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
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?
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?
A population of unicorns is growing over time, but its rate of growth is declining. Which of the following graphs best represents this pattern of growth?
