Análise direta e inversa da dinâmica de populações difusivas via transformações integrais, inferência bayesiana e modelo de erro de aproximação

Abstract

This work presents the direct and inverse analysis of a transient two-dimensional population dynamics problem in the presence of local traps. Initially, the population study is conducted considering the existence of only one species to be later expanded to a prey-predator type system. The formulated direct problem is solved using a hybrid approach via the technique of generalized integral transforms (GITT). Aiming to reduce the computational cost to solve the direct problem, even with GITT, a low-cost solution was implemented. The use of a low-cost solution for the direct problem led to the emergence of modeling error, which was incorporated into the solution of the inverse problem via the approximation error model (AEM). From the perspective of the inverse problem, the estimation of the parameters associated with the models is formulated using the Bayesian formalism, using the maximum a posteriori (MAP) method to generate a point estimate of the posteriori then refined by sampling by Monte Carlo via Markov Chains (MCMC) method. Considering the existence of a priori information of different types, the MAP objective function was optimized by the Luus Jaakola and Gauss-Newton methods, whereas the MCMC was implemented considering classical and adaptive Metropolis-Hastings algorithms. In order to establish satisfactory conditions for estimating the parameters involved, local and global sensitivity analyzes are applied, identifying which parameters can be estimated via inverse problems with Bayesian inference. While the results of the direct problem show that GITT can be used successfully in this type of problem, the results of the inverse problem suggest in which scenarios this methodology can be applied considering real situations.