In this paper, the dynamics of scavenger species predation of both susceptible and infected prey at different rates with prey refuge is mathematically proposed and studied. It is supposed that the disease was spread by direct contact between susceptible prey with infected prey described by Holling type-II infection function. The existence, uniqueness, and boundedness of the solution are investigated. The stability constraints of all equilibrium points are determined. In addition to establishing some sufficient conditions for global stability of them by using suitable Lyapunov functions. Finally, these theoretical results are shown and verified with numerical simulations.

Throughout this paper we study the properties of the composition operator
C
p1  o
p2  o…o
pn  induced by the composition of finite numbers of special
automorphisms of U,
pi  (z)  i
i
p z
1 p z


Such that pi  U, i  1, 2, …, n, and discuss the relation between the product of
finite numbers of automorphic composition operators on Hardy space H2 and some
classes of operators.

The semiconductor ZnO is one of II – VI compound group, it is prepare as thin films by using chemical spray pyrolysis technique; the films are deposited onto glass substrate at 450 °C by using aqueous zinc chloride as a spray solution of molar concentration 0.1 M/L. Sample of the prepared film is irradiating by Gamma ray using CS 137, other sample is annealed at 550°C. The structure of the irradiated and annealed films are analyzed with X-ray diffraction, the results show that the films are polycrystalline in nature with preferred (002) orientation. The general morphology of ZnO films are imaged by using the Atomic Force Microscope (AFM), it constructed from nanostructure with dimensions in order of 77 nm.
The optical properties o

This study presents a mathematical model describing the interaction of gut bacteria in the participation of probiotics and antibiotics, assuming that some good bacteria become harmful through mutations due to antibiotic exposure. The qualitative analysis exposes twelve equilibrium points, such as a good-bacteria equilibrium, a bad-bacteria equilibrium, and a coexisting endemic equilibrium in which both bacteria exist while being exposed to antibiotics. The theory of the Sotomayor theorem is applied to study the local bifurcation around all possible equilibrium points. It’s noticed that the transcritical and saddle-node bifurcation could occur near some of the system’s equilibrium points, while pitchfork bifurcation cannot be accrued at

Objectives: The purpose of this in vitro study was to compare the effect of adding poloxamer surfactant to irrigant solutions on the penetration de..

Phoenix dactylifera l. pinnae (the green leaves of dates palm) were used as natural reinforcing (strengthening) fibers to improve the mechanical properties of polyester as a matrix material, the fibers of the green leaves of dates palm were used in two lengths, 10 and 20mm with five rates of 0, 2.5, 5, 10, and 20% , where the reinforcing with the leaves fibers increases the hardness strength from 76.5 to be about 86.55 , the Impact value raised from about 0.313 to 0.461 , in addition to that the flexural strength from 2.66 to be about 55 , and the thermal conductivity increases from 2.54 𝑤∕𝑚.℃ to 5.41 𝑤∕𝑚.℃. The results of the present search explains that the composite samples reinforced at rate 20% and 10mm fiber length

The reinforced fiberglass in cement slurry reflects the effect on its properties compared to usual additives. Fiberglass is typically used in cement slurry design for one or another of the following goals: (Earth earthquake, bearing storage, and with differential stresses, to enhance cement durability and increase its compressive strength). The main goal is to use glass fiber and ground fiberglass to improve the tensile strength and moderate compressive strength significantly. On the other hand, the use of glass fibers led to a slight increase in the value of thickening time, which is a desirable effect. Eleven glass fiber samples and milled glass fiber were used to show these materials' effect on Iraqi cement with (0.125, 0.25, 0.5

Generally the a.c. conductivity shows a power law in frequency s  ()  where the exponent s ≤ 1. As the frequency goes to zero the conductivity become frequency independent. The a.c. conductivity was studied for the Ge1-xSex thin films to see how the selenium contents affect the permittivity and the permeability for the Ge1-x Sex. The thin films prepared by thermal evaporation at room temperature and under vacuum (~2 x10-5toor) using Edward coating unit model 306A. From the relation between ln conductivity and ln w, the effect of selenium contents in Ge1-x Sex thin films on the exponent value, the relaxation time and the maximum barrier height. An algebric fitting method for circles and circular arcs was used to find the permit

The structural, optical and electrical properties of ZnS films prepared by vacuum evaporation technique on glass substrate at room temperature and treated at different annealing temperatures (323, 373, 423)K of thickness (0.5)µm have been studied. The structure of these films is determined by X-ray diffraction (XRD). The X-ray diffraction studies show that the structure is polycrystalline with cubic structure, and there are strong peaks at the direction (111). The optical properties investigated which include the absorbance and transmittance spectra, energy band gab, absorption coefficient, and other optical constants. The results showed that films have direct optical transition. The optical band gab was found to be in the range t