(Solved): 4.) Construct a graph depicting the general population growth curve that would result from the examp ...

4.) Construct a graph depicting the general population growth curve that would result from the example given in the passage. What would the graph look like for the case of $r<1$ ? (For this question, make sure to research and look for examples to help you with your graph. Make sure you all construct your group's own graph.) In the early isoo's, the reverend thomas Malthus published some of the carliest and most influential work on population growth. He recognized that, left unchecked, populations have the potential to grow very large in a very short period of time. This work had a significant impact on the thinking of Charles Darwin, especially as it related to his ideas on natural selection and "the struggle for existence," Malthus developed a simple mathematical equation, known as the exponential model, in which growth is based on a constant rate of compound interest. This equation, written as ${\mathrm{N}}_{\mathrm{t}}={\mathrm{N}}_0{e}^{\mathrm{rt}}$, allows for the prediction of future population size $\left({\mathrm{N}}_2\right)$, given current population size $\left({\mathrm{N}}_0\right)$ and the per-capita exponential growth nate ( $r$, sometimes called the Malthusian parameter). A very important assumption of this model is that nesoueres are unlimited. Under these conditions, the growth rate of a population can be calculated as $\mathrm{dN}/\mathrm{dt}=\mathrm{rN}$. Note that $\mathrm{dN}/\mathrm{dt}$ is directly proportional to population size. Keep Malthus's equation in mind as you consider the following seenario: The year is 2376 and you are the owner of an inter-galactic company specializing in providing biological solutions to detoxify waste. You have just received an order from Alpha Centurion for a population of the waste-detoxifying microbe Detoxiffcatium complcteinm that is equivalent to the volume of the earth. Thar's right, this is a big project; after all, Alpha Centurion is a very large planet, and its inhabitants generate more waste than anyone else in the galaxy. This is a very lucrative order, but you have only a week to deliver or else the offer will be withdrawn and given to your competifor. To make matters worse, you have only a single cell of $D$. completeitum in your microbial inventory. Under optimal conditions, this mierobe will divide every 20 minutes. You have in storage a hollow carbon shell that holds the equivalent volume of the earth, $1.09\times 10^{21}{\mathrm{m}}^3$, and you have the necessary nutrients in stock. A single microbe occupies a volume of $1\times 10^{?18}{\mathrm{m}}^3$, no you estimate the number of microbes needed to fill the shell as $\left(1.09\times 10^{21}{\mathrm{m}}^3\right)\left(1\times 10^{?18}{\mathrm{m}}^3\right)=1.10\times 10^{39}$ IIIII The magnitude of that number makes you extremely nervous. After all, you have only a weed to fill this order using a single bacterial cell with which to start. You steady your nerves and sit down to make some more calculations. Since your microbe reproduces by binary fission (in which one cell gives rise to two), you estimate its Malthusian growth rate, $r$, to be $\ln (2)=0.69$. Your initial population size is ${\mathrm{N}}_0=1$, and the final population size is ${\mathrm{N}}_1=1.10\times 10^{39}$. You are now in a position to solve for $S_{\text{, }}$ Which is the number of rounds of cell division that will be necessary to accomplish the task. After a little algebra, you have the cquation you need: $t=\left(\ln \left(N_j\right)?\ln \left(N_0\right)\right)r$. You solve the equation, remembering that bs in units of 20 minutes, then calemly reach for your laser-phose to contact Alpha Centurion to discuss their order.