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

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March 2008, 9(2): 415-429. doi: 10.3934/dcdsb.2008.9.415

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

1.	Department of Mathematics and Statistics, Box 41042, Texas Tech University, Lubbock, TX 79409-1042, United States

Received July 2007 Revised October 2007 Published December 2007

Abstract
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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.

Keywords: local stability, dynamical consistency, competition, Lotka-Volterra, NSFD., nonstandard finite-difference methods.

Mathematics Subject Classification: Primary: 34-0.

Citation: Lih-Ing W. Roeger. Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 415-429. doi: 10.3934/dcdsb.2008.9.415

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