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Homework Help: Differentials: Population moving model

  1. Jan 22, 2010 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations
    R' = −mR + nU
    U' = mR − nU.

    3. 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.
  2. jcsd
  3. Jan 23, 2010 #2
    the fraction of people who live in the rural area should be
    Z(t) = R(t) /(R(t)+U(t))
  4. Jan 23, 2010 #3
    Z' = -mZ + n(1-Z)

    you can take from here
  5. Jan 23, 2010 #4


    User Avatar
    Science Advisor
    Homework Helper

    Welcome to PF!

    Hi missavvy! Welcome to PF! :smile:

    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? :wink:
  6. Jan 23, 2010 #5
    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 :)
  7. Jan 23, 2010 #6
    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?
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