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Homework Help: Biological Differential Equation

  1. Jan 25, 2010 #1
    1. The problem statement, all variables and given/known data
    I have a system of coupled differential equations of the form
    [tex] \frac{dR_1}{dt} = R_1^0 \cdot g\left( \frac{R_2}{K_R} \right) - R_1 [/tex]
    [tex] \frac{dR_2}{dt} = R_2^0 \cdot g\left( \frac{R_1}{K_R} \right) - R_1 [/tex]
    [tex] g\left( \frac{ R_i}{K_r} \right) = \frac{ 1 + f\cdot \left[ \frac{ R_i}{K_R} \right]^2 }{1 + \left[ \frac{R_i}{K_r} \right]^2 } [/tex]
    where f << 1 is a constant, [itex] R_1^0 [/itex] is the steady state level of [itex] R_1 [/itex] in the absence of [itex] R_2 [/itex] and vice versa.

    I need to show (graphically) that if we are free to manipulate [itex] R_1^0, R_2^0 [/itex] then this can lead to one or three solutions that simultaneously satisfy both equations.

    3. The attempt at a solution
    It seems to me that an obvious choice for a single solution would be to set [itex] R_i^0 =0 [/itex] which will decouple the systems and make them decreasing exponentials. However, other than this I am unsure how to determine that there are three solutions, let alone what it means to do this graphically.
  2. jcsd
  3. Jan 25, 2010 #2
    Nevermind, solved it.

    The equilibrium points are the intersections of the steady state solutions.
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