Leslie matrix

The Leslie matrix (also known as the Leslie model ) is a mathematical model for analyzing population growth that is used in the field of theoretical ecology to describe populations . It was developed by PH Leslie and is one of the best-known methods for describing population growth, referring to the individual age groups.

In ecology, it describes the changes in an organism population over a certain period of time. In a Leslie model, the population is divided into groups or into age groups and life stages. In order to determine the next generation of a population, one forms the product of the Leslie matrix and a vector describing the previous population .

To set up a matrix , the following information about the population must be available:

${\ displaystyle n_ {k}}$ : the number of individuals in the -th age group ${\ displaystyle k}$
${\ displaystyle s_ {j}}$ : the proportion of individuals who transition from the -th to the -th age group (survive) ${\ displaystyle j}$ ${\ displaystyle (j + 1)}$
${\ displaystyle f_ {k}}$ : the birth rate in the -th age group ${\ displaystyle k}$

Exist age groups (of up to be indexed), the formed with designated population vector at the time from the preceding by the following matrix-vector product: ${\ displaystyle \ nu}$ ${\ displaystyle 0}$ ${\ displaystyle \ nu -1}$ ${\ displaystyle \ mathbf {n} _ {t + 1}}$ ${\ displaystyle t + 1}$ ${\ displaystyle \ mathbf {n} _ {t}}$

{\ displaystyle {\ begin {pmatrix} n_ {0} \\ n_ {1} \\ n_ {2} \\ n_ {3} \\\ vdots \\ n _ {\ nu -1} \\\ end {pmatrix }} _ {t + 1} = {\ begin {pmatrix} f_ {0} & f_ {1} & f_ {2} & \ ldots & f _ {\ nu -2} & f _ {\ nu -1} \\ s_ {0} & 0 & 0 & \ ldots & 0 & 0 \\ 0 & s_ {1} & 0 & \ ldots & 0 & 0 \\ 0 & 0 & s_ {2} & \ ldots & 0 & 0 \\\ vdots & \ vdots & \ vdots & \ ddots & \ vdots & \ vdots \\ {0 & 0 & 0 & \ ldots_ \ nu -2} & 0 \ end {pmatrix}} \ cdot {\ begin {pmatrix} n_ {0} \\ n_ {1} \\ n_ {2} \\ n_ {3} \\\ vdots \\ n_ { \ nu -1} \ end {pmatrix}} _ {t}}

For this one also writes , where denotes the Leslie matrix. ${\ displaystyle \ mathbf {n} _ {t + 1} = \ mathbf {L} \ cdot \ mathbf {n} _ {t}}$ ${\ displaystyle \ mathbf {L}}$

dynamics

In order to be able to determine the dynamics of the population under consideration, one considers the eigenvectors and associated eigenvalues of the matrix. If the absolute greatest eigenvalue is unambiguous, then its associated eigenvector reflects the age structure that appears after a few generations. The age structure converges towards this vector. If the largest eigenvalue in terms of amount is not clear, oscillations in the age structure can occur.

literature

Nicholas F. Britton: Essential Mathematical Biology . 3. printing. Springer, London et al. 2005, ISBN 1-85233-536-X , (Springer undergraduate mathematics series) .