7. Stochastic and Deterministic Models (2/2)

The reason for this is that the gravitational attraction between the Earth and the Moon is not quite strong enough to keep the Moon in a fixed orbit. As the distance between the Moon and the Earth increases, so the Gravitational force will decrease and so the distance between them will increase more rapidly, if everything else stays the same. In practice, everything else does not stay the same, and so the velocity of the Moon will decrease, reducing the centrifugal force.

Another class of deterministic models has to do with animal populations. Initial models of population growth were based on the idea of exponential growth in which the increase in population in a unit of time is a function of the size of the population at that time and the population growth rate.

N1 = N0 · R
Nt = N0 · Rt

in which N0 is the starting population, N1 is the first generation, R is (1.0 plus the rate of growth) and Nt is the tth generation.

Consider cell division. If each cell divides into two at each generation, then R = 2. If the first generation has one cell, then the second shall have two cells, the third shall have four, the fourth eight cells and so forth. This is exponential growth.

However, experimental data have shown that whilst this might be the case under some conditions, under other conditions it did not apply. This observation led to the conclusion that populations become constrained once they reach a certain size, for example by the available space or food, and that this reduces the rate of population growth. This led to the next generation of population growth models based on the Logistic curve.

N t = N o K N o +(K− N o )⋅exp(− r 0 t)

in which K is the upper limit of population growth, often called the Carrying Capacity and ro = R - 1.

The Exponential Curve, the Logistic Curve and an application of the logistic function concerning the spread of Aids in Australia.
Source: Murray Bourne, 1997-2009

The AIDS virus was brought into a city of 50,000 people who were not taking precautions by 100 people. After 10 weeks it was found that 1000 people were infected. Assuming the epidemic follows the Logistic curve, how long will it take for half the population to be affected?

Use the above equation with the data that you have been given and knowing that t = 10 weeks, find ro = 0.23208. Now use this rate in the above equation again to find that half the population will be infected after 26.8 weeks or about 27 weeks.

Questions

  1. What is the gravitational force between you and the Earth? You do not need to calculate it, or rather if you did, it would come out exactly the same as your weight. Your weight is a measure of this force.

  2. What is your weight on the Moon? First answer this question by using the Law of Gravitation by making it into a ratio of the force of gravity on the Earth to the Force of Gravity on the Moon. Simplify this ratio to get the equation
    F Moon F Earth = M Moon M Earth ⋅ R Earth 2 R Moon 2
    Use the mass of the Moon 0.73483·1023 kg and its radius of 1738 km, and the mass of the Earth 59.74·1023 kg and its radius of 6371 km to get this ratio. Now, knowing the weight on the Earth allows you to calculate the weight of the same person on the Moon. Now check your answer by going to this address.