Análise de sistemas biológicos com difusão e controle populacional

Abstract

The use of population models is increasing in agrarian and biological sciences. One of the main interests in the study of these models is to evaluate qualitatively and quantitatively the impact of interactions between different populations with their habitat or between them, such as: atoms or molecules, neurons, bacteria, plants, animals and even pests or infected individuals for example. Mathematical models describe the growth of these populations for the purpose of making predictions, contribuing in the decision making if there is need for the application of control in these biological systems. In this work, we defend the use of biological control, which consists in regulating population growth through human intervention or natural enemies, being very efficient in pest control and at the same time reducing the use of pesticides, becoming less invasive to nature. The aim of this work is to study two population reaction-diffusion models: the Fisher-Kolmogorov equation for a species, and the modified Lokta-Volterra system with diffusion for two species, both in one dimension, with and without control. These models describe the dispersion of individuals in which the population growth occurs in the form of a wavefront, advancing at a constant velocity to the empty regions until reaching the supporting capacity of the medium, thus, admitting a traveling wave type solution, which allows a qualitative analysis of the solution through the theory of stability and the finding of approximate solutions. From the study carried out on both models without the influence of regulators, the theory of linear optimal control is applied considering biological systems that need the introduction of control strategies to achieve the wanted balance or stabilize below economic and/or biological damages. This procedure, an innovative part of this work, constitutes an effective methodology in the application of control in biological systems, whose behavior is modeled by the reaction-diffusion equation, which will be illustrated through computational simulations.