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Abstract

A stochastic hybrid system (SHS) is a dynamic system that exhibits both a continuous and a discrete dynamical behavior. In the scientific literature on SHS the principle of causality is commonly assumed, but more realistic models should account on history dependence. In this thesis we developed and analyze a new mathematical model called Hereditary Stochastic Hybrid System which, with the help of a generalization of stochastic time change, allows for general history dependence. We apply the above model to the study of the movement of the bacterium E. coli: when these bacteria swim in a field of chemoattractant molecules, they perform a random walk characterized by a run-and-tumble motion. This random walk is biased towards regions of higher concentration of the chemoattractant. These bacteria are, however too small to sense spatial gradients along their body axis. Instead they developed a history-dependent strategy to search of food, namely the use of memory of previous mea- surements of chemical concentrations. In this way the bacteria are able to infer whether they swim up or down a chemical gradient. By using our framework we are able to generalize the existing models for E. coli movement and have a better fitting of the experimental data. We developed also a software that simulates and compares the different models existing in the literature, with special care for high performance and usability.