Nonhomogeneous Solutions Research Articles

Hopf bifurcation of a diffusive SIS epidemic system with delay in heterogeneous environment

This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive SIS epidemic system with delay in heterogeneous environment subject to homogeneous Neumann boundary condition. Firstly, we explore the principal eigenvalue to obtain the stability of the disease-free equilibrium (DFE) and the effect of the nonhomogeneous coefficients on the stable region of the DFE. Secondly, we obtain the existence, multiplicity and explicit structure of the endemic equilibrium (EE), i.e., spatially nonhomogeneous steady-state solutions, by using the implicit function theorem and Lyapunov-Schmidt reduction method. Furthermore, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of EE and the existence of Hopf bifurcations at EE are given. Finally, the direction of Hopf bifurcation and stability of the bifurcating periodic solution are obtained by virtue of normal form theory and center manifold reduction.

Pattern formations of a delayed diffusive predator–prey model with predator harvesting and prey social behavior

In this research, we are interested in analyzing the behavior of a diffusive predator–prey model in the presence of predator rivalry and prey social behavior. The predator rivalry affects hugely the behavior of predator and the prey, so we are interested in analyzing the complex dynamics generated by the presence of predator harvesting and its influence on the behavior of the solution. For the mathematical purpose, we will concentrate on analyzing the system in the absence of diffusion where the existence of Hopf bifurcation is investigated; also, we proved that the presence of the time lags generates interesting patterns, which it can lead to instability even Hopf bifurcation. The stability of the homogeneous and nonhomogeneous periodic solution generated by the presence of Hopf bifurcation is established using the normal form. The mathematical obtained results are checked numerically using graphical representations.

Oscillatory and Stationary Patterns in a Diffusive Model with Delay Effect

In this paper, a reaction–diffusion model with delay effect and Dirichlet boundary condition is considered. Firstly, the existence, multiplicity, and patterns of spatially nonhomogeneous steady-state solution are obtained by using the Lyapunov–Schmidt reduction. Secondly, by means of space decomposition, we subtly discuss the distribution of eigenvalues of the infinitesimal generator associated with the linearized system at a spatially nonhomogeneous synchronous steady-state solution, and then we derive some sufficient conditions to ensure that the nontrivial synchronous steady-state solution is asymptotically stable. By using the symmetric bifurcation theory of differential equations together with the representation theory of standard dihedral groups, we not only investigate the effect of time delay on the pattern formation, but also obtain some important results on the spontaneous bifurcation of multiple branches of nonlinear wave solutions and their spatiotemporal patterns.

NONUNIVERSAL PROPERTIES OF SELF-INTERACTING POLYMER IN NON-HOMOGENEOUS ENVIRONMENT MODELED BY 3-SIMPLEX FRACTAL LATTICE

We have studied lattice self-avoiding polygons with attractive interaction between contacts which are nonconsecutively visited nearest neighboring sites. The lattice of choice is 3-simplex fractal lattice and the model represents a ring polymer in non-homogeneous solution whose quality depends on the interaction parameter. It has already been shown, by the renormalization group approach, that polymer on this lattice at any nonzero temperature can exist only in the extended phase. Universal critical exponents, which do not depend on the interaction strength, have also been determined. In this paper we are concerned with two nonuniversal quantities: the connectivity constant related with the free energy of the model and the mean number of contacts related with the internal energy. We have shown that the connectivity constant is an unboundedly increasing function of the interaction strength, while the mean number of contacts is an increasing function asymptotically approaching a limiting value equal to one half, which is the mean number of contacts in the case of Hamiltonian walks on the same lattice. This limiting value is expected, since in the limit of infinite interaction (or zero temperature) each self-avoiding walk on 3-simplex lattice becomes maximally compact and occupies all lattice points, i.e. becomes Hamiltonian walk.

Hopf bifurcation in a reaction-diffusive-advection two-species competition model with one delay

In this paper, we investigate a reaction-diffusive-advection two-species competition model with one delay and Dirichlet boundary conditions. The existence and multiplicity of spatially non-homogeneous steady-state solutions are obtained. The stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay are obtained by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic orbits are derived. Finally, numerical simulations are given to illustrate the theoretical results.

