Question

Unit I Ordinary Differential Equation Problem:
An African government is trying to come up with a good policy regarding the hunting of oryx. They are using the following model: the oryx population has a natural growth rate of k years−1, and there is assumed a constant harvesting rate of a oryxes/year.
1. Write down a model for the oryx population. [First step: choose symbols and units.]

Answer 1

The first task is:
1. Write down a model for the oryx population. [First step: choose symbols and
units.]

I tried to figure start with:
\[\frac{dx}{dt}=x2^{dt/k}\]
But I know that is wrong.

Answer 2

Then I thought that I should think about the population without the harvest rate:
\(\Delta x\) (population change at t) = (population after \(t+\Delta t\)) - (population at t)
\[\Delta x=x2^{(t+\Delta t)/k}-x2^{t/k}\]
But I still don't have a differential equation.

Answer 3

If I divide by \(\Delta t\) and take the limit \(\Delta t \rightarrow 0\):
\[\lim_{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t}=\frac{C2^{(t+\Delta t)/k}-C2^{t/k}}{\Delta t} \\
\lim_{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t}=\frac{dx}{dt}=\frac{1}{k}C\ln (2)\,2^{t/k}
\]
I hope this is correct.
How do I go about adding the harvest rate? What is the motivation?

Answer 4

O = Oi + ky - 1 - ry
O = Oi + (k - r)y -1
O--oryx population
Oi--initial oryx population
y--years
k rate inc. per year
r rate dec. year

Answer 5

dO/dy = (k -r)

Answer 6

@KenLJW Why is there a -1 in your first equation? And shouldn't there be an exponetial growth rate since the population doubles every k years?

Answer 7

it says has a natural growth rate of k years -1
if you mean EXP{ky] -1
then
O = Oi + EXP{ky] - 1 - ry
O--oryx population
Oi--initial oryx population
y--years
k time constand per year
r rate dec. year
dO/dy = kEXP[ky] -r
if you want a constant population dO/dY = 0

Answer 8

@KenLJW Thank you. I have to think more aobut it and then I will check the answers.

Answer 9

I understand why I couldn't get it. The problem says natural growth meaning the growth rate of an exponential growth model.
And I also understand why @KenLJW used -1. It is because in my question it is written "k years−1", but I copied wrong, it should be \(``\mbox{k years}^{−1}"\).