Some new observations on the classical logistic equation with heredity

摘要

Several new and significant observations are presented pertaining to the classical problem of single-population growth with hereditary influences. In its conventional form, the resulting equation with heredity is mathematically represented by a nonlinear Volterra integro-differential equation. In this paper, we propose a new differential formulation where the dependent variable is now defined in terms of the integral of the unknown population. This formulation allows us to develop novel analyses leading to enlightening results. Some particular findings include: the development and analysis of an integrated phase-plane; the elucidation of the exact value for the extremum of the population and several other important functional relations at that corresponding time; the development of two analytic expressions for determining the time at which the population peaks; the determination of the upper asymptote for the cumulative population; and the development of an accurate early-time solution as obtained from a Riccati equation. Additionally, we illustrate that an analytical solution, based on Taylor Series expansions, can be developed with the aid of MathematicaTM. A pure numerical solution is offered for comparison with the analytic solution.