# Rodent Population as a differential equation.

• cp255
In summary: It would be helpful to see the question.At the moment the question has not been posted or is not visible to me.I can see the screen shot of the question uploaded. I will rewrite the question and update the original post.I can see the screen shot of the question uploaded. I will rewrite the question and update the original post.
cp255

## Homework Statement

https://www.physicsforums.com/attachment.php?attachmentid=65556&stc=1&d=1389570667

For those who can not see the screen shot here is the question...
Suppose the population P of rodents satisfies the diff eq dP/dt = kP^2.
Initially there are P(0) = 2 rodents, and their number is increasing at the rate of dP/dt = 1 rodent per month when P = 10. How long does it take for the population to reach 105 rodents.

## The Attempt at a Solution

First I found k by substituting in 10 for p and 1 for dp/dt.
1 = k * 10^2
k = 1/100

Then to solve the differential equation I integrated both sides with respect to t.

∫dp/dt * dt = ∫0.01 * p^2 dt
p = 0.01p^2 * t + C

I solved for C and found C = 2.

Solving for t gives
t = 100(p - 2) / p^2

I then plunged in 105 for p and the answer was wrong.

Last edited:
It would be helpful to see the question.

At the moment the question has not been posted or is not visible to me.

I can see the screen shot of the question uploaded. I will rewrite the question and update the original post.

cp255 said:
I can see the screen shot of the question uploaded. I will rewrite the question and update the original post.

Integrating ∫0.01 * p^2 dt to get 0.01p^2*t+C is incorrect. That's assuming p is a constant. You need to separate the ODE first.

τ
cp255 said:

## Homework Statement

https://www.physicsforums.com/attachment.php?attachmentid=65556&stc=1&d=1389570667

For those who can not see the screen shot here is the question...
Suppose the population P of rodents satisfies the diff eq dP/dt = kP^2.
Initially there are P(0) = 2 rodents, and their number is increasing at the rate of dP/dt = 1 rodent per month when P = 10. How long does it take for the population to reach 105 rodents.

## The Attempt at a Solution

First I found k by substituting in 10 for p and 1 for dp/dt.
1 = k * 10^2
k = 1/100

Then to solve the differential equation I integrated both sides with respect to t.

∫dp/dt * dt = ∫0.01 * p^2 dt
p = 0.01p^2 * t + C
This is incorrect. You are treating p, on the right, as if it were a constant but it is a function of t.
Instead, write it as $$\int \frac{dp}{p^2}dp= \int .01dt$$
$$-\frac{1}{p}= .01t+ C$$
$$p(t)= -\frac{1}{.01t+ C}$$
$$p(0)= -\frac{1}{C}= 2$$

I solved for C and found C = 2.

Solving for t gives
t = 100(p - 2) / p^2

I then plunged in 105 for p and the answer was wrong.

## 1. What is a differential equation in the context of rodent population?

A differential equation is a mathematical expression that represents the relationship between the population of rodents and the factors that affect it over time. It takes into account various factors such as birth rate, death rate, and migration to model the change in population over time.

## 2. How do scientists use differential equations to study rodent populations?

Scientists use differential equations to create mathematical models that can predict changes in the rodent population over time. By inputting data on various factors such as birth rate, death rate, and migration, scientists can use these models to understand how changes in these factors will impact the rodent population.

## 3. What are the limitations of using differential equations to study rodent populations?

While differential equations are powerful tools for studying rodent populations, they do have some limitations. These equations rely on assumptions and simplifications, and may not always accurately reflect the complex dynamics of real-world rodent populations. Additionally, they require accurate and up-to-date data to be effective.

## 4. How do scientists validate the accuracy of their differential equation models for rodent populations?

Scientists can validate the accuracy of their models by comparing the predicted results to real-world data. If the model accurately reflects the observed changes in the rodent population, it can be considered a valid representation of the population dynamics. Additionally, scientists can also use statistical methods to measure the precision and uncertainty of their models.

## 5. What insights can be gained from studying rodent populations using differential equations?

Studying rodent populations using differential equations can provide valuable insights into the factors that affect the population dynamics. This can help scientists understand the impact of different variables such as climate change, disease outbreaks, and human interventions on the population. It can also aid in developing strategies for managing and predicting changes in rodent populations.

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