Variation of bacteria in a population
(b) The number {X(t) : t E [0, oo)} of bacteria in a population varies continuously in time t on the state space {0,1,2, … } according to the a simple birth-and-death process with immigration defined by the transition probability from state j > 0 to state j + k:
{jA At+ o(At) (v + jA) At + o(At) /33+ko(At) = Prob fAX(t) = kIX(t) = j} = _ _, [ . 1 23A+ v] At+ o(At) o(At),
if k = —1 if k = +1 if k = 0 otherwise
where v > 0 is the constant immigration rate, while the birth and death rates are the same and given by A = jA, with j E {0, 1, 2, … }, and AX(t) = X(t At)- X(t). Initially X(0) = N. i. Show that the master equation of this simple birth and death process with immigration is
d
dt d dtPo = —vpo + Api(t),
= [A(j —1) + v]pi_i+ + 1)pi+i — [2jA + v]pi, for j = 1, 2, …
with p3(t) = 0 when j < 0. Give a brief interpretation of this master equation. ii. Use the master equation to find the differential equation obeyed by m(t) = E(X(t)). Integrate this equation and find m(t)
(c) Consider the symmetric two-player game with pure strategies ec = (1, 0)T (C, “Cooperation”) and eD = (0, 1)T (D, “Defection”), and payoff matrix
A= vs C D (2) C D # 2 0 0 – 1
where 1 < 0 < 3. We assume perfect rationality and an infinite number of games between random pairs of players that can use pure or mixed strategies (no information is retained or exchanged from previous games). i. Choose a value of 0 such that A is an example of payoff matrix of a game in which D, “Defection” is the unique evolutionary stable strategy. What is the “social dilemma” faced by the player of this game? (Justify any answer.) ii. Choose a value of 0 such that A is an example of payoff matrix of an anti-coordination game. Determine all the evolutionary stable strategies of this anti-coordination game. iii. We now consider the symetric two-player game of payoff matrix (2) with 0 = 3/2. Determine the replicator equation for this game and analyse its properties (i.e. find its fixed points and their stability).
