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In this paper we construct nonstandard difference schemes, which are dynamically consistent with a metapopulation model formulated by Keymer et al. in 2000, i.e. preserve all dynamical properties of the differential equations of the model. These properties are: monotone convergence, boundedness, local asymptotic stability and especially, global stability of equilibria and non-periodicity of solutions. Numerical examples confirm the obtained theoretical results of the properties of the constructed difference schemes. ARTICLE HISTORY

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In this paper nonstandard finite difference (NSFD) schemes of two metapopulation models are constructed. The stability properties of the discrete models are investigated by the use of a generalization of Lyapunov stability theorem. Due to this result we have proved that the NSFD schemes preserve all properties of the metapopulation models. Numerical examples confirm the obtained theoretical results of the properties of the constructed difference schemes. The method of Lyapunov functions proves to be much simpler than the standard method for studying stability of the discrete metapopulation model in our very recent paper.

In this paper, we construct explicit nonstandard Runge-Kutta (ENRK) methods which have higher accuracy order and preserve two important properties of autonomous dynamical systems, namely, the positivity and linear stability. These methods are based on the classical explicit Runge-Kutta methods, where instead of the usual h in the formulas there stands a function ϕ(h). It is proved that the constructed methods preserve the accuracy order of the original Runge-Kutta methods. The numerical simulations confirm the validity of the obtained theoretical results. difference schemes; Positive nonstandard finite difference methods; Elementary stable. Our first objective is to construct difference schemes preserving the linear stability of the equilibrium points of System (1) for all finite step-size h > 0. These schemes are called also elementary stable [3,7,8]. It should be emphasized that standard finite difference schemes cannot preserve properties of the differential equations for any step-sizes h > 0, including the linear stability. Mickens called this phenomenon numerical instability [25].The construction of elementary stable difference schemes play especially important role in numerical solution of differential equations and numerical simulation of nonlinear dynamical systems. In 2005, Dimitrov and Kojouharov [7] proposed a method for constructing elementary stable NSFD methods for general two-dimensional autonomous dynamical systems. These NSFD methods are based on the explicit and implicit Euler and the second-order Runge-Kutta methods. Later, in 2007 these results are extented for the general n-dimensional dynamical systems, namely, NSFD schemes preserving elementary stability are constructed based on the θ-methods and the second-order Runge-Kutta methods [8]. One important action in the construction of the elementary stable NSFD schemes is the replacement of the standard denominator function ϕ(h) = h by the nonstandard denominator function, which is Email addresses: dangquanga@cic.vast.vn (Quang A Dang),