Q. 77 - Calculus

Question

 Use the solution of the logistic model

$\frac{dP}{dt}=rP 1-\frac{P}{K}$

to prove that as t →∞, the population P(t) approaches

the carrying capacity L. Assume that the constant k is positive. 

Step-by-Step Solution

Answer

Proved

Step 1. Given

The given logistic model is $\frac{dP}{dt}=rP 1-\frac{P}{K}$

Step 2. Proof

Observe that the differential equation does not involve the independent variable at all, so solve the differential equation by antidifferentiation method

$\int \frac{dP}{P\left(1-{\displaystyle \frac{P}{K}}\right)}=rdt$

The Integrand on the left hand side needs to be simplified by the use of partial fractions. So, first resolve the fraction in to partial fractions by using cover up rule

$$\begin{gathered} \frac{1}{P\left(1-{\displaystyle \frac{P}{K}}\right)}=\frac{A}{P}+\frac{B}{1-{\displaystyle \frac{P}{K}}} \\ A=1 \\ B=\frac{1}{K} \\ H\mathrm{ence}, \mathrm{Integral} \mathrm{on} \mathrm{the} \mathrm{left} \mathrm{hand} \mathrm{side} \mathrm{comprises} \mathrm{of} \mathrm{two} \mathrm{integrals}. \mathrm{Solve} \mathrm{these} \mathrm{integrals} \mathrm{as} \\ \int \frac{1}{\mathrm{P}}\mathrm{dP}+\frac{1}{\mathrm{K}}\int \frac{\mathrm{dP}}{1-{\displaystyle \frac{\mathrm{P}}{\mathrm{K}}}}=\mathrm{r}\int \mathrm{dt} \\ \ln \left|\mathrm{p}\right|+\mathrm{k} \ln \left|1-\frac{\mathrm{P}}{\mathrm{K}}\right|=\mathrm{rt}+{\mathrm{C}}_1 \\ \mathrm{simplifying} \mathrm{the} \mathrm{equations} \\ \mathrm{P}=\frac{\mathrm{K}}{1+{\mathrm{Ae}}^{-\mathrm{rt}}} \\ \mathrm{Now} \mathrm{take} \mathrm{t}=0, \mathrm{P}={\mathrm{P}}_0 \mathrm{in} \mathrm{the} \mathrm{above} \mathrm{result} \mathrm{and} \mathrm{evaluate} \mathrm{the} \mathrm{constant} \mathrm{A} \\ {\mathrm{P}}_0=\frac{\mathrm{K}}{1+\mathrm{A}} \\ \mathrm{A}=\frac{\mathrm{K}-{\mathrm{P}}_0}{{\mathrm{P}}_0} \\ \mathrm{P}\left(\mathrm{t}\right)=\frac{{\mathrm{KP}}_0}{{\mathrm{P}}_0+(\mathrm{K}-{\mathrm{P}}_0){\mathrm{e}}^{-\mathrm{rt}}} \\ \mathrm{Now} \mathrm{use} \mathrm{the} \mathrm{condition} \mathrm{that} \mathrm{the} \mathrm{growth} \mathrm{rate} \mathrm{constant} \mathrm{k} \mathrm{is} \mathrm{positive} \mathrm{and} \mathrm{take} \mathrm{limit} \mathrm{both} \mathrm{sides} \mathrm{as} t\to \infty \\ \underset{t\to \infty }{\lim }\mathrm{P}\left(\mathrm{t}\right)=\underset{t\to \infty }{\lim }\frac{{\mathrm{KP}}_0}{{\mathrm{P}}_0+(\mathrm{K}-{\mathrm{P}}_0){\mathrm{e}}^{-\mathrm{rt}}} \\ =\frac{L{P}_0}{P_0} \\ =L \end{gathered}$$