Estimating Growth Rate and Simulating World Population with ODEs | Matlab Help

[cronoxc]

Homework Statement

The rate of change of the population p is proportional to the existing population at any time t:

dp/dt = k_g*p

where k_g is the growth rate. The world population in millions from 1950 through 2000 was

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000
2555 2780 3040 3346 3708 4087 4454 4850 5276 5686 6079

(a) Assuming that the above equation holds, use the data from 1950 through 1970 to estimate kg.
(b) Use the fourth-order Runge-Kutta method, using the value of determined above, to simulate the world population from 1950 to 2050 with a step size of 5 years. Display your simulation results along with the data on the plot.

Homework Equations

I'm not sure.

The Attempt at a Solution

I have no idea how to solve this. Any help would be greatly appreciated.

 

[KataKoniK]

Hello,

Thank you for posting this problem on the forum. I am a scientist and would be happy to assist you with this question.

To solve this problem, we first need to understand the given equation:

dp/dt = k_g*p

This equation represents the rate of change of the population (dp/dt) as proportional to the existing population (p) at any given time (t). The constant k_g represents the growth rate.

Now, to estimate the value of k_g using the data from 1950 to 1970, we can use the following steps:

1. Choose any two data points from the given population data, for example, (1950, 2555) and (1970, 3708).
2. Substitute the values of t and p in the given equation:
dp/dt = k_g*p
(3708 - 2555)/(1970 - 1950) = k_g*3708
3. Solve for k_g:
k_g = (3708 - 2555)/(1970 - 1950*3708) = 0.0189

Therefore, the estimated value of k_g is 0.0189.

Now, for part (b) of the question, we can use the fourth-order Runge-Kutta method to simulate the world population from 1950 to 2050 with a step size of 5 years. This method is a numerical technique used to solve differential equations.

The steps for using the fourth-order Runge-Kutta method are as follows:

1. Define the initial conditions: In this case, we know that in 1950, the population was 2555 million, so we can set p(1950) = 2555 million.
2. Define the step size (h): In this problem, the step size is given as 5 years.
3. Calculate the number of steps: The number of steps will be (2050 - 1950)/5 = 20.
4. Use the following formula to calculate the population at each time step:
p(t+h) = p(t) + (h/6)*(k_1 + 2*k_2 + 2*k_3 + k_4)
Where,
k_1 = h*f(t, p(t))
k_2 = h*f(t+h/2, p(t) + k_1/2)
k_3 = h*f(t+h