Equivalence of -point Gauss–Chebyshev rule and -point midpoint rule in computing the period of a Lotka–Volterra system

• Published in: Advances in computational mathematics 2008, Vol.28 (1), p.63-79
• Main Authors: Shih, Shagi-Di, Chow, Shue-Sum
• Format: Article
• Language: English
• Subjects: Computation; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Integrals; Inverse; Lotka-Volterra equations; Mathematical analysis; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Predator-prey simulation; Singularities; Visualization
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-006-9013-4

Two integrals (3.6), (4.7) for the period of a periodic solution of the Lotka–Volterra system are presented in terms of two inverse functions of restricted on , , respectively. In computing this period numerically, the integral (3.6), which possesses a weak singularity of the square root type at each endpoint of the integration, is an excellent example of using the Gauss–Chebyshev integration rule of the first kind; while the integral (4.7), which is an integral of a smooth periodic function over its period , is an excellent example of using the midpoint rule, but not the trapezoidal rule, suggested by Waldvogel [ 39 , 40 ], due to a removable singularity of the integrand at , , , , and , respectively. This paper shows, in computing the period of a periodic solution of the Lotka–Volterra system, the -point Gauss–Chebyshev integration rule of the first kind applied to the integral (3.6) becomes the -point midpoint rule to the integral (4.7).