# Leslie matrix model : population transition ( by python program)¶

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## Analysis of the Leslie matrix model¶

Input $\bf L = \begin{bmatrix} 0 & 1 & 2 & 0 \\ 0.5 & 0 & 0 & 0 \\ 0 & 0.6 & 0 & 0 \\ 0 & 0 & 0.1 & 0\end{bmatrix}$, $\; {\bf n}(0) = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0\end{bmatrix}$

Put the followings,

Calculate ${\bf n}(1)= {\bf L\: n}(0)$

$$\begin{pmatrix} 0 & 1 & 2 & 0 \\ 0.5 & 0 & 0 & 0 \\ 0 & 0.6 & 0 & 0 \\ 0 & 0 & 0.1 & 0\end{pmatrix} \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0\end{pmatrix}$$

Using ${\bf n}(1) = \begin{pmatrix} 1 \\ 0 \\ 0.6 \\ 0 \end{pmatrix}$ 、calculate ${\bf n}(2) = {\bf L}\,{\bf n}(1)$

Using ${\bf n}(0) = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}$ 、calculate ${\bf n}(2) = {\bf L}{\bf L} \,{\bf n}(0)$

${\bf L}{\bf L}$ is equibalint to ${\bf L}^2$

Compute ${\bf L}^t$ with arbitrary exponent $t$. For example $t=4$, ${\bf L}^4$.

Compute ${\bf n}(t)={\bf L}^{t}{\bf n}(0), \;\; t=0, \cdots$, and plot population number of each age.