Rare events simulation in models for ecology

In the thesis Large Deviation Theory (LDT) is employed to analyse the problem of rare events entering a random process and to devise a high-efficiency simulation based on Importance Sampling technique (IS). The random process considered is the multiplicative growth process in Markovian environment, which has a relevant role in describing Taylor’s Law (TL). TL in ecology states that variance V and mean M of the population number of species are related by a power-law relationship, V = a M^b. Although theoretical models predict for the power exponent b a broad range of values, depending on details of the models, empirical results from observations and simulations report, almost universally, b bounded, mostly b about 2. This behaviour has been verified in a wide variety of research fields, from ecology to biology and finance. There is a previous work demonstrating that the inefficiency in sampling the rare events of the process (events of extremely low probability) may be the possible cause of the behaviour of b. Starting from this work, the thesis adopts LDT to study the problem of rare events and to implement a high-efficiency simulation, based on IS methods, to detect the these events in the process. Results show the b exponent, when estimated by the IS simulation method, takes a broad range of values as predicted by the model and is not bounded, nor it is constrained to be close to a particular value. Results then show the analysis of rare events by LDT and an IS simulation technique could be crucial to correctly infer regular patterns in ecology and in any other context where Taylor’s Law arises.