Application of Vasileva's singular perturbation method to a problem in ecology

The simultaneous presence of fast and slow processes in ecological models leads to the formulation of model equations in a form amenable for the analysis by singular perturbation method. The method as originally proposed by Vasileva (1963) is applied to a specific species-resource logistic model and an algorithm useful for implementation on a digital computer is given. The solutions are obtained for zeroth, first and second order approximations. The accuracy of the solutions is examined by considering the integral-squared-error between the exact solutions and the perturbed solutions. The variation of this error with respect to the small parameter associated with the perturbation method is studied. An important result due to Tikhnov (1950) in the singular perturbation theory is illustrated. It is seen that Vasileva's singular perturbation method is a powerful analytical tool for investigations of various phenomena in the natural sciences.