List of top Mathematics and Quantitative Ecology Questions on Ecology

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 \(K=9000\), \(\dfrac{dN(t)}{dt}\) is maximized at \(N=\) ______ (Answer in integer)

A bacterium can be approximated as a cylinder with a hemisphere capping each end, as shown in the figure. The cylinder has a height of 1 \(\mu m\) and a diameter of 1 \(\mu m\), so the two end hemispheres share the same radius as the cylinder.

Assuming that the density of the bacterium equals that of water, what is the approximate mass of this bacterium?
Given: density of water \(=10^3\ \text{kg/m}^3\); \(1\ \mu m = 10^{-6}\ m\).
Volume of a cylinder \(= \pi r^2 h\), where \(r\) is the radius and \(h\) is the height of the cylinder.
Volume of a sphere \(= \dfrac{4}{3}\pi r^3\), where \(r\) is the radius of the sphere.

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 \(10^5\)?