Population Question Using Half-Life Equation

In summary, it will take approximately 26 years for the wombat population to reach one pair per km^2.
  • #1
justine411
16
0

Homework Statement


Vancouver Island has an area of 33000km^2 but no wombats. If we release a pair of wombats on Vancouver Island and their annual rate of increase is 50%, how many years will it be before the wombat population reaches one pair per km^2?


Homework Equations



[A]t=[A]oe^-kt t1/2=ln2/k

The Attempt at a Solution


This is in a rate of reaction tutorial package and the only equation I could think to use is the half life one, but have it going up by 1/2 instead of going down by 1/2. I'm not sure how to manipulate the equation though. The answer is 26 years, I'm just not sure how to do that. I'm not sure what to put in for k either. I've tried using 0.5 but that doesn't work. Any suggestions?
 
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  • #2
Okay, I can start off by pointing out this isn't a half-life problem. You're not talking about decreasing populations, you're talking about population growth, so while you're generally on the right track, an equation that calculates half life decay rates isn't going to be adequate. Do you know how that equation was derived so you can adjust it for population growth instead of decline?

The other thing you need to factor into this is that you have to first calculate your population size endpoint to solve for time. You are given all the information you need to calculate this based on the area of the habitat and the desired density of the population. So start there.

Although your question is based on a biological system, you don't need biology knowledge to solve this one, you need to know the math. I'd like to move it over to one of the math homework help sections, but will need your help to make sure I put it in the right one. Have you taken calculus yet (or are you in calculus now)? The equations you need to use are based on calculus derivations, but if you do not yet know what a derivative is, I will put it in the pre-calculus section so you get help based on your level of math knowledge.
 
  • #3
I understand calculus, at least derivatives and integrals, so you can move it there if you want. Thanks SO much for your help. I thought it belonged here because it was in a chemistry tutorial, though it doesn't seem much like chem.
 
  • #4
Okay, I'll make the move to the calculus HW forum. I think the folks there will be able to help you better than I can with this.
 
  • #5
justine411 said:

Homework Statement


Vancouver Island has an area of 33000km^2 but no wombats. If we release a pair of wombats on Vancouver Island and their annual rate of increase is 50%, how many years will it be before the wombat population reaches one pair per km^2?
You don't really need calculus, just a calculator (and a bit of trial and error).

First, find the pattern.

At n = 0 years, P = 1 (P = number of pairs)
At n = 1 years, P = (1.5)
At n = 2 years, P = (1.5)^2
...

Got it?
 
Last edited:

1. What is the half-life equation used for in population studies?

The half-life equation is used to calculate the amount of time it takes for a population to decrease by half. This is useful in understanding population growth and decline, as well as making predictions about future population sizes.

2. How is the half-life equation calculated?

The half-life equation is calculated by taking the natural logarithm of 2 and dividing it by the growth rate of the population. The resulting number is the amount of time it takes for the population to decrease by half.

3. Can the half-life equation be used for any type of population?

The half-life equation is most commonly used for populations that follow an exponential growth or decline pattern. It may not be as accurate for populations that follow other growth models, such as logistic growth.

4. Is the half-life equation affected by external factors?

External factors, such as immigration, emigration, and natural disasters, can affect the accuracy of the half-life equation. These factors can cause changes in the population growth rate, which can impact the results of the equation.

5. How can the half-life equation be used to make predictions about future population sizes?

By using the half-life equation, scientists can determine the rate of population decline and make predictions about future population sizes. This can be especially useful in understanding the impact of certain events or interventions on a population's growth or decline.

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