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With the innovation and integration of the Internet and the financial industry, the third-party payment market has developed greatly and has great potential. This paper discusses the duopoly game between third-party payment service providers and banks, which are the main participants in the mobile payment market. By constructing Nash game model, the conditions of equilibrium point, stability and bifurcation are analyzed. The effects of adjusting parameters and cooperation coefficient on business volume and profit are discussed. The conclusions are as follows: excessive investment will lead to unpredictable fluctuations in the market and fall into chaos; By strengthening cooperation, all participants in the mobile payment industry chain can improve business volume and profits while curbing chaos in the mobile payment market.

In this article, our aim is to establish a generalized version of Lions-type theorem for the p-Laplacian. As an application of this theorem, we consider the existence of ground state solution for the quasilinear elliptic equation with the critical growth.

To investigate the joint impact of infection age, spatial diffusion and SARS-CoV-2 infection processes on the threshold dynamic of covid-19 transmission among population, we formulate a space-age structured covid-19 model at individual and population levels. The next generation operator ℛ and the basic reproduction number R 0 are presented. The disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the age-dependent endemic steady state is globally asymptotically stable when R 0 > 1. Finally, we perform numerical simulations based on the confirmed cases from February 21 to March 30, 2022 in Hong Kong, which suggest that (1) the spacial diffusion and infection age play the crucial role in the transmission of covid-19, so they cannot be ignored; (2) it is particularly important to quarantine and treat infected individuals with symptoms, so as to make better use of medical resources and effectively control the deterioration of covid-19. The control of asymptomatic infections cannot be relaxed which can effectively reduce the potential risk of the disease outbreaks; (3) timely and effective treatment can significantly reduce the proportion of severely infected individuals in confirmed cases, which is also one of primary goals to be achieved by taking control measures. • We formulate a space-age structured COVID-19 model. • We study the global stability of the solutions of the system. • The numerical simulation is constructed based on the reported cases in Hong Kong. • We put forward some suggestions for COVID-19 prevention and control in Hong Kong. [ABSTRACT FROM AUTHOR]

The shallow-water wave equations with different forms of delays are presented in this work, such as no delay, local delay and nonlocal delay, which are described in the form of convolutions with different kernels. These shallow-wave equations satisfy the asymptotic integrability condition and include the Korteweg–de Vries equation, Camassa–Holm equation and Degasperis–Procesi equation as particular cases. The existence and non-existence of solitary wave are established by the invariant manifold theory and geometric singular perturbation theory. It is found that different delays have various effects on the existence of solitary waves. In particular, the Melnikov functions with divergence free or not are derived for different delays to measure the separation of stable and unstable manifolds, so that the existence of solitary waves could be justified.

In this paper, we are concerned with quasilinear equations with indefinite nonlinearity and explore the existence of infinitely many solutions.

In this article, we are concerned with multiple solutions of Schrödinger-Choquard-Kirchhoff equations involving the fractional pp-Laplacian and Hardy-Littlewood-Sobolev critical exponents in RN{{\mathbb{R}}}^{N}. We classify the multiplicity of the solutions in accordance with the Kirchhoff term M(⋅)M\left(\cdot )and different ranges of qqshown in the nonlinearity f(x,⋅)f\left(x,\cdot )by means of the variational methods and Krasnoselskii’s genus theory. As an immediate consequence, some recent related results have been improved and extended.

In this paper, an advertising competition model with sales promotion is constructed and investigated. Conditions of the existence and stability of period- T solutions are obtained by means of the discrete map. Flip bifurcation is analyzed by using the center manifold theory and three sales promotion strategies are discussed. Example and numerical simulations are illustrated which agree well with our theoretical analysis. [ABSTRACT FROM AUTHOR]

The Roe scheme is known to have high resolution but to be afflicted by the notorious shock anomalies (such as the carbuncle phenomena) and the unphysical expansion shocks. This paper aims to develop a new Roe-type scheme which is robust for strong shock waves and low dissipative for contact waves and shear layers. Based on the mechanism analysis of shock instability, a dissipation-controlling strategy is adopted to adjust the entropy wave viscosity and shear wave viscosity of the transverse numerical flux in the numerical shock layer and the robustness of the Roe scheme is thus enhanced. The violation of entropy condition and the problem of expansion shocks are also fixed to broaden the scheme's application scope. In addition, the THINC reconstruction procedure and a BVD algorithm are implemented to further improve the resolution for contact waves and shear layers. Numerical results of a series of classical one-, two- and three-dimensional test problems fully demonstrate that the proposed new Roe-type scheme has not only better robustness for strong shock waves but also higher resolution for contact waves and shear layers than the original Roe scheme. • Shock instability of the Roe solver is analyzed by employing two analysis tools. • A simple strategy is adopted to remedy the shock instability of the Roe solver. • Resolution for contact waves and shear layers is improved via the BVD algorithm. [ABSTRACT FROM AUTHOR]

Abstract: Topological structure of the living space is one of the most important parameters for cellular automata (CA), which can be applied to various scientific areas. In order to study CA in larger extent, we define a new type of cellular automata on countable groups and investigate some general dynamical properties of attractors and periodicity. It is shown that if such cellular automaton on a countable group is transitive, it is either sensitive or composed of a single periodic orbit. By using of the form of attractors, classification of this type of cellular automata is given on countable groups. The set of eventually periodic points of a cellular automaton on a countable residually finite group is proved to be dense.

