Question

Find the solution to the following equation for an
age-structured population:

∂/∂t P(t,a)+ ∂/∂a P(t,a)=−δ(a)P(t,a), a≥0,t≥0

P(0,a) = P0(a)

P(t,0) = C

where C > 0 is a constant. Bonus: What is the asymptotic
behavior of the population (total size, age-distribution if
possible) as t → ∞? If limit exists, is it a solution to the PDE
and, if so, what kind of solution?

Answer #1

8. Find the solution of the following PDE:
utt − 9uxx = 0
u(0, t) = u(3π, t) = 0
u(x, 0) = sin(x/3)
ut (x, 0) = 4 sin(x/3) − 6 sin(x)
9. Find the solution of the following PDE:
utt − uxx = 0
u(0, t) = u(1, t) = 0
u(x, 0) = 0
ut(x, 0) = x(1 − x)
10. Find the solution of the following PDE:
(1/2t+1)ut − uxx = 0
u(0,t) = u(π,t) =...

a logistic equation for a pipulation P(T) in which there is
harvesting can be written as
(dP/dt)=aP(1-(P/M))-h, P(0)=P0
(a) if a=22, M=11, h=48, what are the equilibrium solutions for the
logistic equation?
(b) if P(0) = 4, what will the long-term behavior of the
population?

The population P(t) of bacteria grows according to the logistics
equation dP/dt=P(12−P/4000), where t is in hours. It is known that
P(0)=700. (1) What is the carrying capacity of the model? (2) What
is the size of the bacteria population when it is having is fastest
growth?

Why a model like P(t) = P 0 e kt, where
P0 is the initial population, would not be plausible?
What are the virtues of the logistic model?

Find the solution to the initial value problem
(y′−e−t+5)/y=−5, y(0)=−4
Discuss the behavior of the solution y(t)y(t) as tt becomes
large. Does limt→∞y(t)limt→∞y(t) exist? If the limit exists, enter
its value. If the limit does not exist, enter DNE.

Find the solution to the initial value problem
(y′−e−t+5)/y=−5, y(0)=−4
Discuss the behavior of the solution y(t)y(t) as tt becomes
large. Does limt→∞y(t)limt→∞y(t) exist? If the limit exists, enter
its value. If the limit does not exist, enter DNE.

Suppose that the certain population obeys the logistics
equation
dP / dt = 0.025·P ·(1−P / C)
where C is the carrying capacity. If the initial population P0 =
C/3, ﬁnd the time t∗ at which the initial population has doubled,
i.e., ﬁnd time t∗ such that P(t∗) = 2P0 = 2C/3.

Suppose that Newton’s method is applied to find the solution p
= 0 of the equation
e^x −1−x− (1/2)x^2 = 0. It is known that, starting with any p0
> 0, the sequence {pn} produced by the Newton’s method is
monotonically decreasing (i.e., p0 >p1 >p2 >···)and
converges to 0.
Prove that {pn} converges to 0 linearly with rate 2/3. (hint:
You need to have the patience to use L’Hospital rule repeatedly. )
Please do the proof.

The population of the world was about 5.3 billion in 1990 (t =
0) and about 6.1 billion in 2000 (t = 10). Assuming that the
carrying capacity for the world population is 50 billion, the
logistic differential equation
dP =kP(50−P)dt
models the population of the world P(t) (measured in billions),
where t is the number of years after 1990. Solve this differential
equation for P(t) and use this solution to predict what the
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Please solve these 2 questions step by step trying to learn for
an exam.
1.Find an explicit solution of the following differential
equation. y' =xy-xy^2, y(0) =3
2. Suppose a population satisfies, dP/dt = 0.02P-0.00005P^2;
P(0) =40 =P0, Where t is measured in Years.
a) what is the carrying capacity M?
b)for what values of P is the population increasing the
fastest?
c)Given the solution of the differential equation. P(t)
=M/1+Ae^-0.02t, where M is the carrying capacity and A =...

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