Mickens [Applied Numerical Mathematics, 45, pp. 309 -314, 2003] constructed a nonstandard finite difference scheme for the Lotka-Volterra predator-prey system x 0 ¼ x 2 xy; y 0 ¼ 2y þ xy which, for positive initial conditions, gives rise to periodic solutions. His method amazingly preserves the oscillation feature of the Lotka-Volterra system while many numerical methods give solutions that spiral into or out of the positive-valued fixed-point. The reason is that Mickens's scheme is noncanonically symplectic. Applying the same idea, we generalize Mickens's method and produce a class of nonstandard symplectic numerical methods for the above Lotka-Volterra system. These methods are all symplectic with respect to a noncanonical symplectic structure. They all have the property that the computed points do not spiral. Hence, these methods also preserve the oscillation feature of the predator-prey system.

Discrete-time Lotka-Volterra competition models are obtained by applying nonstandard finite difference (NSFD) schemes to the continuous-time counterparts of the model. The NSFD methods are noncanonical symplectic numerical schemes when applying to the predator-prey model x ′ = x − xy and y ′ = −y + xy. The local dynamics of the discrete-time model are analyzed and compared with the continuous model. We find the NSFD schemes that preserve the local dynamics of the continuous model. The local stability criteria are exactly the same between the continuous model and the discrete model independent of the step size. Two specific discrete-time Lotka-Volterra competition models by NSFD schemes that preserve positivity of solutions and monotonicity of the system are also given. The two discrete-time models are dynamically consistent with their continuous counterpart.1. Introduction. Mathematical models are used to represent phenomena in the biological, ecological, and physical sciences, to name a few. Differential equations are used when the model represents continuous variables. However, when working with discrete variables, difference equations are most appropriate. For example, in ecology, predator-prey models can be formulated as discrete-time models. Difference equations are appropriate when organisms have discrete, non-overlapping generations [1,5].Different numerical schemes can be used to convert differential equations into difference equations. If the corresponding difference equations possess the same dynamical behavior as the continuous equations, such as local stability, bifurcations, and/or chaos, then they are said to be dynamically consistent [6]. More specifically, Mickens [7] defines dynamic consistency (DC) as the following: Definition 1.1. Consider the differential equation x ′ = f (x, t). Let a finite difference scheme for the equation be x k+1 = F (x k , t k , h). Let the differential equation and/or its solutions have property P. The discrete model equation is dynamically consistent with the differential equation if it and/or its solutions also has property P.

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We construct nonstandard finite-difference (NSFD) schemes that provide exact numerical methods for a first-order differential equation having three distinct fixed-points. An explicit, but also nonexact, NSFD scheme is also constructed. It has the feature of preserving the critical properties of the original differential equation such as the positivity of the solutions and the stability behavior of the three fixedpoints.

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