Environmental Engineering Reference
In-Depth Information
Chapter 9
Leslie Matrices
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Dagmar S
ondgerath
Abstract This chapter introduces matrix models - used to describe the dynamics of
populations classified by age or other criteria like size or stage. It will be shown how
characteristic values of the Leslie matrix, i.e. eigenvalues and eigenvectors, are
used to determine the asymptotic behaviour of the population. Elasticity analysis
deals with the effects of small parameter changes on population growth. As a result,
values for the relative importance of specific life history parameters for the popula-
tion dynamics are given. For example, these values can be used in conservation to
identify those parts of an organism's life history where management methods
should focus. Finally, an extended Leslie model for populations with both age
and stage structure will be introduced and used to forecast the effects of climate
change on the voltinism and range of occurrence of a dragonfly species.
9.1
Introduction
Matrix models are used to simulate structured populations. The origin of these
models dates back to a paper by Leslie (1945). Originally, they were used for
demographic purposes; i.e. to describe the development of human populations. For
this purpose the population is divided into classes according to age. With an
appropriate projection matrix composed of age-specific fertility, survival rates
and a given initial distribution, it is possible to project the age distribution for
every future time point. Since development, these models have been modified in
multiple ways and have been applied in many fields.
For plant life cycles the vital rates cannot be regarded as functions of age
because reproduction strongly depends on size and more complex life cycles have
to be regarded (Caswell 1986). Therefore, stages instead of age are usually
incorporated in the model to divide the population into classes. A frequently cited
