Classroom Note:Numerical and Analytical Solutions of Volterra's Population Model

Abstract

To illustrate various mathematical methods which may be used to solve problems of engineering, mathematical physics, and mathematical biology, Volterra's model for population growth of a species in a closed system is solved using several methods familiar to junior- or senior-level students in applied mathematics. Volterra's model is a first-order integro-ordinary differential equation where the integral term represents the effect of toxin accumulation on the species. The solution methods used are (i) numerical methods for solving a first-order initial value problem supplemented with numerical integration, (ii) numerical methods for solving a coupled system of two first-order initial value problems, and (iii) phase-plane analysis. A singular perturbation solution previously presented is also outlined. While conclusions drawn using the four methods are correlated, the student may analyze and solve the problem using any of the methods independently of the others.