removal indicates that real-time monitoring could stably track changes chemical composition in wastewater influent over 24 h and introducing ASIS facilitate the efficient utilization of internal carbon sources, thereby enhancing denitrification efficiency and improving TN removal efficiency. Finally, the greenhouse gas (GHG) emissions from the JVM and air diffuser were 9.39401 and 19.60488 tCO2eq year-1, respectively, representing a 52% reduction. Therefore, JVM and ASIS successfully reduced energy consumption and enhanced both nitrification and denitrification efficiencies.

Pulsating waves in a multidimensional reaction-diffusion system of epidemic type

In this work we investigate the existence of front propagation for a two-component reaction -diffusion system of epidemic type posed in a multi-dimensional periodic medium. This system is a spatially heterogeneous version of the well-known Kermack–McKendrick epidemic model with Fickian diffusion. We derive sufficient conditions for the existence of pulsating travelling waves propagating in an arbitrarily given unit direction. Specifically, we prove that the set of admissible wave speeds contains a semi-infinite interval. Then, for each direction of propagation and each admissible speed, there exists a pulsating travelling wave solution of the system which is globally bounded.