We can express the previous idea in another way, if we plot the natural logarith

Question

We can express the previous idea in another way, if we plot the natural logarithm of N (the logarithm of N to the base e), written ln N against time, it would give us a straight line, with the slope equal to r. See the figure below. In the previous equation, r (the per capita rate of population growth) is analogous to R (the net reproductive rate). In populations with a stable age distribution (i.e. populations in which the proportions of different age classes remain the same from year to year), r = (ln R)/T_e where T_e is the generation time - the time taken for new offspring to grow and have their own offspring. This equation essentially means that, for a continuously breeding population, the net reproductive rate divided by the generation lime gives an approximation of the instantaneous rate of population growth. We can also use the geometric growth model to estimate the doubling time for a population growing at a certain rate. According to the rules of calculus. integral^dN_dv = integral^N or N = N e^n where e is the mathematical constant 2.71828 the base of natural logarithms. If you can rearrange the equation so that the population at a specific time, N_ is divided by the original population size N_ you can figure how long it takes the population to double. First of all. rearrange the equation according to the previous sentence. What should Nt/No equal if the population doubles? So replace your answer that you got in (9) in the equation you came up with in (8). What do you get? Now, lets get the superscript rt by itself. To do this we must multiply both sides by the natural log (In). What do you get? Divide by r on both sides and what is your equation then?

Explanation / Answer

Answer:

Based on the given information, the equation for geometric growth model to estimate the doubling time for a population growing at a certain rate is:

Nt = N0en

Where e = 2.71828 (mathematical constant)

1) Rearranging the equation Nt / N0 = ert

2) What should Nt / N0 equal if the population doubles?

If the population doubles then Nt / N0 = 2

3) Replace answer in equation

Nt / N0 = ert = 2

4) To determine the value of rt, we will take natural log (ln) on both sides

ln(ert)  = ln(2)

rt*ln(e) = ln(2)

rt = 0.6931 (natural log of e = 1)

t = 0.6931 / r

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