Periodic Galerkin approximations for Lotka–Volterra equations using trigonometric basis functions

Our main interest in this study is to obtain approximate time-dependent solutions of the Lotka–Volterra predator–prey model given the populations of both species at a certain time. We achieve this by considering an earlier scheme by Shinohara and Yamamoto and modifying it so as to address the initial conditions. An important feature of this scheme is that it treats the system’s period as an unknown variable, making it possible to calculate it as a side product. In this method, the Galerkin procedure is utilized to convert the problem to a system of nonlinear algebraic equations, where sine and cosine waves are chosen as the basis functions in order to ensure the periodicity of solutions. The resulting overdetermined system is then solved in a least squares sense, which yields the coefficients making up the Galerkin approximations. After a detailed presentation of the numerical scheme, we apply it to two example problems and evaluate the results according to several accuracy criteria.