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

The purpose of this paper is to apply different transportation models in their minimum and maximum values by finding starting basic feasible solution and finding the optimal solution. The requirements of transportation models were presented with one of their applications in the case of minimizing the objective function, which was conducted by the researcher as real data, which took place one month in 2015, in one of the poultry farms for the production of eggs

The research aims to study some of the human characteristics of the state of Singapore to know the impact of these characteristics on the strength of the state, its development and. The research included two aspects, theoretical and analytical, using the descriptive analytical method, force analysis method, as well as the historical method. The data was analyzed according to mathematical equations, including the size of the country's population, the extraction of the population growth rate and the concept of age structure, where some indicators related to this concept have been explained. The researcher reached a set of results, the most important of which were: that the population size of the state of Singapore in the period between (19

Eleven hymenopterous species: two braconids, seven eulophids and two pteromalids
parasitized the larvae of Phytomyza horticola Goureau. The activity of the parasites began at
the end of April and evidently increased during May. The female of the parasites fed and
parasitized mostly on the second and third larval instars of P. horticola . Diglyphus iseae
Walker and Cirrospilus vittatus Walker were dominant larval parasites. Chrysocharis
pentheus Walker and Pediobius acantha Walker were main pupal parasites.

This research aimed to examine the correlation and influence of leader-member exchange in the organizational citizenship behavior in the existence of organizational trust as mediator variable among workers in ministry of migration and displaced, has been collecting data and information relating to research by designing questionnaire was distributed to the stratified random sample included (56) member of the workers in the ministry mentioned, and through the use of several statistical method were the main findings of research was the existence of a correlation relationship and influence between leader- member exchange and organizational trust, while the results confirmed no correlation and influen

The research aims at clarifying the relationship between innovative marketing skills and broad recommendation as a vital and important issue for organizations in general and service organizations in particular to demonstrate how innovative marketing skills contribute to broad adoption and to determine the relationship between interdependence and the impact of innovative marketing skills on the broad recommendation. Some questions are posed by the research problem. Is there a clear awareness among individuals in the company about the concept of marketing marketing skills and how do innovative marketing skills affect the broad recommendation of the surveyed company? How innovative marketing skills relate to the broad recommendation