ABSTRACT
Models of population dynamics with extinction, stable points and small stochastic effects lead to singular perturbations problems for differential equations with singular points and turning points. Both singular points and turning points give rise to resonance phenomena. The complete or exponential asymptotic expansions are viewed from the perspective of multiple scaling techniques, such that the complete expansion is recovered as a Poincaré expansion in an extended multiple variable space. This chapter includes discussions of the notation of exponential precision asymptotics, the multiple scaling approach to complete asymptotic expansions. The Liouville-Green or LG-WKB approximation can be used to discover the appropriate exponentially ordered pair. The successful matching of terms may depend on knowing both exponentially large and small terms to exponential precision. The chapter provides a brief review of the dual uniform method of Wazwaz and Hanson, and presents a method for constructing purely exponential pairs from given complete expansions.
