A Logistic Non-linear Difference Equation with Two Delays

  • Francisco BalibreaEmail author
Part of the Understanding Complex Systems book series (UCS)


In this chapter we analyze the state of art of logistic first and second order difference equations with two delays. They model the evolution of populations with respect to seasons in time \(n \in \mathbb {N}\). Of special interest are those non-linear equations with two delays, particularly due to the effect of food in the evolution of the population. As an adequate tool to understand the behaviors of solutions of the equation, we use an unfolding of it obtaining a discrete dynamical system of dimension two, defined in the unit square. We review some dynamical properties already known like periodic solutions and local linear analysis around the fixed points of the unfolding. Besides we also introduce new results and analysis of behavior of invariant curves. All analysis depend on a parameter a. Our study is mainly devoted to the range \(a \in (0,2]\), the setting where we give some revised results. Some open problems remain for the range when \(a \ge 2.27\) where 2.27 is a critical value.


Non-linear difference equations Unfolding Periodic solutions Invariant curves 



This research has been supported by the project MTM2014-51891-P from Spanish Ministerio de Economía (MINECO). This paper is dedicated to the memory of Wieslaw Szlenk, good master and better person.


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Authors and Affiliations

  1. 1.Group of Dynamical Systems, Facultad de MatemáticasUniversidad de MurciaMurciaSpain

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