Estudo da Estabilidade de um Modelo Populacional com Retardo Fuzzy

Abstract

The aim of this work is to numerically solve the Montroll Model with delay, considered as a real and fuzzy number, whose parameters are obtained through data from the Brazilian population between the years 1991 and 2022. In addition, we seek to analyze the stability of the Montroll Model with fuzzy delay. The delay is incorporated into the classic Montroll Model, allowing the insertion of temporal effects that impact the population dynamics in a way that is closer to reality. The numerical solution of this model is obtained using the Step Method, which is combined with the fourth-order Runge-Kutta Method. This process is implemented through a proprietary computer program, developed specifically for this purpose. The stability studies of the equilibrium points of the Montroll Models are carried out in three cases: without delay, with deterministic delay and using a triangular fuzzy number. These analyses are developed for general and specific parameters, considering the data from the Brazilian population, resulting in convergence to the maximum population of the model. The uncertainties of the fuzzy delay are incorporated into the model solution through the Zadeh Extension Principle. The center of gravity method is used to defuzzify the solution at each instant. The arithmetic mean of the relative errors between the defuzzified solution with fuzzy delay and the Brazilian population data is lower than the same metric applied to the model without delay. Thus, the defuzzification of the model solution at each instant is closer to the Brazilian population data, published by the Brazilian Institute of Geography and Statistics.