First Advisor

S.D. Tuljaapurkar

Date of Publication


Document Type


Degree Name

Master of Science (M.S.) in Physics






Population -- Mathematical models



Physical Description

1 online resource (63 p.)


The prediction and analysis of changes in the numbers of biological populations rest on mathematical formulations of demographic events (births and deaths) classified by the age of individuals. The development of demographic theory when birth and death rates vary statistically over time is the central theme of this work. A study of the standard Leslie model for the demographic dynamics of populations in variable environments is made. At each time interval a Leslie matrix of survival rates and fertilities of a population is chosen according to a Markov process and the population numbers in different age classes are computed. Analytical bounds are developed for the logarithmic growth rate and the age-structure of a population after long times. For a two dimensional case, it is shown analytically that a uniform distribution results for the age-structure if the survival rate from the first to the second age-class is a uniformly distributed random quantity with no serial auto correlation. Numerical studies are made which lead to similar conclusions when the survival rate obeys other distributions. It is found that the variance in the survival parameter is linearly related to the variance in the age structure. An efficient algorithm is developed for numerical simulations on a computer by considering a time sequence of births rather than whole populations. The algorithm is then applied to an example in three dimensions to calculate a sequence of births when the survival rate from the first to the second age-class is a random parameter. Numerical values for the logarithmic growth rate and the logarithmic variance for a population and the probability of extinction are obtained and then compared to the analytical results reported here and elsewhere.


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