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Homework Help: Differential, Levin's Model, Steady-states

  1. Jan 21, 2010 #1
    1. The problem statement, all variables and given/known data
    A piece of land is seperated in equal parts. There is one species on this land.

    P ∈ [0, 1] = fraction of parts suitable for the species
    x(t) = fraction of parts currently occupied by the species

    The individuals from an occupied part colonize an empty part at rate c, and occupied parts
    become extinct at a rate e.

    a) Find the steady states(equilibrium solutions) of this equation.
    Set c/e = 1/2 and draw the steady states as a function of P ∈ [0, 1].

    b) Find the stability condition for each of the steady states.
    2. Relevant equations

    x'(t) = cx(h − x) − ex.


    3. The attempt at a solution

    a)

    Ok, so I have determined the steady-states:

    cx(h-x) = ex
    h - x = e/c
    so I found that x = h - e/c and x = 0 are the equilibrium solutions.
    Now I don't know how exactly to draw the steady states.. If c/e = 1/2, then how would I go about plugging that in for the steady states?

    Does this mean to draw a direction field, with 0 and h - e/c as straight lines since they are the steady-states? Or does it mean to draw x(t) as a function ?

    b)

    x'(t) > 0 = unstable, and x'(t) < 0 = stable
    So cx(h - x) - ex > 0, cx(h - x) > ex, h - x > ex/cx, h - e/c > x
    and then cx(h - x) - ex < 0, h - e/c < x

    then the condition is x > h - e/c for stability, and unstable when x < h - e/c

    Okay that was my attempt. Please help if you can!
     
  2. jcsd
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