Problems on Evolutionary dynamics Doctoral Programme in Physics José A. Cuesta Lausanne, June 10 13, 2014
Replication 1. Consider the Galton-Watson process defined by the offspring distribution p 0 = 1 where 0 < p < 1 and 0 < b < 1 p. (a) Obtain the generating function f (s) = j=0 p js j. b 1 p, p n = bp j 1, j N, (b) Calculate the mean of this distribution, m, and the extinction probability q for m < 1, m = 1 and m > 1. (c) Prove that for arbitrary s, u and v, (d) Using this identity show that f (s) f (u) f (s) f (v) = s u 1 pv s v 1 pu. f (s) q f (s) 1 = m 1 s q s 1. (e) Use this equation to obtain the generating function of the Galton-Watson process F n (s). Write it in the form F n (s) = 1 a n + b ns 1 c n s. (f) Obtain the probability distribution of extinction times. What can you deduce about the occurrence of extinctions from the behavior of this distribution? HINT: Once obtained m as a function of p and b, it will prove convenient to eliminate b in terms of m in the expression for f (s). 2. (Percolation) A Cayley tree is an infinite graph in which any given node has z neighbors, each of which has z 1 new neighbors, and so on. It is a tree because there are no loops. The site percolation problem on a graph amounts to tagging each node of the graph with probability p, independently of each other. The percolation threshold is the value p c such that if p > p c there is an infinitely large cluster with probability 1. Another important magnitude related to percolation is P, the probability that a given node belongs to an infinite cluster. Describe the percolation problem on the Cayley tree as a Galton-Watson process and calculate both p c and P. 3. (Birth-death process) Suppose bacteria duplicate at a constant rate λ and die at a constant rate µ.
2 (a) Calculate F(s,t) for this branching process. (b) Express F(s,t) in the form F(s,t) = 1 a(t) + b(t)s 1 c(t)s. (c) Determine P n (t), the probability that there are n individuals at time t. 4. The characteristic function of the random variable W(t) = Z(t)e u (1)t in a continuous-time Markov branching process is given by ( φ(k,t) = E e ikw(t)) ( { }) ( { } ) = E exp ike u (1)t Z(t) = F exp ike u (1)t,t, where F(s, t) is the probability generating fraction of the branching process. Thus the characteristic function of the random variable W = lim W(t) will be φ(k) = lim φ(k,t). Determine φ(k) for the supercritical t t birth-death process and find P(w) = Pr{W w}.
Competition 1. In order to gain insight into the mechanism underlying the phenomenon referred to as error catastrophe in a quasispecies consider the following simplified model. There are two species: the wildtype 0 (with fitness f 1 ) and the mutant 1 (with fitness f 0 > f 1 ). Assume that a fraction µ of a wildtype individual s offspring are mutants, but mutants never produce the wildtype (this reflects the fact that, in a long chain, it is very unlikely that a new mutation corrects the chain back to the wildtype). (a) Write down the quasispecies equation for this model. (b) Find the composition of the population in the steady state as a function of µ. (c) Can you explain the two regimes that you find? 2. Prove that in population genetics Fisher s fundamental theorem holds provided f i j = f ji (i.e., fitness does not depend on which parent one receives the allele from something that often is not true). 3. Consider a case in which there are two alleles at a given locus, A and a, and write down the evolution equation for x, the frequency of the allele A, assuming a sexual population under random mating. Show that ẋ φ (x), so that the negative of the mean fitness of the population, φ(x), becomes a sort of potential. Discuss all the steady states that can be found in this model and provide an evolutionary interpretation of the results.
Evolutionary game theory 1. (Stability of rest points of the replicator equation) Let us denote S n = {(x 1,...,x n ) R n : x 1 + + x n = 1, x i 0} the n 1 dimensional simplex. By Γ we are going to denote a face of S n (i.e., a subset of S n such that one or more of the coordinates x i is set to zero). In an abuse of notation Γ will also denote the set of indexes corresponding to the nonzero coordinates of the face. Consider the replicator equation ẋ i = x i [(Ax) i xax], i = 1,...,n, A = (a i j ). A. Let p be an interior rest point of the replicator equation. Define matrices B = (b i j ) and C = (c i j ), where c i j = p i a i j and b i j = c i j p i (pa) j. Now express x = p + ξ, where ξ 1 = 0 and ξ 1. (a) Show that ξ i = (Bξ) i + o( ξ ). (b) Show that p is an eigenvector of both B and C. (c) Prove that if w p is an eigenvector of C, then v = w (w 1)p is an eigenvector of B with the same eigenvalue and such that v 1 = 0. (d) Provide a sufficient criterion for the stability of the interior rest point p. B. Consider now a rest point of the replicator equation p Γ. (a) Obtain the evolution equations for ξ i up to terms o( ξ ). (HINT: separate i Γ from i / Γ.) (b) Provide a sufficient criterion for the stability of the boundary rest point p. 2. (Hawks-doves-bullies) Maynard Smith proposed a variant of the hawk-dove game in which a third type comes in. This type is called bully and it behaves as follows: it starts escalating the conflict, but retreats if the opponent also escalates. (a) Write down the payoff matrix for this game. (b) Replace this matrix by another one which maintains the dynamics (perhaps at a different time scale) but has zeros in the diagonal. (c) Find the rest points of the replicator equation. (d) Determine their stability. (e) Sketch the phase map of this dynamical system in the simplex S 3. 3. (Iterated games) In iterated games players play another round of the same game with probability w, and payoffs accumulate round after round. Consider the donor game a particular wording of the prisoner s dilemma in which a player can choose between two actions: donate c (cooperate) so that the opponent
6 receives b > c, or not to donate (defect) so that the opponent receives nothing. We will consider tree strategies: (i) cooperate every round irrespective of what the opponent does (AllC), (ii) defect every round irrespective of what the opponent does (AllD), (iii) start cooperating and do what the opponent did in the previous round, i.e., tit-for-tat (TFT). (a) Calculate the payoff matrix of this iterated game. (b) Discuss the phase map of the replicator dynamics for this game as function of the ration b/c and the continuation probability w.
