# Mathematical analysis of the asymptotic behavior of the Leslie population matrix model

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## Abstract

*L*be a Leslie population matrix. Leslie (1945) and others have shown that the matrix

*L*has a leading positive eigenvalue

*λ*

_{0}and that in general:

**X**

_{ λ }

_{0}is an eigenvector corresponding to

*λ*

_{0},

**X**is any initial population vector, and γ is a scalar quantity detormined by

**X**.

In this article we generalize (1) exhaustively by removing the mild restrictions on the fertility rates which most writers impose. The result is an oscillatory limit of a kind first noted by Bernardelli (1941) and Lewis (1942) and described by Bernardelli as “population waves”. We calculate in terms of*λ* _{0} and the entries of the matrix*L* the values of this oscillatory limit as well as its time-independent average over one period. This calculation includes as its leading special case the result of (1), confirming incidentally that γ is nonzero.

To stabilize a population, the matrix*L* must be adjusted so that*λ* _{0}=1. The limits calculated for the oscillatory and non-oscillatory cases then have maximum significance since they represent the limiting population vectors. We discuss a simple scheme for accomplishing stanbilization which yields as a byproduct an easily accessible scalar measure of*L*'s tendency to promote population growth. The reciprocal of this measure is the familiar net reproduction rate.

### Keywords

Population Distribution LESLIE Model Lower Term Population Wave Oscillatory Limit## Preview

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