In this paper, complex dynamics of a two-compartment model of production and regulation of the circulating blood neutrophil number are investigated. It is shown that the proliferative disorders may be possible due to factors of the apoptosis rate rsof the haematopoietic stem cell and the cell cycle duration τs. Applying a recent geometrical criterion for the Hopf bifurcation and transient behaviors of delay systems to this model, we separate the stable regime from the unstable regime on the rs- τsplane. Numerically, regimes of patterned periodic oscillations with low periodicity in the number of circulating blood cells appear on the rs- τsplane. It is found that the dominated period-adding bifurcation mechanism leads transitions from period-nto period-(n + 1), eventually changes to the complex attractor with high-periodicity or chaos.

Abstract: There is the widespread existence of wave phenomena in physics, chemistry and biology. This clearly necessitates a study of traveling waves in depth and of the modeling and analysis involved. In the present paper, we study a nonlinear reaction-diffusion equation, which can be regarded as a generalized Fisher equation. Applying the Cole–Hopf transformation and the first integral method, we obtain a class of traveling solitary wave solutions for this generalized Fisher equation. [Copyright &y& Elsevier]

Abstract: In this paper, we study a Korteweg-de Vries-Burgers equation with a higher-order nonlinearity. An asymptotic analysis of proper traveling wave solutions is presented by means of the qualitative theory of differential equations. An approximate solution with arbitrary velocity is obtained by using the Adomian decomposition method, which agrees well with the phase plane analysis. A class of solitary wave solutions is also derived which covers existing traveling wave solutions in the literature.

Abstract: In this paper, we give a qualitative study to the damped Duffing’s equation by means of the qualitative theory of planar systems. Under certain parametric conditions, the monotonous property of the bounded non-oscillations is obtained. Explicit exact solutions are obtained by a direct method and application of this approach to a reaction–diffusion equation is presented. [Copyright &y& Elsevier]

Abstract: Many nonlinear evolution equations can be converted into first- or high-order ordinary differential equations. In this paper, we illustrate a connection between the elliptic integrals and some nonlinear evolution equations such as the two-dimensional Boussinesq equation, and demonstrate that traveling wave solutions and periodic wave solutions to these nonlinear equations can be obtained by means of the standard forms of elliptic integrals directly. [Copyright &y& Elsevier]

In this paper, a kind of explicit exact solutions to the Lie´nard equation is obtained, and the applications of the result in seeking traveling solitary wave solution of the nonlinear Schro¨dinger equation are presented. [Copyright &y& Elsevier]

Many physical phenomena in one- or higher-dimensional space can be described by nonlinear evolution equations, which can be reduced to ordinary differential equations such as the Lienard equation. Thus, to study those ordinary differential equations is of significance not only in mathematics itself, but also in physics. In this paper, a kind of explicit exact solutions to the Lienard equation is obtained. The applications of the solutions to the nonlinear RR-equation and the compound KdV-type equation are presented, which extend the results obtained in the previous literature. [Copyright &y& Elsevier]

The Burgers–Korteweg–de Vries equation arises from many different physical contexts as a nonlinear model equation incorporating the effects of dispersion, dissipation and nonlinearity. In the present paper, we present a more general traveling wave solution to the Burgers–Korteweg–de Vries equation by applying an exact solution of an ordinary differential equation. [Copyright &y& Elsevier]

In this paper, we are concerned with the generalized Boussinesq equation including the singularly sixth-order Boussinesq equation, which describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number less than but very close to 1/3. By the means of two proper ansatzs, we obtain explicit traveling solitary wave solutions to the generalized Boussinesq equation. [Copyright &y& Elsevier]

• Establish the existence of traveling wave solutions for degenerate reaction-diffusion equations with the nonlocal effect. • Demonstrate the existence of monotone traveling wave solutions by using the monotone iteration and operator theory, and investigate how wave profile (e.g. being non-monotone or periodic) is affected by the enhancement of nonlocality, parameter values and variation of initial values for two different forms of convolution kernel functions. This paper deals with traveling wave phenomena of a degenerate reaction-diffusion equation with the nonlocal effect. We study the existence of traveling wave solutions which may be non-monotonic based on the two-point boundary value problem and Schauder's fixed point theorem. We are excited to find that the unknown positive steady state is exactly a unique positive equilibrium for the large wave speed and the monotonicity of traveling waves depends on the wave speed. On the other hand, we demonstrate the existence of monotone traveling wave solutions by using the monotone iteration and operator theory, and investigate how wave profile (e.g. being non-monotone or periodic) is affected by the enhancement of nonlocality, parameter values and variation of initial values for two different forms of convolution kernel functions. [ABSTRACT FROM AUTHOR]

