# Find an exponential function of the form P(t)=P_0n^{frac{t}{T}} that models the situation, and then find the equivalent exponential model of the form P(t)=P_0e^{rt} 1) Doubling time of 25 weeks, initial population of 1300 Question
Exponential models Find an exponential function of the form $$P(t)=P_0n^{\frac{t}{T}}$$ that models the situation, and then find the equivalent exponential model of the form $$P(t)=P_0e^{rt}$$
1) Doubling time of 25 weeks, initial population of 1300 2020-12-14
Given: Initial population $$=P_0=1300$$ at t=0
Population $$=2\timesP_0$$ at t=26 week
Use the given model
$$P(t)=P_0n^{\frac{t}{T}}$$
Substitute $$P_0=1300$$
$$P(t)=1300n^{\frac{t}{T}}$$
Substitute $$P(t)=2\times1300$$ and $$t=25$$ weeks
$$2\times1300=1300n^{\frac{25}{T}}$$
$$n^{\frac{25}{T}}=\frac{2\times1300}{1300}$$
$$n^{\frac{25}{T}}=2$$
This equation holds true only if n=2 and T=25
Therefore, the possible model for the given situation is $$P(t)=1300(2)^{\frac{t}{25}}$$ where t is in weeks and P(t) is the population at any time t.
Since
The equation $$P(t)=P_0n^{\frac{t}{T}}$$ is equivalent to the equation $$P(t)=P_0e^{et}$$ thherefore
$$P_0n^{\frac{t}{T}}=P_0e^{rt}$$
$$n^{\frac{t}{T}}=e^{rt}$$
Take ln both sides of the equation
$$\ln(n)^{\frac{t}{T}}=\ln e^{rt}$$
$$\frac{t}{T}\ln(n)=rt$$
$$\frac{1}{T}\ln(n)=r$$
Here, substitute the value if n=2 and T=25
$$\frac{1}{25}\ln(2)=r$$
$$r=0.0277$$
Hence, the required model is $$P(t)=1300e^{0.0277t}$$ where t is in weeks.

### Relevant Questions Find an exponential function of the form $$\displaystyle{P}{\left({t}\right)}={P}_{{0}}{n}^{{{\frac{{{t}}}{{{T}}}}}}$$ that models the situation, and then find the equivalent exponential model of the form $$\displaystyle{P}{\left({t}\right)}={P}_{{0}}{e}^{{{r}{t}}}$$
1) Doubling time of 25 weeks, initial population of 1300 Several models have been proposed to explain the diversification of life during geological periods. According to Benton (1997), The diversification of marine families in the past 600 million years (Myr) appears to have followed two or three logistic curves, with equilibrium levels that lasted for up to 200 Myr. In contrast, continental organisms clearly show an exponential pattern of diversification, and although it is not clear whether the empirical diversification patterns are real or are artifacts of a poor fossil record, the latter explanation seems unlikely. In this problem, we will investigate three models fordiversification. They are analogous to models for populationgrowth, however, the quantities involved have a differentinterpretation. We denote by N(t) the diversification function,which counts the number of taxa as a function of time, and by rthe intrinsic rate of diversification.
(a) (Exponential Model) This model is described by $$\displaystyle{\frac{{{d}{N}}}{{{\left.{d}{t}\right.}}}}={r}_{{{e}}}{N}\ {\left({8.86}\right)}.$$ Solve (8.86) with the initial condition N(0) at time 0, and show that $$\displaystyle{r}_{{{e}}}$$ can be estimated from $$\displaystyle{r}_{{{e}}}={\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{N}{\left({t}\right)}}}{{{N}{\left({0}\right)}}}}\right]}\ {\left({8.87}\right)}$$
(b) (Logistic Growth) This model is described by $$\displaystyle{\frac{{{d}{N}}}{{{\left.{d}{t}\right.}}}}={r}_{{{l}}}{N}\ {\left({1}\ -\ {\frac{{{N}}}{{{K}}}}\right)}\ {\left({8.88}\right)}$$ where K is the equilibrium value. Solve (8.88) with the initial condition N(0) at time 0, and show that $$\displaystyle{r}_{{{l}}}$$ can be estimated from $$\displaystyle{r}_{{{l}}}={\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{K}\ -\ {N}{\left({0}\right)}}}{{{N}{\left({0}\right)}}}}\right]}\ +\ {\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{N}{\left({t}\right)}}}{{{K}\ -\ {N}{\left({t}\right)}}}}\right]}\ {\left({8.89}\right)}$$ for $$\displaystyle{N}{\left({t}\right)}\ {<}\ {K}.$$
(c) Assume that $$\displaystyle{N}{\left({0}\right)}={1}$$ and $$\displaystyle{N}{\left({10}\right)}={1000}.$$ Estimate $$\displaystyle{r}_{{{e}}}$$ and $$\displaystyle{r}_{{{l}}}$$ for both $$\displaystyle{K}={1001}$$ and $$\displaystyle{K}={10000}.$$
(d) Use your answer in (c) to explain the following quote from Stanley (1979): There must be a general tendency for calculated values of $$\displaystyle{\left[{r}\right]}$$ to represent underestimates of exponential rates,because some radiation will have followed distinctly sigmoid paths during the interval evaluated.
(e) Explain why the exponential model is a good approximation to the logistic model when $$\displaystyle\frac{{N}}{{K}}$$ is small compared with 1. The population of a region is growing exponentially. There were 10 million people in 1980 (when t=0) and 75 million people in 1990. Find an exponential model for the population (in millions of people) at any time tt, in years after 1980.
P(t)=?
What population do you predict for the year 2000?
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