Análise teórica e computacional de processos estocásticos inspirados em sistemas biológicos.

Abstract

The aim of this work is to present two methodologies based on the theoretical and computational analysis of continuous time stochastic processes inspired by biological systems, whose dynamics are influenced by the stochastic nature of their constituent entities.

In the first part, we studied a particle system called the frog model (MS), in which there are two types of particles: the inactive and the active, so that each active particle runs a random walk, running through a finite graph $ \G $. Among the quantities of interest we have the proportions of each possible state over time and the final proportion of vertices visited or not visited by active particles. In this part of the thesis, we look for information about this proportion for different finite graphs. The effectiveness of the modeling techniques were analyzed using the following three approaches: Density-dependent Markov chains method (CMDD), approaching the mean field approaches (ACM) and computer simulations (SC). In the first two theoretical cases, their systems of equations were also obtained at a deterministic limit. These approaches were evaluated for the complete graphs $ \mathcal{K}_n $, complete bipartites graphs $ \mathcal{K}_{n_1, n_2} $, and for the cyclic graphs $ C_{n, c} $. The comparative results suggest a relationship between the density of the graph and the performance of the approaches in the MS and in this case, indicate that the three approaches are suitable for the M.S. for densely connected graphs. For cases considered sparse, the computational approach SC was presented as the most indicated.

In the Part \ref{part: EDE_pred_presa}, a model based on stochastic differential equations is applied, using an ecological system consisting of a predator specialized in hunting a type of prey, only in its adult stage. Parallel to this, we assume that the predator's mortality rate is affected by a randomness of the environment. We discuss the influence of this premise on the dynamic behavior of the model through a theoretical and computational analysis and show that the stochastic differential equations provide a more adequate model for this system.