## DanielM_113 2 years ago 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.]

1. DanielM_113

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.

2. DanielM_113

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.

3. DanielM_113

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?

4. KenLJW

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

5. KenLJW

dO/dy = (k -r)

6. DanielM_113

@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?

7. KenLJW

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

8. DanielM_113

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

9. DanielM_113

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 $$\mbox{k years}^{−1}"$$.