The dog-flea model was discussed in lecture. It concerned two dogs, named A and

Question

The dog-flea model was discussed in lecture. It concerned two dogs, named
A and B, and 100 trained, and identified by number, fleas that jumped from
one dog to the other when their number was called. The numbers were
selected at random, and there is a jump every two minutes. What is the net
transfer rate, in fleas per minute, to dog B, when
a) there are 70 fleas on dog A and 30 fleas on dog B?
b) there are 50 fleas on dog A and 50 fleas on dog B?
What was the point of this model (50 words or less)?

Explanation / Answer

a) if there are more fleas on dog A to start with the initial transfer rate is (70to - 30away)/200min = 40fleas/200min = 4fleas/20min = 0.2 fleas/min This is a hypothetical situation: how many fleas join us minus how many fleas leave us for greener pastures, divided by the time it would take for 100 fleas to jump This is just the initial rate, in time both dogs will have the same number of fleas and then the net transfer rate will be 0 fleas per minute. b) Here the chance that we lose a flea is the same as the chance that we gain a flea, so the net transfer rate is zero. c) This is like a Markov chain. No matter how unbalanced we start out, eventually only the transition probability matters. This is a self-regulating problem, the more lopsided we start, the bigger the chance that we gain. If we start out with lots of fleas, odds are we will lose more than we gain, evening out the populations.

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