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A dynamic population system is often modelled by a deterministic difference equation model to obtain average estimates. However, there is a risk of the results being distorted because unexplained (random) variations are left out and because entities in the population are described by continuous quantities of an infinitely divisible population so that irregularly occurring events are described by smooth flows. These distortions have many aspects that cannot be understood by only regarding a deterministic approach. However, the reasons why a deterministic model may behave differently and produce biased results become visible when the deterministic model is compared with a stochastic model of the same structure. This paper focuses first on demographic stochasticity, i.e. stochasticity that refers to random variations in the occurrence of events affecting the state of an individual, and investigates the consequences of omitting this by deterministic modelling. These investigations reveal that bias may be strongly influenced by the type of question to be answered and by the stopping criterion ending the analysis or simulation run. Two cases are identified where deterministic models produce unbiased state variables: (1) Dynamic systems with stable local linear dynamics produce unbiased state variables asymptotically, in the limit of large flows; and (2) linear dynamic systems produce unbiased state variables as long as all state variables remain non-negative in both the deterministic and the stochastic models. Both cases also require the question under study to be compatible with a solution over a fixed time interval. Stochastic variability of initial values between simulation runs because of uncertainty or lack of information about the initial situation is denoted initial value stochasticity. Elimination of initial value stochasticity causes bias unless the model is linear. It may also considerably enlarge bias from other sources. Unknown or unexplained variations from the environment (i.e. from outside the borders of the studied system) enter the model in the form of stochastic parameters. The omission of this environmental stochasticity almost always creates biased state variables. Finally, even when a deterministic model produces unbiased state variables, the results will be biased if the output functions are not linear functions of the state variables., (Copyright © 2013 Elsevier Inc. All rights reserved.)

A population system can be modelled using a micro model focusing on the individual entities, a macro model where the entities are aggregated into compartments, or a state-based model where each possible discrete state in which the system can exist is represented. However, the concepts, building blocks, procedural mechanisms and the time handling for these approaches are very different. For the results and conclusions from studies based on micro, macro and state-based models to be consistent (contradiction-free), a number of modelling issues must be understood and appropriate modelling procedures be applied. This paper presents a uniform approach to micro, macro and state-based population modelling so that these different types of models produce consistent results and conclusions. In particular, we demonstrate the procedures (distribution, attribute and combinatorial expansions) necessary to keep these three types of models consistent. We also show that the different time handling methods usually used in micro, macro and state-based models can be regarded as different integration methods that can be applied to any of these modelling categories. The result is free choice in selecting the modelling approach and the time handling method most appropriate for the study without distorting the results and conclusions., (Copyright 2010 Elsevier Inc. All rights reserved.)

Population models concern collections of discrete entities such as atoms, cells, humans, animals, etc., where the focus is on the number of entities in a population. Because of the complexity of such models, simulation is usually needed to reproduce their complete dynamic and stochastic behaviour. Two main types of simulation models are used for different purposes, namely micro-simulation models, where each individual is described with its particular attributes and behaviour, and macro-simulation models based on stochastic differential equations, where the population is described in aggregated terms by the number of individuals in different states. Consistency between micro- and macro-models is a crucial but often neglected aspect. This paper demonstrates how the Poisson Simulation technique can be used to produce a population macro-model consistent with the corresponding micro-model. This is accomplished by defining Poisson Simulation in strictly mathematical terms as a series of Poisson processes that generate sequences of Poisson distributions with dynamically varying parameters. The method can be applied to any population model. It provides the unique stochastic and dynamic macro-model consistent with a correct micro-model. The paper also presents a general macro form for stochastic and dynamic population models. In an appendix Poisson Simulation is compared with Markov Simulation showing a number of advantages. Especially aggregation into state variables and aggregation of many events per time-step makes Poisson Simulation orders of magnitude faster than Markov Simulation. Furthermore, you can build and execute much larger and more complicated models with Poisson Simulation than is possible with the Markov approach.