Spatiotemporal Regimes in the Kuramoto–Battogtokh System of Nonidentical Oscillators

We consider the spatiotemporal states of an ensemble of nonlocally coupled nonidentical phase oscillators, which correspond to different regimes of the long-term evolution of such a system. We have obtained homogeneous, twisted, and nonhomogeneous stationary solutions to the Ott–Antonsen equations corresponding to key variants of the realized collective rotational motion of elements of the medium in question with nonzero mesoscopic characteristics determining the degree of coherence of the dynamics of neighboring particles. We have described the procedures of the search for the class of nonhomogeneous solutions as stationary points of the auxiliary point map and of determining the stability based on analysis of the eigenvalue spectrum of the composite operator. Static and breather cluster regimes have been demonstrated and described, as well as the regimes with an irregular behavior of averaged complex fields including, in particular, the local order parameter.

Analysis on a diffusive multiple Allee effects predator–prey model induced by fear factors

This paper is concerned with a diffusive predator–prey system with multiple Allee effects induced by fear factors subject to Neumann boundary conditions. Firstly, we introduce a priori estimates of non-negative and non-trivial solutions. Moreover, nonexistence of non-constant positive solutions are shown for certain parameters ranges. Secondly, we study the stability of non-negative constant solutions by the linearized theory. Then, by choosing the Allee threshold θ as the bifurcation parameter, we investigate the spatially homogeneous and non-homogeneous periodic solutions as well as non-constant steady-state solutions. Finally, some theoretical results are illustrated by numerical simulations.

Hopf bifurcation in a reaction-diffusion-advection equation with nonlocal delay effect

This paper investigates the dynamics of a general reaction-diffusion-advection equation with nonlocal delay effect and Dirichlet boundary condition. The existence and stability of positive spatially nonhomogeneous steady state solution are shown. By analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized equation, the existence of Hopf bifurcation is proved. We introduce the weighted space to overcome the hurdle from advection term. We also show that the effect of adding a term advection along environmental gradients to Hopf bifurcation values for a Logistic equation with nonlocal delay.

Existence of solutions for a nonhomogeneous Kirchhoff-Schrödinger type equation in $\mathbb{R}^{2}$ involving unbounded or decaying potentials

In this paper, we consider the following nonhomogeneous Kirchhoff-Schrodinger equation: $$ m\bigg(\int_{\mathbb{R}^{2}}|\nabla u|^2{d}x +\int_{\mathbb{R}^{2}}V(|x|)u^2{d}x \bigg) [-\Delta u + V(|x|)u] = Q(|x|)f(u) + \varepsilon h(x), $$% for $ x\in\mathbb{R}^2$, where $m$, $ V$, $ Q$ and $f$ are continuous functions, $\varepsilon$ is a small parameter and $h\neq 0$. When $f$ has exponential growth by means of a Trudinger-Moser type inequality, the Mountain Pass Theorem and Ekeland's Variational Principle in weighted Sobolev spaces are applied in order to establish the existence of at least two weak solutions for this equation.

The Rr Form of the Kedem–Katchalsky–Peusner Model Equations for Description of the Membrane Transport in Concentration Polarization Conditions

The paper presents the matrix form of Kedem–Katchalsky–Peusner equations for membrane transport of the non-homogeneous ternary non-electrolyte solutions. Peusner’s coefficients and det [] (i, j ∈ {1, 2, 3}, r = A, B) occurring in these equations, were calculated for Nephrophan biomembrane, glucose in aqueous ethanol solutions and two different settings of the solutions relative to the horizontally oriented membrane for concentration polarization conditions or homogeneity of solutions. Kedem–Katchalsky coefficients, measured for homogeneous and non-homogeneous solutions, were used for the calculations. The calculated Peusner’s coefficients for homogeneous solutions depend linearly, and for non-homogeneous solutions non-linearly on the concentrations of solutes. The concentration dependences of the coefficients and det [] indicate a characteristic glucose concentration of 9.24 mol/m3 (at a fixed ethanol concentration) in which the obtained curves for Configurations A and B intersect. At this point, the density of solutions in the upper and lower membrane chamber are the same. Peusner’s coefficients were used to assess the effect of concentration polarization and free convection on membrane transport (the ξij coefficient), determine the degree of coupling (the coefficient) and coupling parameter (the coefficient) and energy conversion efficiency (the coefficient).

Global stability of nonhomogeneous equilibrium solution for the diffusive Lotka–Volterra competition model

A diffusive Lotka-Volterra competition model is considered for the combined effect of spatial dispersal and spatial variations of resource on the population persistence and exclusion. First it is shown that in a two-species system in which the diffusion coefficients, resource functions and competition rates are all spatially heterogeneous, the positive equilibrium solution is globally asymptotically stable when it exists. Secondly the existence and global asymptotic stability of the positive and semi-trivial equilibrium solutions are obtained for the model with arbitrary number of species under the assumption of spatially heterogeneous resource distribution. A new Lyapunov functional method is developed to prove the global stability of a non-constant equilibrium solution in heterogeneous environment.

Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system

<p style='text-indent:20px;'>This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive Lotka-Volterra type competition system with advection terms under the homogeneous Dirichlet boundary condition. First, we obtain the existence, multiplicity and explicit structure of the spatially nonhomogeneous steady-state solutions by using implicit function theorem and Lyapunov-Schmidt reduction method. Secondly, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of spatially nonhomogeneous positive steady-state solutions and the non-existence of Hopf bifurcations at spatially nonhomogeneous positive steady-state solutions are given. Finally, two concrete examples are provided to support our previous theoretical results. It should be noticed that an elliptic operator with advection term is not self-adjoint, which causes some trouble in the spatial decomposition, explicit expressions of steady-state solutions and some deductive processes related to infinitesimal generators. Moreover, unlike other work, the advection rate here depends on the spatial position, which increases some difficulties in the investigation of the principal eigenvalue.

Global stability of nonhomogeneous steady-state solution in a Lotka–Volterra competition–diffusion–advection model

A classical two species Lotka–Volterra competition–diffusion–advection model is considered in this paper, where the diffusion coefficients, advection coefficients, resource functions and competition rates are all spatially heterogeneous. The global stability of nonhomogeneous steady states in the advective heterogeneous environment is studied by introducing a weighted Lyapunov functional associated with advection term.

Incorporating physical constraints in braneworld black-string solutions for a Minkowski brane in scalar-tensor gravity

In the framework of a general scalar-tensor theory, where the scalar field is non-minimally coupled to the five-dimensional Ricci scalar curvature, we investigate the emergence of complete brane-world solutions. By assuming a variety of forms for the coupling function, we solve the field equations in the bulk, and determine in an analytic way the form of the gravitational background and scalar field in each case. The solutions are always characterized by a regular scalar field, a finite energy-momentum tensor, and an exponentially decaying warp factor even in the absence of a negative bulk cosmological constant. The space-time on the brane is described by the Schwarzschild solution leading to either a non-homogeneous black-string solution in the bulk, when the mass parameter $M$ is non-zero, or a regular anti-de Sitter space-time, when $M=0$. We construct physically-acceptable solutions by demanding in addition a positive effective gravitational constant on our brane, a positive total energy-density for our brane and the validity of the weak energy condition in the bulk. We find that, although the theory does not allow for all three conditions to be simultaneously satisfied, a plethora of solutions emerge which satisfy the first two, and most fundamental, conditions.

Patterns in a Modified Leslie–Gower Model with Beddington–DeAngelis Functional Response and Nonlocal Prey Competition

In this paper, we present the theoretical results on the pattern formation of a modified Leslie–Gower diffusive predator–prey system with Beddington–DeAngelis functional response and nonlocal prey competition under Neumann boundary conditions. First, we investigate the local stability of homogeneous steady-state solutions and describe the effect of the nonlocal term on the stability of the positive homogeneous steady-state solution. Lyapunov–Schmidt method is applied to the study of steady-state bifurcation and Hopf bifurcation at the interior of constant steady state. In particular, we investigate the existence, stability and multiplicity of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous periodic solutions. Furthermore, we present a simple description of the dynamical behaviors of the system around the interaction of steady-state bifurcation curve and Hopf bifurcation curve. Finally, a numerical simulation is provided to show that the nonlocal competition term can destabilize the constant positive steady-state solution and lead to the occurrence of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous time-periodic solutions.

Bifurcation on diffusive Holling–Tanner predator–prey model with stoichiometric density dependence

This paper studies a diffusive Holling–Tanner predator–prey system with stoichiometric density dependence. The local stability of positive equilibrium, the existence of Hopf bifurcation and stability of bifurcating periodic solutions have been obtained in the absence of diffusion. We also study the spatially homogeneous and nonhomogeneous periodic solutions through all parameters of the system, which are spatially homogeneous. In order to verify our theoretical results, some numerical simulations are carried out.

