Understanding Population Growth Dynamics
The problem describes population growth using the logistic differential equation:
$ \frac{dn}{dt} = rn \left(1 - \frac{n}{K}\right) $
Here, $n$ is the population density, $r$ is the intrinsic growth rate, and $K$ is the carrying capacity. The equation given is $dn/dt = rn(1-n/1000)$, so the carrying capacity $K = 1000$. We need to find the population density ($n$) where the growth rate ($dn/dt$) is maximum.
Finding Maximum Growth Rate
Let the growth rate function be $G(n) = dn/dt$.
$ G(n) = rn \left(1 - \frac{n}{1000}\right) = rn - \frac{r}{1000} n^2 $
To find the maximum value of $G(n)$, we take the derivative with respect to $n$ and set it equal to zero.
$ \frac{dG}{dn} = \frac{d}{dn} \left( rn - \frac{r}{1000} n^2 \right) $
$ \frac{dG}{dn} = r - \frac{r}{1000} (2n) = r - \frac{2rn}{1000} $
Set the derivative to zero to find critical points:
$ r - \frac{2rn}{1000} = 0 $
Since $r \ne 0$, we can simplify:
$ r = \frac{2rn}{1000} $
$ 1 = \frac{2n}{1000} $
$ 2n = 1000 $
$ n = \frac{1000}{2} $
$ n = 500 $
Conclusion on Population Density
The second derivative test confirms this is a maximum:
$ \frac{d^2G}{dn^2} = \frac{d}{dn} \left( r - \frac{2rn}{1000} \right) = -\frac{2r}{1000} $
Since $r > 0$, the second derivative is negative, confirming that $n = 500$ yields the maximum growth rate. The growth rate is maximum at a population density of 500.
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 __________
