Systems and Signals Models - population model

[satchmo05]

Homework Statement

At the beginning of the year 2000, country A had a population p of 100 million people. The birth rate is 4%/year and the death rate is 2%/year, compounded daily. Every day, 275 immigrants enter country A. Neglect leap-year effects.

Write a difference equation for the population at the beginning of the nth day after January 1,2000 (day 0) with the immigration rate as the input signal of the system.

By finding the zero-input and zero-state responses of the system, determine the population of country A at beginning of the year 2050.

Homework Equations

This is where I am super confused. I am looking at my text, and there isn't great help here. I have been going to class, nothing mentioned about it here. Is this simpler than I am making it out to be? I am thinking that in order to get the nth day of the population's output signal, I would need a summation of some sort. What is a difference equation?? That is my big question and how do I place this information into one to find the answer?

The Attempt at a Solution

I know this is not the final solution, but my summation model is:
4%/365 days = 0.01096% birth rate/day
-2%/365 days = -0.00548% death rate/day

($$\sum$$ from m=-infinity to n) [(1e8)(1+0.01096)^m(1-0.00548)^m + x[m]]

where x[m] = 275*m

However, solving the second part of this problem, I am finding that the population is around 2e50, which cannot be right! Please help! I appreciate all help in advance. Best.

 

[KataKoniK]

Dear student,

Thank you for your post. It seems like you are on the right track with your attempt at a solution. Let me explain what a difference equation is and how to apply it to this problem.

A difference equation is an equation that describes the relationship between the input and output signals of a system. In this case, the input signal is the immigration rate and the output signal is the population at the beginning of each day. The difference equation will help us determine the population at any given day based on the immigration rate and the previous day's population.

To start, let's define some variables:
- P(n) = population at the beginning of the nth day
- I(n) = immigration rate on the nth day (in this case, it is a constant value of 275)
- B = birth rate (in this case, it is 4%)
- D = death rate (in this case, it is 2%)

Now, let's break down the problem into smaller steps. First, we need to determine the population at the beginning of the first day (day 1). We know that at the beginning of day 1, the population is 100 million, so we can write:
P(1) = 100 million

Next, we need to determine the population at the beginning of the second day (day 2). To do this, we need to consider the factors that affect the population on day 2:
- The population on day 1 (P(1))
- The immigration rate on day 2 (I(2))
- The birth and death rates on day 1 (B and D)

We can write this as:
P(2) = P(1) + I(2) + B*P(1) - D*P(1)

Notice that we are adding the immigration rate and the birth rate, but subtracting the death rate. This is because immigration and birth increase the population, while death decreases it.

We can use this same logic to write a general difference equation for the population on any given day (day n):
P(n) = P(n-1) + I(n) + B*P(n-1) - D*P(n-1)

Now, we can use this difference equation to determine the population at the beginning of the year 2050 (which is day 18,263). We know that the immigration rate is constant at 275, so we