Logistic Model.In Section 3.2 we discussed the logistic equation\(\frac{{{\bf{dp}}}}{{{\bf{dt}}}}{\bf{ = A}}{{\bf{p}}_{\bf{1}}}{\bf{p - A}}{{\bf{p}}^{\bf{2}}}{\bf{,p(0) = }}{{\bf{p}}_{\bf{o}}}\)and its use in modeling population growth. A more general model might involve the equation\(\frac{{{\bf{dp}}}}{{{\bf{dt}}}}{\bf{ = A}}{{\bf{p}}_{\bf{1}}}{\bf{p - A}}{{\bf{p}}^{\bf{r}}}{\bf{,p(0) = }}{{\bf{p}}_{\bf{o}}}\)where r>1. To see the effect of changing the parameter rin (25), take \({{\bf{p}}_{\bf{1}}}\)= 3, A= 1, and \({{\bf{p}}_{\bf{o}}}\)= 1. Then use a numerical scheme such as Runge–Kutta with h= 0.25 to approximate the solution to (25) on the interval\(0 \le {\bf{t}} \le 5\) for r= 1.5, 2, and 3What is the limiting population in each case? For r>1, determine a general formula for the limiting population.

Here\({p_1} = 3\), \(A = 1\), and \({p_o} = 1\) then the equation is:

\(\frac{{dp}}{{dt}} = 3p - {p^r}\)

Suppose\(r > 1\). So,

\(\begin{array}{c}\int {\frac{1}{{3p - {p^r}}}dp} = \int {dt} \\\smallint \frac{1}{{{p^r}\left( {3{p^{1 - r}} - 1} \right)}}dp = t + c\\\int {\frac{1}{{3\left( {1 - r} \right)u}}du} = t + c\\\frac{{\ln \left| u \right|}}{{3\left( {1 - r} \right)}} = t + c\\\frac{{\ln \left| {3{p^{1 - r}} - 1} \right|}}{{3(1 - r)}} = t + c\\\ln \left| {3{p^{1 - r}} - 1} \right| = 3\left( {1 - r} \right)t + c\\{p^{1 - r}} = \frac{{C{e^{3(1 - r)t}} + 1}}{3}\\p = {\left( {\frac{{C{e^{3(1 - r)t}} + 1}}{3}} \right)^{\frac{1}{{1 - r}}}}\end{array}\)

Here \(p\left( 0 \right) = {p_o} = {\rm{ }}1\) then

\(\begin{array}{c}{\bf{1 = }}{\left( {\frac{{{\bf{C + 1}}}}{{\bf{3}}}} \right)^{\frac{{\bf{1}}}{{{\bf{1 - r}}}}}}\\{\bf{C = 2}}\\{\bf{p = }}{\left( {\frac{{{\bf{2}}{{\bf{e}}^{{\bf{3(1 - r)t}}}}{\bf{ + 1}}}}{{\bf{3}}}} \right)^{\frac{{\bf{1}}}{{{\bf{1 - r}}}}}}\end{array}\)

\(\begin{array}{c}\mathop {{\bf{lim}}}\limits_{t \to \infty } {\bf{p(t) = }}\mathop {{\bf{lim}}}\limits_{t \to \infty } {\left( {\frac{{{\bf{2}}{{\bf{e}}^{{\bf{3(1 - r)t}}}}{\bf{ + 1}}}}{{\bf{3}}}} \right)^{\frac{{\bf{1}}}{{{\bf{1 - r}}}}}}\\{\bf{ = }}{\left( {\frac{{2\mathop {{\bf{lim}}}\limits_{t \to \infty } {{\bf{e}}^{{\bf{3(1 - r)t}}}}{\bf{ + }}1}}{3}} \right)^{\frac{{\bf{1}}}{{{\bf{1 - r}}}}}}\\{\bf{ = }}{\left( {\frac{{\bf{1}}}{{\bf{3}}}} \right)^{\frac{{\bf{1}}}{{{\bf{1 - r}}}}}}\\{\bf{ = }}{{\bf{3}}^{\frac{{\bf{1}}}{{{\bf{r - 1}}}}}}\end{array}\)

Apply this method in Mat Lab.

Therefore, the general formula for the limiting population is\({{\bf{3}}^{\frac{{\bf{1}}}{{{\bf{r - 1}}}}}}\).