Question
Do the question completely. Especially part C
thanks
4. A population P of organisms dies at a constant rate of a organisms per unit time, and the growth rate is proportional to t
C. Find lim,0 P(t) if Po>
4. A population P of organisms dies at a constant rate of a organisms per unit time, and the growth rate is proportional to the population size with the proportionality constant k. A. Assume the initial population P(0) = Po. Write a differential equation that models the size of the population P(t) at ay time t. B. Write the equation from part A in standard form, and solve. (The terms Po, a, k but not a constant C.) answer must contain the
C. Find lim,0 P(t) if Po>
0 0
Add a comment Improve this question Transcribed image text
Answer #1

4

A Let P(t) denote the population size at time t. We are given that the growth rate of the population is proportional to population size with proportionality constant k. Therefore, the growth rate at time t is kP(t).

The death rate is constant and equals a organisms per unit time.

The rate of change of the population size is given by the derivative dP/dt. We also know that the change in population size comes from births and deaths. So the rate of change of population size is the birth rate - death rate. Writing this in mathematical notation gives us the required differential equation:

dP kP a dt. The given initial population size P_o is the initial condition for this equation.

Combining the two, we find the model for our population as

dP kP-а. = dt Р() — Р,

-----------------------------------------------------------------------------------------------------------------------------------

B Now let us try to solve the given differential equation  dP kP a dt.

If we write this equation in standard form, we find

dP - kP =-a dt

The method to solve this type of first order equations is to multiply by an integrating factor and then integrating.

The integrating factor can be found as -kdt -kt . Multiply the equation throughout by this integrating factor.

-ktdP - kt -ae - kPekt- dt

We can rewrite the equation again as

(Pe kt) -kt -ae

Now we can integrate both sides with respect to t to find

-ktC Pe kt e

Multiply throughout by kt to find the expression for P(t).

P(t) Cekt

Next we apply the given initial condition P(0) = Po to find the constant C.

P(0) Po C k a C Po- I

Substituting for C gives us the final solution

P(t) P)et (Po ekt

------------------------------------------------------------------------------------------------------------------

C In this part, we look at the asymptotic behavior of the solution.

We are given Po> or  > 0 P-.

With this in mind, let us look at the solution.

The first term in the solution is a constant.

The second term has an exponential term. Since it is the proportionality constant for growth rate, k is positive. So the term kt e as t \rightarrow \infty . This exponential term is also multiplied by a constant Po . We were given that this constant is also positive, so the entire term  (P-ekt goes to infinity as t does.

We found the behavior of each term in the solution as t \rightarrow \infty . Combining what we found

a lim P(t) lim (Pekt tooLk a kt too lim lim (Po -e too k a oo oo t+o0

We have found that

lim P(t) o.

Add a comment
Know the answer?
Add Answer to:
Do the question completely. Especially part C thanks 4. A population P of organisms dies at a constant rate of a or...
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for? Ask your own homework help question. Our experts will answer your question WITHIN MINUTES for Free.
Similar Homework Help Questions
ADVERTISEMENT
Free Homework Help App
Download From Google Play
Scan Your Homework
to Get Instant Free Answers
Need Online Homework Help?
Ask a Question
Get Answers For Free
Most questions answered within 3 hours.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT