Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes

Abstract

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.

A class of nonstandard numerical methods for autonomous dynamical systems

“…If the system (1) is locally stable atx, then Re(λ) < 0. Using (6), which gives a relation between the eigenvalues of differential system (1) and the numerical method (3), the eigenvalues, 1 + φ(h)λ, of B satisfy…”

“…[2,3,4,5,6,7,8,9]). Here, a finite difference method is constructed in Section 2 which preserves the aforementioned qualitative properties for the system (1).…”

Global dynamics of a discretized SIRS epidemic model with time delay

“…Traditional numerical schemes such as the Euler and Runge-Kutta sometimes fail, by generating oscillations, bifurcations, chaos and false steady states (see [2,5]). However, as one of numerical schemes, the nonstandard finite-difference scheme is well known and has been applied to various problems in science (for example [1,3,15,16,13]). Mickens [12] concludes that the use of this scheme leads to asymptotic dynamics and numerical results that are always qualitatively the same as the corresponding solutions of several ordinary differential equations for any positive step size.…”

Periodic solutions preserved by nonstandard finite-difference schemes for the Lotka–Volterra system: a different approach

“…Numerical simulations show that the methods of the three classes produce periodic solutions for the predator -prey model (1). Similar pattern of the nonstandard terms in the three classes when applied to Lotka -Volterra competitive system produce numerical methods that preserve local stability [2,7]. However, they do not preserve monotonicity of the competitive system.…”

“…This manuscript is organized as follows. We show that the above three classes of symplectic methods are not necessary conditions for periodic solutions to appear in the NSFD method (7). More general NSFD methods that preserve periodic solutions can be found and they are given in Section 2.…”