Controllability of linear and nonlinear systems governed by Stieltjes differential equations

In this manuscript, controllability of vector valued g-derivative Stieltjes differential system is considered. For linear system, explicit solution of homogeneous case is obtained by constructing a generalized matrix exponential function and nonhomogeneous solution is achieved with transferring the corresponding problem into a class of Stieltjes impulsive differential equations. Based on the analysis techniques of linear system theory, the Gramian criterion and the rank criterion for linear case are established, respectively. Existence of local and global solution for nonlinear system are presented by the well-knonw Banach contraction principle and the controllability result for nonlinear systems is shown by Krasnoselskii’s fixed point theorem. Simulations for linear and nonlinear systems are tackled respectively to verify the efficiency of theoretical results.

Fast multiphase analysis: Self-separation of mixed solution by a wettability-controlled CuO@Ag SERS substrate and its applications in pollutant detection

Surface-enhanced Raman spectroscopy (SERS) is widely used in chemical analysis, with very promising application prospects in daily life. However, there are still lots of problems in actual operation, for instance, pollutant detection from sewage, in which the analyte is often dispersed in nonhomogeneous solutions, leading to inaccurate Raman signal collection. Adjusting surface wettability to realize selective self-separation of multiphase liquids in situ on SERS substrate is an effective and challenging approach. In this work, a complex of CuO@Ag nanowires on Cu mesh modified by different wettability, converting from hydrophilic to hydrophobic by simple infrared illumination, is prepared as a multifunctional SERS substrate. Benefitting from the controllable wettability of the substrate, the multiphase solution can be self-separated and adsorbed at the two opposite terminals of it, and the 10−9 M of malachite green and 10−5 M of thiram used as two kinds of probe molecules can be separately detected at these two terminals. In addition, after fully hydrophobic modification, the substrate can be also used to extract the organic phase in the ‘water/oil’ mixture and perform in situ detection of the probe molecules in the ‘oil’.

Stress state of a rectangular domain with the mixed boundary conditions

The problem stated at the paper is not new, and some analytical methods and a lot of numerical methods are used to solve it. The approach proposed in the paper Popov G., Vaysfeld N. (2011) is based on the integral transform method and reduce the stated problem to a singular integral equation. This paper is continuation of paper Pozhylenkov O. V. (2019), where a rectangular domain with conditions of ideal contact at the lateral sides was considered. The novelty of the presented paper consists of a new statement of the problem when the conditions of the second main elasticity problem are given at the lateral sides. With the help of the Fourier transform, the one-dimensional vector boundary problem in the transform’s domain is derived. The solution of the homogeneous problem is found using matrix differential calculations, the fundamental solution matrix is constructed in the form of the contour integral, which is found using the residue theorem. As a result, we get the non-homogeneous one-dimensional problem, so the final solution is found with help of Green’s matrix Popov G., Abdimanapov S., Efimov V. (1999), which is constructed as a combination of the fundamental basis solution matrix. The solution is constructed as a superposition of the homogeneous and non-homogeneous solutions and contains the unknown derivatives of the displacement. The aim is to find the unknown function, which must satisfy the boundary condition on the upper edge of the domain. It leads to the singular integral equation which is solved with the help of the orthogonal method. The stress state of a domain was investigated depending on load properties and domain size.

Cross-Contaminations in Powder Bed Fusion: Influence of Copper Alloy Particles in Nickel-Base Alloy Feedstock on Part Quality

When two or more different metal powders are processed on a single additive manufacturing (AM) machine, cross-contaminations can occur. This is particularly relevant to the consecutive processing of different materials on a powder bed fusion (PBF) machine through material changes as well as simultaneous processing of different materials via multi-material PBF. However, uncertainty about tolerable foreign particle percentages in metal powder feedstock limits the applicability of material changes and multi-material PBF. Two alloys which are of particular relevance to the aerospace industry are nickel-base alloy 2.4668 and copper alloy CW106C. In multi-material applications, 2.4668 mainly serves as a structural, load-bearing material. Therefore, this study investigates the influence of defined quantities of copper alloy particles in nickel-base alloy feedstock on metallurgical structure and static tensile strength. Foreign particle inclusions were dissolved in the matrix material and formed a solid solution. No material deteriorations were observed for contamination levels up to 20 particle percent (part.%). Etching revealed a nonhomogeneous solid solution with Cu-rich areas. Contamination levels up to two particle part.% CW106C in 2.4668 showed no influence on ultimate tensile strength and a limited influence on fracture elongation. At five part.% contamination, both properties deteriorated and inferior material qualities were observed. Fractography showed a similar fracture behavior for all of the contamination levels examined. Implications for the aerospace industry by the material combination examined are made on the basis of the results presented.