A population shows exponential growth of the form $N_t = N_0e^{rt}$ where $N_t$ is the population at time $t$, $N_0$ is the initial population size and $r$ is the rate of increase. If $r = 0.1$, then the doubling time for this population is____________. (Round off to two decimal places.)
Objective: Calculate the doubling time for a population with exponential growth.
Doubling Time Calculation
The population growth is described by the formula $N_t = N_0e^{rt}$.
Doubling time ($t_d$) occurs when the population reaches twice its initial size, meaning $N_t = 2N_0$.
We set the final population to be twice the initial population:
$2N_0 = N_0e^{rt_d}$
Exponential Growth Model Analysis
To find the doubling time, we solve the equation for $t_d$:
- Divide by the initial population $N_0$: $2 = e^{rt_d}$
- Take the natural logarithm of both sides: $\ln(2) = \ln(e^{rt_d})$
- Simplify the right side using logarithm properties ($\ln(e^x) = x$): $\ln(2) = rt_d$
- Isolate the doubling time $t_d$: $t_d = \frac{\ln(2)}{r}$
Growth Rate Substitution
The problem states the rate of increase is $r = 0.1$. Substitute this value into the formula for $t_d$:
$t_d = \frac{\ln(2)}{0.1}$
Using the approximate value $\ln(2) \approx 0.693147$:
$t_d \approx \frac{0.693147}{0.1}$
$t_d \approx 6.93147$
Result and Rounding
Rounding the result to two decimal places gives:
$t_d \approx 6.93$
This value lies between 6.8 and 7.0, consistent with the provided answer range.
Important Questions from Population growth curves
- A flask containing nutrient-rich media is seeded with 100 isogenic bacteria. Assuming that no bacteria die in the flask, after approximately how many generations will the population reach a size of $10^5$?
Population growth of a species can be modelled as $$ \frac{dN(t)}{dt} = rN(t)\left(1 - \frac{N(t)}{K}\right) $$ where $N(t)$ is the population size at time $t$; $r$ is the growth rate; and $K$ is the carrying capacity of the environment.
For $K = 9000$, $\frac{dN(t)}{dt}$ is maximized at $N =$ _____
(Answer in integer)
- The population size at which net recruitment is the highest is also when the greatest amount can be harvested, while ensuring the long-term survival of the population. The amount harvested at this population size is known as
The graphs shown represent the relationship between population size ($N$) and population growth rate ($\frac{dN}{dt}$). Which one of the following growth curves represents a density-dependent population that experiences a strong Allee effect?
Overfishing reduced food availability for sea lions in California, causing a decline in their population size. In 1972, under the US Endangered Species Act, fishing was banned from sea lion foraging areas. Subsequently, the population of sea lions increased in a logistic form as shown in the figure.
The per capita growth rate is highest in the interval __________ and the population growth rate is highest in the interval __________
