| Numerical methods based on finite difference approximations, Taylor series expansions and interpolations, such as Euler, Runge-Kutta and Adams methods are widely used in practice. However, their use raises questions about the truncation errors, the stability regions and, from a dynamical point of view, the accuracy at which the dynamics of the continuous system are represented by the discrete system. The class of nonstandard finite difference schemes, developed by Mickens, 1994, has laid the foundation for designing methods that preserve the dynamical behavior of the approximated differential system. The techniques are based on a nonlocal numerical treatment of the right-hand side function and more sophisticated discretizations of time derivatives. Nonstandard numerical methods are relatively easy to implement and have much greater computational efficiency as compared to standard numerical schemes.; In this dissertation we develop new classes of nonstandard methods for solving dynamical systems numerically. Among the newly designed schemes are nonstandard finite-difference methods for ordinary differential equations of the form dx/dt = f(x), where f is a polynomial, elementary stable nonstandard (ESN) finite-difference methods for general multi-dimensional autonomous dynamical systems, based on the standard theta and second-order Runge-Kutta methods and positive and elementary stable non-standard (PESN) finite-difference methods for Rosenzweig-MacArthur predator-prey systems, for predator-prey systems with Beddington-DeAngelis functional response and for phytoplankton-nutrient systems with nutrient loss. Applications of those methods on various biological systems are also presented and analyzed. The advantages of the newly designed methods by numerical simulations are demonstrated. |
