91Ó°ÊÓ

In her work as a population biologist for Idaho Fish and Game, Leila is assigned the task of modeling a population p(t) of prairie dogs that ranchers in the eastern part of the state are complaining about. She uses the so-called logistic growth model, which quickly results in an expression of the form

In her work as a population biologist for Idaho Fish and Game, Leila is assigned the task of modeling a population p(t) of prairie dogs that ranchers in the eastern part of the state are complaining about. She uses the so-called logistic growth model, which quickly results in an expression of the form

In her work as a population biologist for Idaho Fish and Game, Leila is assigned the task of modeling a population p(t) of prairie dogs that ranchers in the eastern part of the state are complaining about. She uses the so-called logistic growth model, which quickly results in an expression of the form

$r\left(t\right)={∫}_{p_0}^{p\left(t\right)}\frac{1}{q\left(1-{\displaystyle \frac{q}{K}}\right)}dq$

where K represents the carrying capacity of the land where the prairie dogs live, r is the rate of reproduction, and $p_0$is the initial population of the prairie dogs. Use the Fundamental Theorem of Calculus to calculate r(t) and then use your answer to solve for p(t) in terms of r(t), assuming that $p_0$and p(t) both lie between 0 and K.

Step by step solution

01

Step 1. Given information

Logistic growth model is$r\left(t\right)={∫}_{p_0}^{p\left(t\right)}\frac{1}{q\left(1-{\displaystyle \frac{q}{K}}\right)}dq$

02

Step 2. Explanation

$\begin{array}{rcl}r\left(t\right)& =& {∫}_{p_0}^{p\left(t\right)}\frac{K}{q\left(K-q\right)}dq\\ & =& {∫}_{p_0}^{p\left(t\right)}\frac{1}{q}dq-{∫}_{p_0}^{p\left(t\right)}\frac{1}{q-K}dq\\ & =& {\left[\ln q\right]}_{p_0}^{p\left(t\right)}-{\left[\ln \left(q-K\right)\right]}_{p_0}^{p\left(t\right)}\\ & =& \ln \left|\frac{p\left(t\right)\left(K-p_0\right)}{p_0\left(K-p\left(t\right)\right)}\right|\end{array}$

Take exponential on both sides.

$\begin{array}{rcl}{e}^{r\left(t\right)}& =& \left|\frac{p\left(t\right)\left(K-p_0\right)}{p_0\left(K-p\left(t\right)\right)}\right|\\ p\left(t\right)\left[K-p_0+p_0{e}^{r\left(t\right)}\right]& =& p_{0}K{e}^{r\left(t\right)}\\ p\left(t\right)& =& \frac{p_{0}K{e}^{r\left(t\right)}}{K-p_0+p_0{e}^{r\left(t\right)}}\\ & & \end{array}$

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