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Math Help - a messy looking ODE system

  1. #1
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    a messy looking ODE system

    I have 6 ODES:

     W\frac{dl_E}{dt} = -l_E\rho_F + \psi_Ll_B
     \frac{dl_B}{dt} = \rho_Fl_E - (\psi_L + \chi_L)l_B
     \frac{dl_I}{dt} = \chi_Ll_B - \omega_Ll_I
     \sigma\frac{d\rho_F}{dt} = \gamma_{rr}\sigma p_I - \chi_0p_F - ml_E\rho_F + m\psi_Ll_B - \frac{m\chi_Ll_B\rho_F}{1 - \rho_F}
     \sigma\frac{d\rho_I}{dt} = \frac{\gamma_S\sigma}{K + c} + \chi_0f\rho_F + f(1 + \frac{\rho_F}{1 - \rho_F})m\chi_Ll_B - \sigma\gamma_{rr}\rho_I
     \frac{dc}{dt} = \gamma(\omega_LR_Ll_I) - \lambda(c - 1)

    for which I'm trying to find the steady states, so setting each equation equal to zero:

     -l_E\rho_F + \psi_Ll_B = 0....(1)
     \rho_Fl_E - (\psi_L + \chi_L)l_B = 0....(2)
     \chi_Ll_B - \omega_Ll_I = 0....(3)
     \gamma_{rr}\sigma p_I - \chi_0p_F - ml_E\rho_F + m\psi_Ll_B - \frac{m\chi_Ll_B\rho_F}{1 - \rho_F} = 0 ....(4)
     \frac{\gamma_S\sigma}{K + c} + \chi_0f\rho_F + f(1 + \frac{\rho_F}{1 - \rho_F})m\chi_Ll_B - \sigma\gamma_{rr}\rho_I = 0....(5)
     \gamma(\omega_LR_Ll_I) - \lambda(c - 1) = 0....(6)

    Then:

    From (3)  l_I = \frac{\chi_Ll_B}{\omega_L} ....(7)

    From (6)  c = 1 + K_1l_I where  K_1 = \frac{\gamma\omega_LR_L}{\lambda}....(8)

    Assume  \chi_0 = 0

    From (1)  l_E = \frac{\psi_Ll_B}{\rho_F} ....(9)

    From (5)  \rho_I =  \frac{1}{\sigma\gamma_rr}(\frac{\gamma_S\sigma}{K + c} + f(1 + \frac{\rho_F}{1 - \rho_F})) ....(10)

    Subbing (10) into (4):

     \gamma_{rr}\sigma(\frac{1}{\sigma\gamma_rr}(\frac{  \gamma_S\sigma}{K + c} + f(1 + \frac{\rho_F}{1 - \rho_F}))) - m\psi_Ll_B + m\psi_Ll_B - \frac{m\chi_Ll_B\rho_F}{1 - \rho_F} = 0

    Subbing (7) into (8):

     c = 1 K_1 \frac{\chi_Ll_B}{\omega_L}

     \gamma_{rr}\sigma(\frac{1}{\sigma \gamma_{rr}}(\frac{\gamma_S\sigma}{K + 1 + K_1\frac{\chi_Ll_B}{\omega_L}} + f(1 + \frac{\rho_F}{1 - \rho_F}))) - m\psi_Ll_B + m\psi_Ll_B - \frac{m\chi_Ll_B\rho_F}{1 - \rho_F} = 0

    From (2)  l_B = \frac{\rho_Fl_E}{\psi_L + \chi_L} ....(11)

     \gamma_{rr}\sigma(\frac{1}{\sigma\gamma_{rr}}(\fra  c{\gamma_S\sigma}{K + 1 + \frac{K_1\rho_Fl_E\chi_L}{\omega_L(\psi_L + \chi_L)}} + f(1 + \frac{\rho_F}{1 - \rho_F}))) - \frac{m\chi_L\rho_F^2l_E}{(1 - \rho_F)(\psi_L + \chi_L)} = 0 ....(*)

    and due to the nature of the question I can assume  l_E = 0 and hence have an equation for  \rho_F .

    Was just hoping that someone could please check this over for me, and also tell me if it possible to solve (*) in Maple, (never used it before so not sure).

    Thanks in advance!
    Last edited by hunkydory19; December 16th 2009 at 05:52 AM.
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