Assume that two colonies each have $700$ members at time $t = 0$ and that each evolves with a constant relative birth rate $k = r_{b} - r_{d}$. For colony 1, assume that individuals migrate into the colony at a rate of $70$ individuals per unit time. Assume that this immigration occurs for $0 \leq t \leq 1$ and ceases thereafter. For colony 2, assume that a similar migration pattern occurs but is delayed by one unit of time; that is, individuals immigrate at a rate of $70$ individuals per unit time, $1 \leq t \leq 2$. Suppose we are interested in comparing the evolution of these two populations over the time interval $0 \leq t \leq 2$. The initial value problems governing the two populations are

$$\begin{array}{lllll} \displaystyle \frac{dP_1}{dt} = k P_1 + M_1(t), && P_1(0) = 700, && M_1(t) = \left\lbrace \begin{array}{rcl} 70, && 0 \leq t \leq 1, \\ 0, && 1 < t \leq 2. \\ \end{array} \right. \\ \\ \displaystyle \frac{dP_2}{dt} = k P_2 + M_2(t), && P_2(0) = 700, && M_2(t) = \left\lbrace \begin{array}{rcl} 0, && 0 \leq t < 1, \\ 70, && 1 \leq t \leq 2. \\ \end{array} \right. \end{array}$$

1. Solve both problems to find $P_1$ and $P_2$ at time $t = 2$.
   $P_1(2) =$
   $P_2(2) =$
   
   
2. Show that $\displaystyle P_1(2) - P_2(2) = \frac{70}{k} \left( e^k - 1 \right)^2$.
   If $k > 0$, which population is larger at time $t = 2$? Choose colony 1 colony 2 cannot be determined
   If $k < 0$, which population is larger at time $t = 2$? Choose colony 1 colony 2 cannot be determined
   
   
3. Suppose that there is a fixed number of individuals that can be introduced into a population at any time through migration and that the objective is to maximize the population at some fixed future time. Do the calculations performed in this problem suggest a strategy (based on the relative birth rate) for accomplishing this?