Analyse long-run behaviour of a Leslie model
An orchard model records expected numbers of trees in successive production stages. The recorded quantities are expected counts, so fractional values may occur in calculations
All modelled items are counted. A first-row entry rate means the expected number entering the first stage per current item in one year. Survival entries are proportions for moving to the next stage
The stages, in order, are young trees, established trees, fruiting trees, and seed-producing trees. One application of the Leslie matrix represents one year
\[L = \begin{bmatrix} \frac{3}{5} & 0 & 0 & \frac{18}{5} \\ \frac{4}{5} & 0 & 0 & 0 \\ 0 & \frac{3}{5} & 0 & 0 \\ 0 & 0 & \frac{3}{5} & 0 \end{bmatrix}\]
\[S_0 = \begin{bmatrix} 576 \\ 384 \\ 192 \\ 96 \end{bmatrix}\]
Use \(S_{n+1} = L S_n\). The displayed probe periods are 0, 1, 2. Give finite ratios or percentage changes to 1 decimal place where requested