BRANCHING PROCESSES

Vejitveni proces je matematični model razvoja in rasti neke populacije. Opazujemo lahko razvoj populacije po generacijah ali pa po časovnih intervalih. Takim vejitvenim procesom pravimo časovno odvisni. Obravnavali smo jih s pomočjo rodovne funkcije in s pomočjo markovskih verig. Najprej so razloženi osnovni pojmi, kot so neodvisne naključne spremenljivke, rodovna funkcija in markovske verige, ki so potrebni za razumevanje teorije o vejitvenih procesih. Pri tem gre predvsem za izračun verjetnosti izumrtja populacije. Poleg teoretične obravnave izumrtja, je na konkretnih primerih predstavljen vejitveni proces z začetno geometrijsko porazdelitvijo, za katerega je narejena tudi računalniška simulacija. Program je narejen v programskem jeziku C# in vejitveni proces simulira z gradnjo drevesa in izračunom verjetnosti izumrtja. Simulacija da rezultate, ki so primerljivi s teoretično izračunanimi.The branching process is a mathematical model of development and growth of a population. We can suppose that a population evolves in generations or we attach another random variable, called age. We name them age-dependent branching processes. For describing such processes we use probability generating functions and markov chains. At rst are explained basic concepts, such as dicrete random variables, proba- bility generating functions and markov chains, which are necessary to understand the theory of branching processes. Especially we are interested in expressing proba- bility of ultimate extinction. In addition to the theoretical treatment of extinction, are also exemples which presents the geometrical distributed processes. Those are included in computer simulation. The program, designed in C #, simulate growth of a population with family tree of a branching process and calculate the probability of extinction. Results of the simulation are comparable to the theoretical results