1. A second-order nonstandard finite difference method for a general Rosenzweig–MacArthur predator–prey model.
- Author
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Hoang, Manh Tuan and Ehrhardt, Matthias
- Subjects
- *
FINITE difference method , *NONLINEAR dynamical systems , *LOTKA-Volterra equations , *FINITE differences , *CONTINUOUS time models , *MATHEMATICAL analysis - Abstract
In this paper, we consider a general Rosenzweig–MacArthur predator–prey model with logistic intrinsic growth of the prey population. We develop the Mickens' method to construct a dynamically consistent second-order nonstandard finite difference (NSFD) scheme for the general Rosenzweig–MacArthur predator–prey model. The second-order NSFD method is based on a novel nonlocal approximation using right-hand side function weights and nonstandard denominator functions. Through rigorous mathematical analysis, we show that the NSFD method not only preserves two important and prominent dynamical properties of the continuous-time model, namely positivity and asymptotic stability independent of the values of the step size, but also is convergent of order 2. Therefore, it provides a solution to the contradiction between the dynamic consistency and high-order accuracy of NSFD methods. The proposed NSFD method improves positive and elementary stable nonstandard numerical schemes constructed in a previous work of Dimitrov and Kojouharov (2006). Moreover, the present approach can be extended to construct second-order NSFD methods for some classes of nonlinear dynamical systems. Finally, the theoretical insights and advantages of the constructed NSFD scheme are supported by some illustrative numerical simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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