Associative Property of Interactive Addition for Intervals: Application in the Malthusian Model

Abstract

This paper presents a study about the properties of the interactive arithmetic sum$$+:{0.5}$$ for intervals. This particular arithmetic operator is defined by a family of joint possibility distributions, which gives raise to the notion of interactivity. This work shows that the class of intervals$$\mathcal {I}$$ under the sum$$+:{0.5}$$ is a commutative monoid, that is, the operation$$+:{0.5}$$ satisfies the commutative and associative properties and$$\mathcal {I}$$ has an identity element. In order to illustrate the properties of this arithmetic sum, the Malthusian problem is investigated from the perspective of numerical methods, such as the Euler’s and Runge-Kutta methods. Finally, a comparison between the interactive arithmetic$$+:{0.5}$$ and the standard arithmetic is presented in order to provide the advantages of the sum$$+:{0.5}$$.