The simplest model of a population is Malthus model. It states that the rate of change of the number of individuals in a population is proportional to the number of individuals so that populations modeled by this law grow or decay exponentially. Malthus model describes an unstructured population, that is a population in which individuals are not distinguishable. Models for structured populations may also display an overall exponential behavior. This is the case with the Lotka-McKendrick model for age-structured populations, where the so called persistent solutions occur, namely solutions that can be expressed as a product of a function of age (the stable distribution) and an exponentially growing function of time. The Lotka-McKendrick model has played an important role in demography, due to its simplicity and relatively good fit to data on a short time scale. Consequently, it remains the preferred theoretical paradigm in the study of human demography. The Malthus and the Lotka-McKendrick models are, however, linear. Usually non-linear population models are designed to avoid unlimited growth. In fact, through non-linear rates, it is possible to introduce those mechanisms that, in nature, produce containment of the population size.
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