List of top Mathematics and Quantitative Ecology Questions

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)

The genome of a diploid species has 10 genes. In a population of this species, if each gene has two unique alleles, then the number of possible unique genotypes in this population is ______ (Answer in integer)

Bob is studying the effect of coral and sponge species on reef ecosystems using experiments in artificial square tanks. In each tank, he places 3 species of corals and 2 species of sponges. If there are 6 species of corals and 5 species of sponges to choose from, the minimum number of tanks required to test all combinations of 3 coral and 2 sponge species is ______ (Answer in integer)

Students M and N sampled body lengths of two populations of a fish species using the same study design. The table below summarizes their data as mean \(\pm\) standard error.

Body length (cm)	Population I	Population II
Student M	\(8.0 \pm 1.2\)	\(6.2 \pm 0.7\)
Student N	\(10.0 \pm 0.6\)	\(5.1 \pm 0.3\)

Given that the data are normally distributed, when comparing the means of the populations using a t-test, which student will find a lower p-value?

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.

In a population, the frequency of an allele changes with time as shown in the table below.

Year (x)	0	10	20	30	40
Allele frequency (y)	0.1	0.2	0.3	0.4	0.5

Which one of the following describes how allele frequency changes with year?

The scatter plot below shows annual rainfall (in mm) on the x axis and the number of plant species recorded at a site on the y axis. Each point is one location, and the points trace out a clear increasing pattern that looks close to a straight line as rainfall rises.

Based on the pattern in the graph, which one of the following statistical methods would be most appropriate to model the relationship between annual rainfall and the number of plant species?

The table below gives the abundance of species 1 to 10 in three communities, P, Q, and R, each with a total of 100 individuals. Simpson's diversity index for a community is
\[ D = 1 - \sum_i p_i^2, \] where \(p_i\) is the proportion of individuals in the community belonging to the \(i^{th}\) species.

Species	Community P	Community Q	Community R
1	69	45	70
2	18	43	21
3	5	4	4
4	4	3	5
5	1	1	0
6	1	1	0
7	0	0	1
8	1	1	0
9	0	0	1
10	1	1	0

Which one of the following statements is true about the comparison of Simpson's index between these communities?

In a population of birds, the proportion of blue individuals is 0.4 and the proportion of males is 0.3. Assuming that colour and sex are independent of each other, the probability that a randomly sampled individual from this population is a blue male is ______ (Round off to two decimal places)

Assume that the human haploid genome is \(3 \times 10^{9}\) base pairs long. The mutation rate, when DNA is copied, is \(5 \times 10^{-11}\) per base pair per replication. Based on these numbers, the expected number of mutations that take place per replication of the human haploid genome is _______ (Round off to two decimal places)

In a null hypothesis significance test, the significance level \(\alpha\) was set to 0.01. For which one or more of the following p values would the null hypothesis be rejected?

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 \(10^5\)?

In a diploid population of \(N\) individuals, the probability of fixation of a new neutral mutation (assuming no other mutation occurs) is _________.