Approximate solutions of the logistic equation and Ulam stability (Q1992209)

The stability problem of functional equations originated from a question of Ulam in 1940, concerning the stability of group homomorphisms. In 1941, D. H. Hyers gave a first affirmative answer to the question for Banach spaces. The logistic growth model, proposed by P. F. Verhulst in the first half of the 19th century, represents not only a cornerstone in the study of population dynamics and mathematical biology, but is also a typical example of a nonlinear dynamical system. The governing equation \[ \frac{dP}{d\tau}= rP(1 -\frac{P}{K}) \tag{1} \] describes the time evolution of the size \(P\) of a population whose growth obeys a self-limiting version, which accounts for the scarcity of available resources, of the Malthusian law. The growth rate \(r(1 -\frac{P}{K})\) decreases linearly as a function of the population size and becomes zero as \(P\) reaches the maximum carrying capacity \(K>0\) of the environment. In this paper the authors use the concept of conditional Ulam stability to investigate the stability of the dimensionless form of the following logistic model: \[ \frac{dy}{dt}= y - y^2. \]