I'm a biologist in the process of modeling a fairly simple biological system using a system of ODEs. To verify the simulations, I'm attempting to obtain an analytical steady-state solution that I can check the simulations against. My attempts so far haven't borne fruit, so I thought I'd toss the question out to mathematicians. This is my first post, so apologies if the question isn't right for this site. 


The equation is of the form:


$${dS\_3\over dt} = 2Xv\_{max} {S\_1 - {S\_3^2S\_7^4\over K\_{eq,3}}\over K\_m+S\_1+{S\_3^2S\_7^4\over K\_{eq,3}}} + D(S\_{3,in} - S\_3)$$
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$${dS\_4\over dt} = Xv\_{max} {S\_1 - {S\_4S\_7^2\over K\_{eq,4}}\over K\_m+S\_1+{S\_4S\_7^2\over 
K\_{eq,4}}} + D(S\_{4,in} - S\_4)$$
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$${dS\_1\over dt} = -Xv\_{max} \Bigg[{S\_1 - {S\_3^2S\_7^4\over K\_{eq,3}}\over K\_m+S\_1+{S\_3^2S\_7^4\over K\_{eq,3}}} + {S\_1 - {S\_4S\_7^2\over K\_{eq,4}}\over K\_m+S\_1+{S\_4S\_7^2\over 
K\_{eq,4}}}\Bigg] + D(S\_{1,in}-S\_1)$$
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$${dX\over dt} = Xv\_{max}Y \Bigg[4{S\_1 - {S\_3^2S\_7^4\over K\_{eq,3}}\over K\_m+S\_1+{S\_3^2S\_7^4\over K\_{eq,3}}} + 3{S\_1 - {S\_4S\_7^2\over K\_{eq,4}}\over K\_m+S\_1+{S\_4S\_7^2\over 
K\_{eq,4}}}\Bigg] + D(X\_{in}-X)$$
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$${dS\_7\over dt} = Xv\_{max} \Bigg[4{S\_1 - {S\_3^2S\_7^4\over K\_{eq,3}}\over K\_m+S\_1+{S\_3^2S\_7^4\over K\_{eq,3}}} + 2{S\_1 - {S\_4S\_7^2\over K\_{eq,4}}\over K\_m+S\_1+{S\_4S\_7^2\over 
K\_{eq,4}}}\Bigg] + D(S\_{7,in}-S\_7)$$
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Where S1, S3, S4 and S7 and X are variables 


and 


Km, Keq,3, Keq,4, vmax, S1,in, S3,in, S4,in, S7,in, Xin, D and Y are constants. 


This system models the change in the substrate Sn or the microbial population X in a perfectly-stirred vessel with microbes acting upon a substrate S1 to produce S3, S4 and S7 when the kinetics of the chemical reactions are thermodynamically reversible. 


Sn,in is the input concentration of Sn. Km and vmax are constants that describe the "affinity" of the microbe to S1 and the maximum rate of the reaction respectively and Keq,n is the thermodynamic equilibrium constant for the reaction S1 -> A Sn + B S7. I need to solve this system for Sn where n=1,3,4,7. 


Is this even possible, or am I barking up the wrong tree here?



Analytical steady-state solution of a complex ODE
As Andrey mentioned, you shouldn't expect an analytical solution since you're dealing with a system of algebraic equation in several variables. (Just to be clear, what we're envisioning here is the system of equations you get by setting all the left-hand-sides to be zero). In your case, I believe you have 5 variables (X, $S\_i$) and 5 equations, whose denominators can be cleared to make everything polynomial. 


Such systems can be solved numerically (once you specify numerical values for your constants). One tool that I've used before is [PHCpack](http://www.math.uic.edu/~jan/download.html), though for your purposes maybe Mathematica or something similar will be just fine.


Perhaps an expert can describe how to calculate resultants of your system of polynomials which will give you information on when the nature of the roots of this system change as you change your parameters...


