To determine which graph represents the relationship between population size (N) and population growth rate (\(dN/dt\)) for a population showing exponential growth, we need to understand the principles of exponential growth in population ecology.
Exponential growth occurs when the resources in the environment are unlimited and conditions do not restrict the population's growth. This growth can be described by the exponential growth equation:
\(dN/dt = rN\)
Where:
- \(dN/dt\) is the population growth rate.
- \(r\) is the intrinsic rate of natural increase.
- \(N\) is the population size.
This equation implies that the population growth rate (\(dN/dt\)) is directly proportional to the population size (N). As the population size increases, the growth rate also increases.
In graphical terms, this relationship is depicted as a straight line that passes through the origin and shows a positive slope. This is because as the population size increases, the growth rate increases linearly with it.
Now, let's evaluate the options:
- : This appears to show no relationship or possibly a plateau, which does not correspond to exponential growth.
- : This graph shows a straight line passing through the origin with a positive slope, indicating a direct proportional relationship between N and \(dN/dt\). This is characteristic of exponential growth.
- : This graph likely represents logistic growth, where the growth rate eventually decreases as the population approaches carrying capacity.
- : This graph is not consistent with the direct linear relationship required for exponential growth.
Therefore, the correct graph that represents the relationship between population size (N) and population growth rate (\(dN/dt\)) for a population showing exponential growth is: 
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?
