In this paper, a spatio-temporal model as systems of ODE which describe two-species Beddington-DeAngelis type predator-prey system living in a habitat of two identical patches linked by migration is investigated. It is assumed in the model that the per capita migration rate of each species is influenced not only by its own but also by the other one's density, i.e., there is cross diffusion present. We show that a standard (self-diffusion) system may be either stable or unstable, a cross-diffusion response can stabilize an unstable standard system and destabilize a stable standard system. For the diffusively stable model, numerical studies show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation and the cross migration response is an important factor that should not be ignored when pattern emerges.

Inspired by the work of Górnicki in his recent article [J. Górnicki, Fixed Point Theory Appl., \({\bf 2017}\) (2017), 10 pages], where he introduced a new class of self mappings called \(F\)-expanding mappings, in this paper we introduce the concept of \(F_{m}\)-contractive and \(F_{m}\)-expanding mappings in \(M\)-metric spaces. Also, we prove the existence and uniqueness of fixed point for such mappings.

In this study, we present a new approximation method to give an explicit solution of a laminar flow using a Sisko model. This is a problem of a generalized Newtonian fluid with slip boundary conditions. The proposed method is based on the variational iteration method (VIM) combined with an approximation step. This method is validated where the exact solution is available. In addition, in order to enrich the discussion, a numerical method is presented. The results illustrate that the VIM may be more effective that the finite difference method for a dilatant fluid. However, the VIM will be inappropriate for pseudoplastic fluid cases.

This paper deals with the concepts of upper and lower \((\tau_1,\tau_2)\)-precontinuous multifunctions. Some characterizations of upper and lower \((\tau_1,\tau_2)\)-precontinuous multifunctions are investigated. The relationships between upper and lower \((\tau_1,\tau_2)\)-precontinuous multifunctions and the other types of continuity are discussed.

The main goal of this paper is to develop a mathematical model to study the dynamic of malaria transmission, and the direct effects of congenital malaria on the spread of malaria. In this study, we have clarified the significant impact of malaria on the human community through their impact on the newborn, and that directly increases spread of the malaria in the human community, especially in the newborns with the lower and inexperienced immunity systems. The existence and stability of the disease-free points of the system is analyzed. We established that the disease-free equilibrium point is locally asymptotically stable when the reproduction number \(R_{0}<1\) and the disease always dies out. For \(R_{0}>1\) the disease-free equilibrium becomes unstable and there exists a unique endemic equilibrium.

Multi-layered QR (MLQR) codes are created by superimposing many black and white QR codes, all of which are assigned their white areas with the colors in RGB color space. These colors must be different enough to enable distinguishing each layer of a MLQR code. This makes a MLQR code be able to hold more data than a common QR code. In this work, the procedures for generating and un-layering MLQR codes were proposed and the according graphical user interfaces (GUIs) were created with MATLAB. They use the property of a partition of the number 255, constructed by the geometric sequence \(\left\{2^{n-1}\right\}_{n\in\mathbb{N}}\), to compute the collection of suitable colors for assigning to black and white QR codes in the generating process. Our developed procedures can promote better intercommunication between human and computers, therefore ensure easier computer programming and being more flexible in the number of layers of created MLQR codes. We found that the developed GUIs could work accurately up to 15 layers because more QR code layers require the use of more colors, which diminish an ability to clearly distinguish the color of each QR code layer.

In this paper, we first apply properties of the wedge product and continuous finite element methods to prove that the linear, quadratic element methods are symplectic algorithms to the linear Hamiltonian systems, i.e., the symplectic condition \(dp_{j+1}\wedge dq_{j+1}=dp_{j}\wedge dq_{j}\) is preserved exactly and the linear element method is an approximately symplectic integrator to nonlinear Hamiltonian systems, i.e., \(dp_{j+1}\wedge dq_{j+1}=dp_{j}\wedge dq_{j}+O(h^2)\), as well as energy conservative.

A reaction diffusion equation with a Caputo fractional derivative in time and with time-varying delays is considered. Stability properties of the solutions are studied via the direct Lyapunov method and arbitrary Lyapunov functions (usually quadratic Lyapunov functions are used). In this paper we give a brief overview of the most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. These derivatives are applied to various types of reaction-diffusion fractional neural network with variable coefficients and time-varying delays. We show the quadratic Lyapunov functions and their Caputo fractional derivatives are not applicable in some cases when one studies stability properties. Some sufficient conditions for stability are obtained and we illustrate our theory on a particular nonlinear Caputo reaction-diffusion fractional neural network with time dependent delays.

The aim of this paper is to prove common fixed point theorems for compatible mappings of type (A) for three self mappings satisfying certain contractive conditions and its topological properties in partial metric spaces.