Bacterial Growth Calculation: Generations to Reach Target Population
This problem involves calculating the number of generations for a bacterial population to grow exponentially from an initial size to a target size.
Understanding Exponential Growth
Bacteria reproduce through binary fission, where one cell divides into two. This results in exponential growth. The formula relating the final population ($N_t$), initial population ($N_0$), and the number of generations ($n$) is:
$ N_t = N_0 \times 2^n $
Solving for Number of Generations
We are given:
- Initial population, $N_0 = 100$
- Target population, $N_t = 10^5$
We need to find the number of generations, $n$. Substitute the values into the formula:
$ 10^5 = 100 \times 2^n $
Step 1: Isolate the Growth Factor
Divide both sides by the initial population ($N_0 = 100$):
$ \frac{10^5}{100} = 2^n $
$ 1000 = 2^n $
Step 2: Solve for 'n' using Logarithms
To find $n$, we can take the logarithm base 2 of both sides:
$ n = \log_2(1000) $
Step 3: Approximate the Result
We know that $2^{10} = 1024$. Since 1024 is very close to 1000, the number of generations ($n$) is approximately 10.
Therefore, after approximately 10 generations, the population will reach $10^5$.
Important Questions from Population growth curves
The population of whirligig beetles in a lake grows or declines exponentially i.e.
$N(t) = N(0)e^{rt}$
where $N(t)$ is the population size at time $t$, $N(0)$ is the initial population size and $r$ is the per capita rate of population change, occurring only due to birth and death.
A researcher tracks population sizes for a year and finds the following:Time interval Number of beetles at start Number of beetles at end January - March 1000 150 April – June 150 3013 July – September 3013 100 October - December 100 2009 Assuming that the individual birth rates remain constant throughout the year and only death rates are affected, which one or more of the following statements is/are true?
(In your calculations, round off the birth and date rates to two decimal places)- 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 __________
Consider the logistic population growth model, given by $$ \frac{dn}{dt} = rn \left(1 - \frac{n}{k}\right) $$ where $r$ is the intrinsic growth rate, $n$ is the population size and $k$ is the carrying capacity. Which one or more of the following is/are assumption(s) of the model?
