Differentials: Population moving model

[missavvy]

Homework Statement

The population of a country is divided in two groups:

People who live in rural areas (R(t)) and people who live in urban areas (U(t)).
People move from rural to urban areas with a rate m and from
urban to rural areas with rate n.

a) Introduce the fraction of people (Z(t)) who live in rural
areas as a new variable and derive an equation for it.

b) Find the steady state(s) of the equation for Z and the
stability condition.

Homework Equations

R' = −mR + nU
U' = mR − nU.

The Attempt at a Solution

a) So I don't really get this, but is the fraction Z(t) different from R(t)??
Anyways, I think it goes:
Z(t) = R(t) - U(t) / R(t)
how would i go about deriving it?

I am extremely confused. If anyone can just somehow reword it or something so that I can understand it that would be great.

b) If I could get Z(t) into differential form I can easily find the steady state, by just setting it = to 0.. so my problem really lies in the first part.

 

[payumooli]

the fraction of people who live in the rural area should be
Z(t) = R(t) /(R(t)+U(t))

 

[payumooli]

Z' = -mZ + n(1-Z)

you can take from here

 

[tiny-tim]

Welcome to PF!

The population of a country is divided in two groups:

People who live in rural areas (R(t)) and people who live in urban areas (U(t)).
People move from rural to urban areas with a rate m and from
urban to rural areas with rate n.

a) Introduce the fraction of people (Z(t)) who live in rural
areas as a new variable and derive an equation for it.
…
a) So I don't really get this, but is the fraction Z(t) different from R(t)??
…
b) If I could get Z(t) into differential form I can easily find the steady state, by just setting it = to 0.. so my problem really lies in the first part.

Hi missavvy! Welcome to PF!

You're confusing totals with proportions.

R and U are totals, Z is only a proportion …

so just ask yourself, what two things is it a proportion of?

 

[missavvy]

ah sorry i meant z= r/r + u!
thanks tiny-tim! yeah that's what i had thought but I wasn't sure, thanks for clarifying that :)

 

[missavvy]

Z' = -mZ + n(1-Z)

you can take from here

Just wondering how did you derive the Z(t)?
Did you derive Z(t) = R(t) / R(t) + U(t), and then just plug in the R' and U' for those values?