Let's back to the question, how does the number of inhabitants looks like many years later. Before we perform the calculation, let's make some hypotheses and check them out. Why do we make some hypotheses? Since it is the fun of mathematics. If I predict something based on mathematics, then I can check it out by calculation. If my prediction is correct, that's the fun. It is like my plan succeeded in my chess game.
One thing is clear, the total number is 1000 since no one is born or die and these people always stay in one of the cities.
- Hypothesis 1: One day, all the inhabitants move to Berlin since the ratio of staying Berlin (0.8) is larger than that of the Potsdam (0.7).
- Hypothesis 2: the inhabitant of Potsdam increased \(100 \rightarrow 250 \rightarrow 325\) in these two years. However, one point, the number of inhabitant of Potsdam becomes enough large, then inhabitant of Potsdam will decrease. Some years later of that, the inhabitants of Berlin becomes larger than Potsdam. Repeating that, the number of inhabitants of the cities may oscillates.
Let's try some computation with octave to check this hypothesis 2. (Nowadays I may rely on computer too much. In old time, people first ``think'' and prove the following...)
octave:5> M^3 * p
octave:6> M^4 * p
octave:7> M^10 * p
octave:8> M^100 * p
It seems there is no oscillation, but the number of inhabitants converges to a specific value. 100 years later, Berlin has 600 people, and Potsdam has 400 people. Figure 11 shows this change.
|Figure 11: Population history of Berlin and Potsdam.|
You might notice the intermediate results has fraction. We computed number of people and you might think what is 600.29 people means. Because we first assume the ratio of moving people 0.2. This doesn't happen in reality. In reality, some people move and then we can think how much percentage people was moved. However, this is not completely nonsense. For example, moving out ratio of a city or birth ratio of a city don't change so drastically in a year. It is a good guess next year is the same ratio. The calculus has more sophisticated guessing method, but, we will stick this guess. Current guess is called first order approximation. Making a plan with such guess is not so bad compare to assuming completely different number. We could interpret the fraction as some people spent a few months in Berlin and rest of the months in Potsdam.
By the way, is this result depends on the initial state? In other words, are these conversing numbers the property of the matrix only, or the property of both the matrix and the initial population vector? Let's see this in the next article.