In this paper, we present a Lotka-Volterra predator-prey model with Allee effect. This system with general functional response has an Allee effect on prey population. A nonstandard finite difference scheme is constructed to transform the continuous time predator-prey model with Allee effect into the discrete time model. We use the Schur-Cohn criteria which deal with coefficients of the characteristic polynomial for determining the stability of discrete time system. The proposed numerical schemes preserve the positivity of the solutions with positive initial conditions. The new discrete-time model shows dynamic consistency with continuous-time model.
This study introduces us to a new model developed for computer viruses. The model is presented to remove the protective restriction on the total number of computers connected to the Internet. This model is nonlinear differential equation system. Therefore, finding analytical solutions is very difficult. This means that we have to apply numerical methods in order to find the solution. The behavior of numerical solution has been investigated for the discretized system. By using Nonstandard Finite Difference Scheme (NSFD), it is aimed to preserve both the positivity of the solutions for positive initial points and the local asymptotic stability of the equilibrium point.
This study aims to explain the dynamics of a competitive problem affected by toxicants. The effect of toxicants on ecological systems is an interesting topic for mathematical modelling. Discretization of the nonlinear problem is inevitable for right approximation of its solutions due to the difficulty of finding analytical solutions. In this work, a continuous time two species competitive problem was transformed into a discrete time problem. Because, it is very important to create a discrete model that will protect the properties of the original continuous model and the dynamics will be independent of step size. Also, in this study, the dynamic behaviour of a competitive system under the influence of toxicants were investigated. Lastly, the stability properties of each fixed point of the corresponding discrete problem have been examined using some theoretical results.
In this paper, we introduced nonstandard finite difference scheme (NSFD) for solving the continuos model with Michaelis-Menten harvesting rate. We have seen that the proposed scheme preserve local stability and positivity. Stability analysis of each fixed point of the discrete time model has been proven. Also, numerical comparisons were made between the nonstandard finite difference method and the other methods.
Physics and engineering problems require a detailed knowledge of applied mathematics and an understanding of special functions such as gamma and beta functions. The topic of special functions is very important and it is constantly expanding with the existence of new problems in the applied sciences. In this article, we describe the basic theory of gamma and beta functions, their connections with each other and their applicability to engineering problems.
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