• We formulate a diffusive waterborne pathogen model with the distinct diffusion rates and spatial heterogeneity. • We define the basic reproduction number and show its threshold role. • The asymptotic profiles of the endemic steady state are discussed when the diffusion rates of the susceptible populations approach zero. To explore the influence of environmental pollution on the spread of waterborne diseases, we formulate a diffusive waterborne pathogen model with the distinct diffusion rates and spatial heterogeneity. We define the basic reproduction number R 0 and show its threshold role: if R 0 < 1 , the disease-free steady state is globally asymptotically stable; if R 0 > 1 , the model system is uniformly persistent. The asymptotic profiles of the endemic steady state are discussed when the diffusion rates of the susceptible populations approach zero. [ABSTRACT FROM AUTHOR]

In this paper, we study an epidemic model in a periodic patchy environment. It is assumed that all the parameters share the same period N ∈ N. When the basic reproduction number R 0 > 1 , we first establish the spreading speed of traveling wave, which is denoted by c *. Then we show that for each c ≥ c *, the system admits a nontrivial traveling wave solution with the speed c if R 0 > 1. We also show that there is no nontrivial traveling wave solution when R 0 ≤ 1 and c > 0, or R 0 > 1 and c ∈ (0, c *). This indicates that the spreading speed coincides with the minimum wave speed of traveling waves. Finally, we discuss the dependence of R 0 and c * on the heterogeneity and amplitude of the parameters of our model. [ABSTRACT FROM AUTHOR]

Highlights • The Kirchhoff equations, in fluid dynamics, are usually used to describe the motion of a rigid body in an ideal fluid. If the body is completely submerged body in an infinitely large volume of irrotational, incompressible, inviscid fluid, the dynamics of the body can be described by various Kirchhoff type equations [17,23]. • From the mathematical point of view, it is very interesting to consider the superlinear Kirchhoff type equations with steep potential well and their positive solutions, which enables us to better understand their fluid dynamics originating from several scientific fields such as eletrorheological fluids and elastic mechanics [23]. • Nowadays, the superlinear case is mainly studied in dimension three while not being considered in higher dimensions (N >=4) [27]. In addition, in other references related to the superlinear case, only the nontrivial solution was discussed for N>=4 [2,3,5]. Motivated by this fact, the main purpose of this paper is to study the multiplicity of nontrivial solutions of nonlinear Kirchhoff type equations with N>=4 and f being superlinear. Abstract In this paper, we study a class of superlinear Kirchhoff type equations with steep potential well: { − (a ∫ R N | ∇ u | 2 d x + b) Δ u + λ V (x) u = f (x , u) in R N , u ∈ H 1 (R N) , where N ≥ 4, a, b > 0, and the parameter λ > 0. By proposing new techniques and introducing new superlinear hypotheses on f , we establish the existence of two positive solutions, one of which will blow up as the nonlocal term vanishes. Some existing results are improved and generalized in the literature. Moreover, the nonexistence of nontrivial solution is also obtained. [ABSTRACT FROM AUTHOR]

Zika virus is an arbovirus transmitted by mosquitoes, which can cause low fever, mild skin rash and eye swelling. In severe cases, it may cause neurological and autoimmune complications, such as microcephaly. Climate factors and weather, especially temperature, have a strong effect on the transmission dynamics of vector-borne diseases, such as Dengue, West Nile virus, Schistosomiasis, and Zika. To study the comprehensive effects of spatial structure, seasonality and the temperature sensitivity of incubation period on the transmission of Zika virus, we propose a PDE model with periodic delay. We present the basic reproduction number R 0 and explore the global dynamics of the model. Our simulations show that (i) it is possible to overestimate R 0 in the periodic system if the spatial averaged system is used; (ii) R 0 can be underestimated in the absence of seasonality; and (iii) shortening the incubation period may reduce the risk of disease transmission, which is critical for modeling the transmission of Zika virus dynamics to understand geographic and seasonal limits of Zika risk. • To study the transmission of Zika virus, we propose a PDE model with periodic delay. • We present the basic reproduction number R 0. • We explore the global dynamics of the model. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an investigation of dynamics of the coupled nonidentical FHN models with synaptic connection, which can exhibit rich bifurcation behavior with variation of the coupling strength. With the time delay being introduced, the coupled neurons may display a transition from the original chaotic motions to periodic ones, which is accompanied by complex bifurcation scenario. At the same time, synchronization of the coupled neurons is studied in terms of their mean frequencies. We also find that the small time delay can induce new period windows with the coupling strength increasing. Moreover, it is found that synchronization of the coupled neurons can be achieved in some parameter ranges and related to their bifurcation transition. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behavior are clarified. [Copyright &y& Elsevier]