Numerical dynamics of nonstandard finite difference schemes for a Logistics model with feedback control

Abstract

In this paper, we construct nonstandard finite difference (NSFD) schemes that preserve essential qualitative properties of a Logistics model with feedback control using the Mickens’ methodology. These properties of the model include positivity of solutions and stability of equilibrium points. Dynamical properties of the proposed NSFD schemes are rigorously investigated. The results show that the constructed nonstandard schemes preserve the essential properties of the continuous model for all finite step sizes. Consequently, we obtain NSFD schemes which are dynamically consistent with the continuous model. Especially, the constructed NSFD schemes become exact finite difference ones in some special cases. Besides, some numerical simulations are performed to validate the theoretical results as well as the advantages of the proposed NSFD schemes. The numerical simulations also indicate that the NSFD schemes are effective and appropriate to solve the continuous model. Meanwhile, the standard finite difference schemes such as the Euler scheme, the second order and fourth order Runge–Kutta schemes can generate numerical solutions that are completely different from the solutions of the continuous model and therefore, they fail to correctly reflect the correct behavior of the continuous model.