# Population Modeling using DE's

• prace
In summary, the conversation is about understanding population modeling. The problem is to determine the time it takes for a population to triple or quadruple, given that it doubles in 5 years. The rate of change is represented by dy/dt = kP, and the solution is P(t) = P(initial)e^(kt). Using the given information, the unknowns can be solved for and the time can be determined.
prace
Hi,

I was wondering if anyone out here on a Friday night could help me understand population modeling. Here is what I have as a problem (this is pretty simple because my goal here is to understand the thinking behind the madness )

-
The population of a certain community is known to increase at a rate proportional to the number of people present at any time. If the population has doubled in 5 years, how long will it take to triple, to quadruple?
-

So I understand that the rate = dy/dt and proportional translates to "something" = "some constant" times "something", or (dy/dt)=Ky, or in this case, dP/dt = kP.

Solving this DE I get the equation P(t)= P(initial)e^(kt). So all I am told is that the population doubles in 5 years. So what can I do? I can't assume an arbitrary number as the initial population can I? So I have one equation with three unknowns, P(initial), P(t), and k.

Any insight would be great. Thanks!

you know that $$\frac{dP}{dt} = kP$$. The solution is:

$$P(t) = P_{0}e^{kt}$$

You also know that $$2P_{0}= P_{0}e^{5k}$$. What can you do from here? Solve for k:

$$e^{5k} = 2$$

$$k = \frac{\ln 2}{5}$$.
Then solve the equations for t:

$$3P_{0} = P_{0}e^{(\frac{\ln 2}{5})t}$$
$$4P_{0} = P_{0}e^{(\frac{\ln 2}{5})t}$$

Last edited:
Oh man... So here the $$P_{0}$$'s will cancel and all I have to do is take the natural logs of both sides to get t? man... I think I was making it a lot harder than it really was. Thank you for taking the time to show me this.

## What is population modeling using DE's?

Population modeling using differential equations (DE's) is a mathematical approach to studying and predicting changes in population size over time. It involves using equations to represent the relationships between different factors that affect population growth, such as birth and death rates, immigration and emigration, and environmental factors.

## What is the purpose of population modeling using DE's?

The purpose of population modeling using DE's is to better understand and predict changes in population size. This can be useful for studying and managing different populations, such as endangered species, human populations, and disease outbreaks.

## What are the advantages of using DE's for population modeling?

DE's allow for a more dynamic and realistic representation of population growth compared to simpler models. They can also incorporate multiple factors and their interactions, making them more accurate and useful for predicting changes in population over time.

## What are the limitations of population modeling using DE's?

DE's rely on assumptions and simplifications, so they may not accurately reflect real-world complexities. They also require a significant amount of data and expertise to construct and analyze, making them more challenging to use compared to simpler models.

## What are some real-world applications of population modeling using DE's?

Population modeling using DE's has been used in various fields, including ecology, epidemiology, and economics. It has been used to study and manage endangered species, predict disease outbreaks, and inform policy decisions related to human populations.

Replies
12
Views
2K
Replies
1
Views
2K
Replies
13
Views
2K
Replies
5
Views
2K
Replies
10
Views
5K
Replies
5
Views
1K
Replies
7
Views
2K
Replies
17
Views
4K
Replies
73
Views
9K
Replies
5
Views
4K