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 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>

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 . 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/dt

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,

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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 . Multiply the equation throughout by this integrating factor.

-kdt -kt

-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 to find the expression for P(t).

kt

P(t) Cekt

Next we apply the given initial condition P(0) = P_o 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

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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