# Using Leslie Matrices as the Application of Eigenvalues and Eigenvectors in a First Course in Linear Algebra

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## Abstract

Leslie matrices may be used to model the age distribution of a population as well as population growth. The dominant eigenvalue tells us the long term population growth and the corresponding eigenvector tells us the long term age distribution. Because the model is so simple, and it does not require any knowledge of physics or chemistry or biology, it’s ideal for presenting in a first course on Linear Algebra as the main application of eigenvalues and eigenvectors.

In this paper we present the Leslie age distribution model and provide accompanying exercises suitable for students. We use Maple for both numerical calculations and symbolic calculations. We include some data for real populations that instructors may use for classroom presentation or for assignments.

## References

- 1.Anton, H., Rorres, C.: Elementary Linear Algebra with Applications. Wiley, Hoboken (1987)zbMATHGoogle Scholar
- 2.Boyd, S., Vandenberghe, L.: Introduction to Applied Linear Algebra. Cambridge University Press, Cambridge (2018)CrossRefGoogle Scholar
- 3.Caughley, G.: Parameters for seasonally breeding populations. Ecology
**48**, 834–839 (1967)CrossRefGoogle Scholar - 4.Laberson, R.H., McKelvey, R., Noon, B.R., Voss, C.: A dynamic analysis of the viability of the northern spotted owl in a fragmented forest environment. J. Conserv. Biol.
**6**, 505–512 (1992) CrossRefGoogle Scholar - 5.Lay, D.C., Lay, S.R., McDonald, J.J.: Linear Algebra and its Applications, 5th edn. Pearson, London (2016)Google Scholar
- 6.Leslie, P.H.: The use of matrices in certain population mathematics. Biometrika
**33**(3), 183–212 (1945)MathSciNetCrossRefGoogle Scholar - 7.Manske, M., Schwarz, C.J., Stobo, W.T.: The estimation of the rate of population change of Grey Seals (Halichoerus grypus) on Sable Island using a Leslie projection matrix with Capture-Recapture data. Unpublished manuscriptGoogle Scholar
- 8.Poole, D.: Linear Algebra, A Modern Introduction. Brooks/Cole, Pacific Grove (2003)Google Scholar
- 9.York, A.E., Hartley, J.R.: Pup production following harvest of female northern fur seals. Can. J. Fish. Aquat. Sci.
**38**, 84–90 (1981)CrossRefGoogle Scholar