Finite populations y 1. Let F(x,y,t) P(x,y,t)dt = Pr{X(t) y X(0) = x}, where P(x,y,t) is the solution of the backward Kolmogorov equation 0 t P(x,y,t) = µ(x) σ(x) P(x,y,t) + x 2 If the process is eventually absorbed either at X = 0 or X = 1, then 2 x 2 P(x,y,t). π 0 (x,t) lim y 0 + F(x,y,t), π 1(x,t) 1 lim y 1 F(x,y,t), are the absorption probabilities at either end by time t. (a) Prove that π 0 (x,t) and π 1 (x,t) satisfy the same backward Kolmogorov equation and provide suitable boundary conditions in each case. (b) Obtain a differential equation for π 0 (x) lim t π 0 (x,t) and provide its solution in terms of µ(x) and σ(x). (c) φ(t,x) = t [π 0(x,t) + π 1 (x,t)] is the probability density that absorption occurs between t and t + dt. Show that it satisfies the backward Kolomogorov equation. (d) Write down a differential equation for τ(x) it. 0 φ(t,x)dt, the mean time to absorption, and solve (e) Obtain the expression for τ(x) for the neutral case µ(x) = 0, σ(x) = x(1 x). 2. The diffusion approximation can be generalized to arbitrary Markov chains. In particular, for a diploid Wright-Fisher model without mutations, where f i j = 1 + ν i j /(2N) is the fitness of indiduals with alleles i j (N is the number of individuals, hence 2N the number of alleles) and x is the fraction of alleles of type A, we can derive the Kolmogorov equations with µ(x) = ν AA x 2 (1 x) + ν Aa x(1 x)(1 2x) ν aa x(1 x) 2, σ(x) = x(1 x). When ν AA = ν aa = 0 and ν Aa = ν > 0 (heterozygote dominance) the deterministic model predicts a polymorphic equilibrium with x = 1/2, whereas if populations are finite, eventually x = 0 or x = 1. Discuss this discrepancy by computing the absorption time from an initial population at equilibrium in the limit ν 1.
Neutral evolution 1. Consider the following simplified model of neutral evolution. A small homogeneous population evolves in the neutral network corresponding to the fittest phenotype for a given environment. There is a set of P possible phenotypes, and from each genotype a subset of K of them can be reached in a single point mutation. Suppose for simplicity that each of these subsets are random and independent selections of the set of possible phenotypes. Every generation there is a probability µ that the population changes to a genotype adjacent to the current one. With probability q this new genotype belongs to the same phenotype, and with probability 1 q it belongs to one of the K accessible phenotypes (q is the robustness of the current phenotype). Initially any other phenotype is lethal, so evolution cannot move the population out of the current phenotype, where it reaches equilibrium. Then, all of a sudden an environmental shift makes one and only one of the P alternative phenotypes more fit. (a) Divide the genotypes of the neutral network in two groups: those that are not neighbor of the new phenotype (A) and those that are (B), and justify why the evolution on this system can be described by the following three-state Markov chain, where state A represents the set of genotypes of the first class, B the set of genotypes of the second class, and C the new fittest phenotype. (b) Write down the Markov transition matrix. (c) Calculate p B, the probability that, before the environmental shift (no transition to C is possible), the population has a class B genotype. (d) C is, of course, an absorbing state. In a Markov chain, if τ i is the mean time to absorption from state i, then τ i = 1 + j Q i j τ j, where Q i j is the probability that there is a transition from state i to state j. Calculate τ, the mean absorption time of a system that was in equilibrium before the environmental shift. (e) Discuss what happens if q 0 or q 1, and show that there is an optimum robustness, q, that minimizes the time to find the phenotype P. Discuss the relationship between q and K in the limit K/P 1.
10 REMARK: This is a simplified version of a model introduced in Draghi et al., Nature 463, 353 355 (2010) (Supplementary Information).