Volterra's Population Model Research Articles

Analytical approximations and Padé approximants for Volterra's population model

In this paper, an analytic approximation for Volterra's model for population growth of a species in a closed system is presented. The nonlinear integro-differential model includes an integral term that characterizes accumulated toxicity on the species in addition to the terms of the logistic equation. The series solution method and the decomposition method are implemented independently to the model and to a related ODE. The Padé approximants, that often show superior performance over series approximations, are effectively used in the analysis to capture the essential behavior of the population u(t) of identical individuals.

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

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.