Colonization and collapse models

• Grant number: 18/05430-7
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: April 01, 2018
• End date: December 31, 2018
• Field of knowledge: Physical Sciences and Mathematics - Probability and Statistics - Probability
• Principal Investigator: Fabio Prates Machado
• Grantee: Luiz Henrique Alves de Oliveira
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 17/10555-0 - Stochastic modeling of interacting systems, AP.TEM

Abstract

We will investigate a basic immigration process where colonies grow, during a random time, following a counting process until a collapse. At the time of collapse, only random amounts of individuals survive. These survivors try to establish new colonies independently in neighboring locations. We will consider this general process subject to two distinct schemes, the first with colony growth following a Poisson process with geometric catastrophe and the second with colony growth following a Yule process with binomial catastrophe. Independently of everything else, the colonies grow during an exponential time, like a Poisson process (or Yule) and after that exponential time, its size is reduced according to the geometric (or binomial) law. Each survivor tries independently, start a new colony on a local neighbor of a homogeneous tree. This colony will thrive until its collapse, and so on. We will study (I) conditions in the space of the parameters for the survival of these processes, (II) limits relevant to the survival probability, the number of vertices that are colonized and the reach of the colonies in comparison with the starting point. As a by-product, we will study (III) convergences of sequences of branching processes. (AU)