Chapter 2.1, Problem 16P

Program Plan Intro

Program Description: Purpose of problem is to calculate the number of months takes for $P\left(t\right)$ to reach 95% of the limiting population ( $M$ ).

Explanation of Solution

Given information:

The logistic equation is $\frac{dP}{dt}=aP-b{P}^2$ .

Here, $P\left(t\right)$ is a rabbit population, $B=aP$ is the time rate at which births occur and $D=b{P}^2$ is the rate at which deaths occur.

Initial population is 120 with rate of 8 births per month and 6 deaths per month.

  $\begin{array}{c}{P}_0=120\\ D_0=6\\ B_0=8\end{array}$

Take initial time as 0.

Explanation:

The solution of the logistic differential equation is shown below.

  $P\left(t\right)=\frac{M{P}_0}{P_0+\left(M-P_0\right)e^{-kMt}}$

Here, $k=\frac{D_0}{P_0{}^2}$ and $M=\frac{B_0{P}_0}{D_0}$ .

Obtain the maximum capacity of the system $M$ as follows:

  $\begin{array}{c}M=\frac{8\cdot 120}{6}\\ =160\end{array}$

Obtain the value of $k$ as follows:

  $\begin{array}{c}k=\frac{D_0}{P_0{}^2}\\ =\frac{6}{{120}^2}\\ =\frac{6}{14400}\\ =\frac{1}{2400}\end{array}$

Now, substitute the known values $k$ and $M$ into the equation $P\left(t\right)=\frac{M{P}_0}{P_0+\left(M-P_0\right)e^{-kMt}}$ as,

  $\begin{array}{c}P\left(t\right)=\frac{160\cdot 120}{120+\left(160-120\right)e^{-\frac{160t}{2400}}}\\ =\frac{19200}{120+40e^{-\frac{t}{15}}}\end{array}$

To obtain the time at which population reaches to 95% of maximum population, substitute $P\left(t\right)=0.95M$ in the above equation and solve for $t$ .

  $\begin{array}{c}0.96\cdot 160=\frac{19200}{120+40e^{-\frac{t}{15}}}\\ 152=\frac{19200}{120+40e^{-\frac{t}{15}}}\\ 120+40e^{-\frac{t}{15}}=\frac{19200}{152}\\ 120+40e^{-\frac{t}{15}}=126.3157\end{array}$

Further, solve the equation as follows:

  $\begin{array}{c}40e^{-\frac{t}{15}}=6.31578\\ e^{-\frac{t}{15}}=0.15789\\ -\frac{t}{15}=\ln 0.158\\ t=-15\cdot \left(-1.84582\right)\\ t\approx 27.69\end{array}$

Conclusion:

Therefore, the number of months takes for $P\left(t\right)$ to reach 95% of the limiting population $M$ is approximately 27.69 months.