(Solved): 9.1 A simple population model. We consider a certain population of fish (say) each (yearly) season. ...

9.1 A simple population model. We consider a certain population of fish (say) each (yearly) season. $x(t)?$ ${\mathbf{R}}^3$ will describe the population of fish at year $t?\mathbf{Z}$, as follows: - $x_1(t)$ denotes the number of fish less than one year old - $x_2(t)$ denotes the number of fish between one and two years old - $x_3(t)$ denotes the number of fish between two and three years (We will ignore the fact that these numbers are integers.) The population evolves from year $t$ to year $t+1$ as follows. - The number of fish less than one year old in the next year $(t+1)$ is equal to the total number of offspring born during the current year. Fish that are less than one year old in the current year $(t)$ bear no offspring. Fish that are between one and two years old in the current year $(t)$ bear an average of 2 offspring each. Fish that are between two and three years old in the current year $(t)$ bear an average of 1 offspring each. - $40\%$ of the fish less than one year old in the current year $(t)$ die; the remaining $60\%$ live on to be between one and two years old in the next year $(t+1)$. fish, and $50\%$ of the old fish. - $30\%$ of the one-to-two year old fish in the current year die, and $70\%$ live on to be two-to-three year old fish in the next year. - All of the two-to-three year old fish in the current year die. Express the population dynamics as an autonomous linear system with state $x(t),i,?$. in the form $x(t+1)=Ax(t)$. Remark: this example is silly, but more sophisticated population dynamics models are very useful and widely used.