A few questions to warm up and check the basics, before the exam-type questions below. They are reshuffled each time, so you can keep retrying.
The following are exam-type questions in the style of the examination paper, with marks at the rates used in the papers. A question totalling fewer than 25 marks would, in the examination, be combined with further parts, often one of the examinable proofs, to make a full 25-mark question. Attempt them in full before reading the worked solutions.
In class we simulated a Moran process for the Hawk-Dove game with payoff matrix
(rows and columns ordered Hawk, Dove), where the fitness of a type in a state with \(v_H\) Hawks and \(v_D\) Doves is \(f_i = (v_i - 1)A_{ii} + \sum_{j \neq i} v_j A_{ij}\).
(a) Describe the two random choices made at each step of a Moran process. [3]
(b) In a population of size \(N = 4\) with two Hawks and two Doves, compute the fitness of a Hawk and of a Dove. [6]
(c) Compute the probability that, at the next step, the number of Hawks increases by one. [7]
(d) Define the fixation probability, and explain why this process has exactly two absorbing states. [5]
(e) Compute the probability that the number of Hawks decreases by one, and hence the ratio \(\gamma = P(\text{decrease})/P(\text{increase})\). What does the value of \(\gamma\) tell you about the direction in which the process tends to drift from this state? [4]
(a) Provide definitions for the following terms:
(b) A single mutant of constant relative fitness \(r\), whose fitness does not depend on the composition of the population, is placed among \(N - 1\) residents of fitness \(1\). Its fixation probability is \(\rho = \dfrac{1 - r^{-1}}{1 - r^{-N}}\).
(i) State the value under neutral drift and explain it. [3]
(ii) Compute \(\rho\) when \(N = 4\) and \(r = 2\). [3]
(iii) Compute \(\rho\) when \(N = 4\) and \(r = \tfrac{1}{2}\). [3]
(iv) Comment on the comparison of the three values, and explain why even a strongly advantageous mutant is far from certain to take over when it starts as a single individual. [5]
(c) Show that as \(N \to \infty\) with \(r > 1\), \(\rho \to 1 - r^{-1}\). [4]
(d) Interpret this limit. [3]
(a) State the formula for the fixation probability \(\rho_i\) of a birth-death process, in terms of the ratios \(\gamma_i = p_{i, i-1}/p_{i, i+1}\). [4]
(b) A Moran process of size \(N = 3\) uses the Hawk-Dove game \(A = \begin{pmatrix} 0 & 3 \\ 1 & 2 \end{pmatrix}\) (first type Hawk). In state \(i\), \(f_H(i) = (i - 1)A_{11} + (N - i)A_{12}\), \(f_D(i) = i A_{21} + (N - i - 1)A_{22}\) and \(\gamma_i = f_D(i)/f_H(i)\).
(i) Compute \(f_H\) and \(f_D\) for \(i = 1\) and \(i = 2\). [3]
(ii) Compute \(\gamma_1\) and \(\gamma_2\). [4]
(iii) Compute the fixation probability \(\rho_1\) of a single Hawk. [4]
(iv) Compute the fixation probability \(\rho_2\) starting from two Hawks. [4]
(c) Interpret the two fixation probabilities, and using the criterion \(\rho_1 > 1/N\) determine whether Hawks are favoured by selection. [4]
Consider a constant-fitness Moran process: a population of \(N\) individuals contains a number of a mutant type, of fitness \(f_M = r\) with \(r > 0\), among residents of fitness \(f_R = 1\). Let \(\rho_i\) be the fixation probability of the mutant type starting from \(i\) mutants.
(a) In state \(i\), write down the transition probabilities \(P_{i \to i+1}\) and \(P_{i \to i-1}\), and hence show that \(\gamma_i = P_{i \to i-1}/P_{i \to i+1} = r^{-1}\) for every \(i\), so the ratio does not depend on the state. [4]
(b) Using the general fixation formula, show that the fixation probability of a single mutant is \(\rho_1 = \dfrac{1 - r^{-1}}{1 - r^{-N}}\) for \(r \neq 1\), and \(\rho_1 = 1/N\) for \(r = 1\). [6]
(c) A mutant is favoured by selection if \(\rho_1 > 1/N\). Writing \(\gamma = r^{-1}\), prove that \(\rho_1 > 1/N\) if and only if \(r > 1\). You may find it helpful to consider \(g(\gamma) = 1 - \gamma^N - N(1 - \gamma)\). [10]
(d) Deduce \(\lim_{N \to \infty} \rho_1\) for \(r > 1\), and explain what it shows about the fate of a single advantageous mutant in a large population, contrasting the role of selection with that of drift. [5]
These exercises are entirely optional: they are extra practice, not extra content. The best way to use an AI assistant is to be its examiner, not its student: checking work is exactly the skill the examination rewards, and the skill you will need to use these tools well. Each exercise gives a transcript in the style of a widely used free AI chatbot answering one of the example questions above. Every transcript contains at least one error. Mark it as an examiner would: find each error, explain what goes wrong, and award a mark out of the marks of the original question. A marker's report for each transcript is on the solutions page.
A student pastes Question 1(b) and (c) into a free AI chatbot, which replies:
Fitnesses: each individual interacts with everyone in the population.
A Hawk meets 2 Hawks and 2 Doves, so f_H = 2(0) + 2(3) = 6; a Dove
meets 2 Doves and 2 Hawks, so f_D = 2(2) + 2(1) = 6.
Probability the Hawks increase: the two types are equally fit, so
selection plays no role here: the process is neutral drift. The number
of Hawks rises when a Hawk reproduces (probability 1/2) and a Dove dies
(probability 2/4), giving (1/2)(1/2) = 1/4.
A student pastes Question 3(b) into a free AI chatbot, which replies:
Using γ1 = 1/2 and γ2 = 2/3, the fixation probabilities are
ρ1 = 1 / (1 + γ1 + γ2) = 1 / (1 + 1/2 + 2/3) = 6/13 ≈ 0.46
ρ2 = (1 + γ1) / (1 + γ1 + γ2) = 9/13 ≈ 0.69
Both exceed 1/N = 1/3, so Hawks are favoured by selection. ✅
You do not need any of this to follow the topic, but the following chapters of the textbook may help if you would like more background:
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