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 __________
The figure displays a population growing in a logistic manner, modeled by the S-shaped curve. We need to identify the intervals where the per capita growth rate and the population growth rate are highest.
Definitions of Growth Rates
- Population Growth Rate ($\frac{dN}{dt}$): This is the absolute change in population size over time, represented by the slope of the population curve ($N$ vs. time).
- Per Capita Growth Rate ($\frac{1}{N} \frac{dN}{dt}$): This is the growth rate relative to the current population size ($N$). In the logistic model, this is expressed as $r(1 - N/K)$, where $r$ is the intrinsic rate of increase and $K$ is the carrying capacity.
Analysis of the Logistic Curve ($K \approx 105$)
1. Population Growth Rate ($\frac{dN}{dt}$)
The population growth rate is maximized where the slope of the curve is steepest. In the logistic model, the slope is maximum when the population size ($N$) is exactly half the carrying capacity ($K$): $N = K/2$.
Since $K \approx 105$ (thousand), $K/2 \approx 52.5$ (thousand).
Looking at the curve, the point of steepest inflection occurs centrally in the rapidly rising phase, which corresponds to **Interval II**.
Thus, the population growth rate is highest in **Interval II**.
2. Per Capita Growth Rate ($\frac{1}{N} \frac{dN}{dt}$)
The per capita growth rate is $r(1 - N/K)$. Since $r$ and $K$ are positive constants, this rate is highest when $N$ is smallest.
The smallest population size $N$ occurs at the beginning of the growth phase shown, which is within **Interval I** (where the population is still low, around 15 thousand).
Thus, the per capita growth rate is highest in **Interval I**.
Conclusion
The structure requested is: (Per capita growth rate is highest in the interval) , (population growth rate is highest in the interval).
The corresponding intervals are: **I, II**.
This matches Option 1.
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?
- A population grows as per the equation $dn/dt = rn (1-n/1000)$ where $n$ is the population density, $r$ is the intrinsic growth rate and 1000 is the carrying capacity. The growth rate of the population is maximum at a population density of ________
