I'm having a bit of trouble understanding what the matrix entries mean practically in this problem:
100 kg of a highly toxic substance is spilled into three lakes. The state, t weeks after the accident can be described by x(t):
What do these terms inside the matrix actually mean in practice?
Thanks!
$$x(t+1) = \begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \begin{bmatrix}x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} = \begin{bmatrix}ax_1(t) +bx_2(t)+cx_3(t)\\ dx_1(t) +ex_2(t)+fx_3(t) \\ gx_1(t) +hx_2(t)+ix_3(t) \end{bmatrix}$$
So the amount of pollution in a particular lake is determined by
1) The amount of its own pollution that it retains. This is likely reduced by either the pollution breaking down or leaving the lake.
2) The amount of pollution this lake receives from the other lakes. Water flows between the lakes and takes pollution with it.
So for Lake Silvaplana, the lake retains $0.6$ of its own pollution, but also receives $0.1$ of the pollution from Lake Sils. Of note is that each column in the matrix adds up to $0.8$, which means that $20\%$ of the pollution is lost in each time period, likely through it breaking down or leaving this 3 lake system.