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September 25th, 2010, 09:50 AM   #1
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differential equations - population model

Animal population P(t) with constant death rate delta = 0.01 (deaths per animal per month) and B proportional to P. Suppose P(0) = 200 and P'(0) = 2. When is P = 1000?
What's got me here is the delta being a constant number... haven't run into that yet. Usually, I have to find delta or beta by using the P(0) and P'(0) together. But since I already have it... so does it mean the DE looks like this:
dP/dt = BP - dP ... since B is proportional to P, can I write B = P? So..
dP/dt = P^2 - 0.01P

Is it correct to do it like that?
Then I can see I'd have to use partial fractions, through which I get:
-100 ln|P| + 100 ln|P - 0.01| = t + E (E = constant of integration from right side minus constant of integration from left side)
-100 (ln|P| - ln|P - 0.01|) = t + E
ln|P| / ln|P - 0.01| = (t+E)/100
|P| / |P - 0.01| = e^(t/-100) * e^(E/-100)
P / (P - 0.01) = e^(t/-100) * +/- e^(E/-100) {+/- e^(E/-100) = F}
P / (P - 0.01) = Fe^(t/-100)
Solving for P then plugging in the initial value P(0) = 200 gives me ~1 for F, so:
P(t) = [-0.01e^(t/-100)] / [1 - e^(t/-100)]
So then to solve for what they're asking, I put P = 1000 and solve for t, which gives me:
t = 0.004

[I've rounded everything here in this post, but kept the actual numbers in my calculator while calculating]

The 0.004 seems like a way too small number. The answer the book gives is 100 ln(9/5) ~ 59 months
Plus, I haven't even used the P'(0) = 2 that they gave me... how was I supposed to use that?

I'd appreciate any help clearing up this problem, thanks!
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September 25th, 2010, 10:05 AM   #2
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Re: differential equations - population model

It's simpler than you're making it:

The death rate is constant, so you will just get a "-d" term in the differential equation.

The birth rate is proportional to P, so the births will be represented by "bP" where b is the per-capita birth rate.


Therefore the equation you need to solve is


(Obviously, this D.E. won't be valid for since can't realistically be negative.)
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September 25th, 2010, 10:33 AM   #3
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Re: differential equations - population model

Oh, okay. So how do I continue from there?
Find b by using P'(0) = 2, right?
Well, that means for dP/dt, I can put in 2, but what do I do with the zero?
dP/dt = bP - d
2 = bP - 0.01... I have no t..?
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September 25th, 2010, 11:32 AM   #4
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Re: differential equations - population model

Sorry, when I said can't be negative, I actually meant can't be negative when is zero .

Anyway, I noticed that you said it was 0.01 deaths per animal per month. This isn't a constant death rate, it's a constant per capita death rate, which is different. Can you clarify which is meant?
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September 25th, 2010, 12:29 PM   #5
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Re: differential equations - population model

I still don't understand where to go next with dP/dt = bP - d.
How do I begin to find b?

The statement is "Consider an animal population P(t) with constant death rate delta = 0.01 (deaths per animal per month)..."
It says constant death rate.
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September 25th, 2010, 12:48 PM   #6
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Re: differential equations - population model

A constant death rate implies that the number of deaths per unit time is constant. This would be appropriate for a species modelled with a fixed predator population, or with an annual cull, and with low natural death rate.

However, when it says 0.01 (deaths per animal per month), I'm rather inclined to believe that what they mean is a constant per capita death rate (i.e. a term in the equation of the form -dP). This would be consistent with natural death only.

What is really puzzling me is that if by 'death rate' they mean 'per capita death rate', then they would also mean 'per capita birth rate' when they say 'birth rate'. If the per capita birth rate is proportional to P, then this implies that the actual birth rate is proportional to P^2. This simply doesn't make any sense in a real world application (at least as far as I have seen). You very often see death rates proportional to P^2 (death due to overcrowding, etc.) but never birth rates - it doesn't make sense for a species to become more fertile as the size of the population grows!
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September 25th, 2010, 01:31 PM   #7
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Re: differential equations - population model

When I use dP/dt = bP - dP and the initial values given, I get t = 100 ln(5) for P(t) = 1000.
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September 25th, 2010, 01:44 PM   #8
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Re: differential equations - population model

I note that in the original question, they state that B is proportional to P, rather than "the birth rate". Even so, MarkFL is right in saying that if we assume the equation is and substitute in the initial values, we end up with P=1000 at t=100 log 5.

Having said that, I note that if we do presume that the differential equation is then we get the answer for the question in the book. Bizarre.
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September 25th, 2010, 01:53 PM   #9
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Re: differential equations - population model

I agree that is a bizarre population model. Most models I've seen assume competition for limited resources, etc., that result in a growth rate that decreases as the population increases.
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