Semigroup structures and properties of some kinds of mappings

Abstract

In this paper, we consider several kinds of maps frequently appearing in the population dynamics, e.g., logistic mapping: $f(x)=rx(1-x)$ $(x\in \mathbb{R}, $ $r>0$ is a parameter), unimodel Allee maps with Allee effect and Sigmoid mapping, etc. We study the persistence of some properties for these maps such as the numbers of fixed point, their positions, stabilities and so on. We prove that they are closed under composition of maps. That means they form a semigroup. The examples are given to illustrate the results.