2. [Smarks] At the right is an infinite grid populated by a large number of counters. The squares are ofthree different colours: white W, light L [light grey) and dark D [dark grey]. At any time, a square might contain no counters, or one or more counters. At one minute intervals there is a transition at which a counter can move or die [be removed) or reproduce. The behaviour of each counter depends on the colour of its square. A counter on a white square dies with probability 1/4 and moves one square to the right, the left, above or below, each with probability 3/16 . A counter on a light square moves one square to the right, the left, above or below, each with probability 1/4. A counter on a dark square dies but also produces 3 offspring counters that occupy three of the four white squares that share a corner with the dark square. Construct a node diagram for this population. [b] Show that this population has a steady state and find the steady state proportions. Check your work as you will need the steady-state proportions for part (d). [c] Calculate bK be the average number of future offspring for an individual on node K, for K = W or L. Note that by, = 3. [d] Assuming that the population above is in steady state. calculate the average age ofa counter on each type of node. Here age is the number of time-steps since birth. Thus an offspring placed on W will have made one move and Will have age 1. It will survive to the next time step with probability 3f8 + 3/8 2 3X4 and at this point it will have age 2