Consider a population that shows logistic growth of the form
$ \frac{dN}{dt} = rN(1 - \frac{N}{K}) $ where $ \frac{dN}{dt} $ is the population growth rate, $r$ is the instantaneous rate of increase, $K$ is the carrying capacity and $N$ is the population size.
For such a population ($N> 0$), which one of the following graphs shows the correct relationship between per capita growth rate ($ \frac{1}{N} \frac{dN}{dt} $) on the y-axis, and population size ($N$) on the x-axis?
To identify the correct graph, we must derive the expression for the per capita growth rate from the given logistic growth equation and analyze its mathematical form.
1. Derivation of the Equation
The standard logistic growth equation is given by:
$$\frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right)$$
The question asks for the relationship between the per capita growth rate \(\left( \frac{1}{N} \frac{dN}{dt} \right)\) and the population size \((N)\). To find this, we divide both sides of the equation by \(N\):
$$\frac{1}{N} \frac{dN}{dt} = r \left(1 - \frac{N}{K}\right)$$
Expanding the right side, we get:
$$\frac{1}{N} \frac{dN}{dt} = r - \left( \frac{r}{K} \right) N$$
2. Mathematical Analysis
The resulting equation is in the form of a linear equation, \(y = mx + c\), where:
- \(y = \frac{1}{N} \frac{dN}{dt}\) (the variable on the vertical axis).
- \(x = N\) (the variable on the horizontal axis).
- \(c = r\) (the \(y\)-intercept, representing the maximum per capita growth rate when population is near zero).
- \(m = -\frac{r}{K}\) (the negative slope of the line).
3. Evaluating the Graphs
- Graph (i): Shows a straight line with a negative slope. This perfectly matches our derived linear equation where the growth rate decreases linearly as \(N\) approaches \(K\).
- Graph (ii): Shows a non-linear increase. Incorrect.
- Graph (iii): Shows a constant rate. This would represent exponential growth, where density has no effect. Incorrect.
- Graph (iv): Shows a parabola. This represents the total growth rate \(\left( \frac{dN}{dt} \right)\) vs \(N\), not the per capita rate. Incorrect.
Conclusion
Since the per capita growth rate decreases linearly with increasing population size in a logistic model, Graph (i) is the correct representation.
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 __________
