# Eigenvalue meaning in a Leslie Model

May 29th, 2015
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The dominant eigenvalue of , denoted , gives the population's asymptotic growth rate (growth rate at the stable age distribution). The corresponding eigenvector provides the stable age distribution, the proportion of individuals of each age

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We take as the number of newly-born individuals at time . Thus where is the maximum age to which individuals can survive. The number of individuals in other age categories is determined purely by the number of individuals that survive from the preceding year. Specifically, This completely specifies the demographics of the population, assuming for the moment that the and don't vary from one year to the next. This can be written in matrix form as More compactlyEigenvalue meaning in a Leslie ModelThe dominant eigenvalue of , denoted , gives the population's asymptotic growth rate (growth rate at the stable age distribution). The corresponding eigenvector provides the stable age distribution, the proportion of individuals of each age within the population. Once the stable age distribution has been reached, a population undergoes exponential growth at rate .PropertiesThe population on finally grows if 1> 1. The population on finally decreases if 1 <1. The population on eventually stabilize if 1 = 1.Application problem This problem was taken from Linear Algebra a Modern Introduction by David Poole chapter 3.Woodland caribou are found primarily in the western provinces of Canada and the American northwest. The average lifespan of a female is about 14 years. The birth and survival rates for each bracket are given in the following table, which shows the caribou cows do not give birth during their first 2 years and give birth to about

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