Wednesday, November 16, 2011

The Logistic Equation

For population growth calculations, Malthusian method was the first to arrive at the scene. This method / model measures an unlimited growth that is only decreased by the natural death rate of the population. However Malthusian method fails to predict the population size correctly after a certain time period.

The logistic equation was an improvement to Malthusian model that gives much accurate results. It does this by calculating another death rate, not proportional to the population, but this time proportional to the "interaction density" of that population.

If population at time t is p(t), we have dp/dt = ap - bp^2 where a and b are constants (they are calculated from historical data).

The interaction density between p people is roughly calculated as p(p-1) / 2, for example between 4 people there are 6 possible peer-to-peer interactions, and simply p^2 can be used and placed in the equation as bp^2 (bcz there is already a p in there, and the rest is handled by constants). Interaction density is supposed to represent death due to contention for shared resources, spread of disease that place denser populace at a disadvantage.

According to our calculations, with natural growth world population in 60 years is supposed to be at around 12 billion. But according to logistic growth prediction, the real number will not get there. It will remain at around 10 billion - and I am sorry to say this - the difference of around 2 billion people are going to die.

This death will occur because of a race toward shared resources, widespread epidemic, or war.

If however contention for shared resources issue is resolved, such as the invention of new energy sources, known and future deadly diseases are cured speedily, disagreement within, between people are resolved piecefully then the story can change.

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