Probability
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- Jan 26, 2012
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Suppose I have a population where an exponential model was assumed. The population is P_n in a community on 1st March n years after 1977. Let me say that the initial population is P0, representing the size on 1st March 1977, which is 8.
Assuming then that the births are 3.7P_n and the deaths are 0.8P_n, a recurrence system that satisfies P_n is;
P_n+1 - P_n = 3.7P_n - 0.8P_n
P_n+1 = 3.9P_n
So a recurrence system would be;
P0 = 8, P_N+1 = 3.9P_n (n=0,1,2...)
OK this is a geometric system. This system states that as the population keeps growing because births are greater than deaths, which we know in reality this could not go on indefinitely as the population would be over crowded.
Now I want to look at setting up a logistic model where the proportionate birth rate decreases linearly with the population size, and the proportionate death rate increases linearly with the population size.
This is what I know.
If the death rate was constant and the birth rate decreased linearly with the population size P, then according to the formula;
(3.9 - 0.0017P) I could find the following recurrence system;
The proportionate growth rate R(P) = (3.9 - 0.0017P) - 0.8
= 3.1 - 0.0017P
So the recurrence system would be;
P0 = 8, P_n+1 - P_n = (3.1 - 0.0017P_n)P_n
Now this example above is where the proportionate birth rate decreases linearly, but what I want to find is the proportionate death rate increase linearly.
So does anyone know how I change the formula above?
In the example above remember the death rate is constant and the birth rate decreases linearly, but what I want to do now is change that to show the birth rate decreases linearly with population size while the death rate increases linearly with population size.
Do I just add (3.9 - 0.0017P) + 0.8 and follow through as subtracting seems to produce a decreasing factor and the death rate is increasing so I am somewhat at a loss how to change it round?
Assuming then that the births are 3.7P_n and the deaths are 0.8P_n, a recurrence system that satisfies P_n is;
P_n+1 - P_n = 3.7P_n - 0.8P_n
P_n+1 = 3.9P_n
So a recurrence system would be;
P0 = 8, P_N+1 = 3.9P_n (n=0,1,2...)
OK this is a geometric system. This system states that as the population keeps growing because births are greater than deaths, which we know in reality this could not go on indefinitely as the population would be over crowded.
Now I want to look at setting up a logistic model where the proportionate birth rate decreases linearly with the population size, and the proportionate death rate increases linearly with the population size.
This is what I know.
If the death rate was constant and the birth rate decreased linearly with the population size P, then according to the formula;
(3.9 - 0.0017P) I could find the following recurrence system;
The proportionate growth rate R(P) = (3.9 - 0.0017P) - 0.8
= 3.1 - 0.0017P
So the recurrence system would be;
P0 = 8, P_n+1 - P_n = (3.1 - 0.0017P_n)P_n
Now this example above is where the proportionate birth rate decreases linearly, but what I want to find is the proportionate death rate increase linearly.
So does anyone know how I change the formula above?
In the example above remember the death rate is constant and the birth rate decreases linearly, but what I want to do now is change that to show the birth rate decreases linearly with population size while the death rate increases linearly with population size.
Do I just add (3.9 - 0.0017P) + 0.8 and follow through as subtracting seems to produce a decreasing factor and the death rate is increasing so I am somewhat at a loss how to change it round?