Question

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 population will be in 2050 according to this model.

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 population will be in 2050 according to this model.

Answer #1

Biologists stocked a lake with 300 fish and estimated the
carrying capacity (the maximal population for the fish of that
species in that lake) to be 7000. The number of fish doubled in the
first year.
(a) Assuming that the size of the fish population satisfies the
logistic equation
dP/dt =
kP(1−PK),
determine the constant k, and then solve the equation
to find an expression for the size of the population after
t years.
k =
P(t) =
(b) How...

Suppose a population P(t) satisfies
dP/dt = 0.8P − 0.001P2 P(0) =
50
where t is measured in years.
(a) What is the carrying capacity?
______
(b) What is P'(0)?
P'(0) = ______
(c) When will the population reach 50% of the carrying capacity?
(Round your answer to two decimal places.)
_____yr
Please show all work neatly, line by line, and justify steps so
that I can learn.
Thank you!

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 =...

The population of the world in 1990 was 50 billion and the
relative growth rate was estimated at 0.5 percent per year.
Assuming that the world population follows an exponential growth
model, find the projected world population in 2018.

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?

In 1990 (t = 0), the world use of natural gas was 73874 billion
cubic feet, and the demand for natural gas was growing
exponentially at the rate of 5.5% per year. If the demand continues
to grow at this rate, how many cubic feet of natural gas will the
world use from 1990 to 2012?

In 1990 (t = 0), the world use of natural gas was 79413 billion
cubic feet, and the demand for natural gas was growing
exponentially at the rate of 6.5% per year. If the demand continues
to grow at this rate, how many cubic feet of natural gas will the
world use from 1990 to 2018?

Suppose that the population develops according to the logistic
equation dP/dt = 0.0102P - 0.00003P^2 where t is measured in days.
What is the P value after 10 days where A = (M-P(0)) / P(0)) if the
P(0) = 40? And P(0) = 20?

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.

The population P(t) of
mosquito larvae growing in a tree hole increases according to the
logistic equation with growth constant k = 0.3
days−1 and carrying capacity A = 800. Find a
formula for the larvae population
P(t), assuming an initial
population of P(0) = 80 larvae.
P(t) =______________

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