The Dynamics of Discrete Populations and Series of Events

Content: Introduction References Statistical Preliminaries Introduction Probability Distributions Moment-Generating Functions Discrete Processes Series of Events Summary Further Reading Markovian Population Processes Introduction Births and Deaths Immigration and the Poisson Process The Effect of Measurement Correlation of Counts Summary Further Reading The Birth-Death-Immigration Process Introduction Rate Equations for the Process Equation for the Generating Function General Time-Dependent Solution Fluctuation Characteristics of a Birth-Death-Immigration Population Sampling and Measurement Processes Correlation of Counts Summary Further Reading Population Processes Exhibiting Sub-Poisson Fluctuations and Odd-Even Effects Introduction A Limited Birth-Death Process: Binomial Number Fluctuations The Effect of Monitoring a Limited Birth-Death Process A Birth Limited Death Immigration Process The Effect of Monitoring a Birth Limited Death Process Odd Even Effects in a Population Driven by Immigrant Pairs Monitoring an Immigrant Pair Death Process Summary Further Reading The Death-Multiple Immigration Process Introduction Rate and Generating-Function Equations for the Process Negative Binomial Fluctuations Revisited Sampling and Measurement Processes Time-Dependent Multiple Immigrations Summary Further Reading Stable Processes and Series of Events Introduction Discrete Stability A Discrete Stable Markov Process Stable Series of Events Construction of Statistical Measures by Clipping Summary Further Reading Coupled Processes Introduction Coupled Death Multiple Immigrations Processes Properties of Populations Coupled by Multiple Immigrations Cyclically Coupled Populations Populations Coupled through Exchange Interactions Summary Further Reading Doubly Stochastic Processes Introduction General Statistical Relationships Commonly Encountered Doubly Stochastic Poisson Processes Continuous and Discrete Stable Variables Doubly Stochastic Stable Processes Summary Further Reading Level Crossings of a Continuous Process Introduction Generalisation of the Van Vleck Theorem Crossing Number Statistics of a Differentiable Process Processes that are Continuous but Not Differentiable Distribution of Intervals between Zeros Summary Further Reading Simulation Techniques Introduction Interevent Intervals Transformation of Random Variables Simulating the Death Immigration Process Extension to More than Two Events Processes with Multiple Immigrations Gaussian Random Process The Doubly Stochastic Poisson Process Summary Further Reading Glossary of Special Functions