In an attempt to explain experimental evidence of chaotic oscillations in blood cell population, A. Lasota suggested in 1977 a discrete-time one-dimensional model for the production of blood cells, and he showed that this equation allows to model the behavior of blood cell population in many clinical cases. Our main aim in this note is to carry out a detailed study of Lasota's equation, in particular revisiting the results in the original paper and showing new interesting phenomena. The considered equation is also suitable to model the dynamics of populations with discrete reproductive seasons, adult survivorship, overcompensating density dependence, and Allee effects. In this context, our results show the rich dynamics of this type of models and point out the subtle interplay between adult survivorship rates and strength of density dependence (including Allee effects).
In the framework of fixed point theory, many generalizations of the classical results due to Krasnosel'skii are known. One of these extensions consists in relaxing the conditions imposed on the mapping, working with k-set contractions instead of continuous and compact maps. The aim of this work if to study in detail some fixed point results of this type, and obtain a certain generalization by using star convex sets.
We study a discrete-time model for a population subject to harvesting. A maximum annual catch H is fixed, but a minimum biomass level T must remain after harvesting. This leads to a mathematical model governed by a continuous piecewise smooth map, whose dynamics depend on two relevant parameters H and T. We combine analytical and numerical results to provide a comprehensive overview of the dynamics with special attention to discontinuity-induced (border-collision) bifurcations. We also discuss our findings in the context of harvest control rules.
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