differential equations - Analytical steady-state solution of a complex ODE

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:

$${dS_3over dt} = 2Xv_{max} {S_1 - {S_3^2S_7^4over K_{eq,3}}over K_m+S_1+{S_3^2S_7^4over K_{eq,3}}} + D(S_{3,in} - S_3)$$

$${dS_4over dt} = Xv_{max} {S_1 - {S_4S_7^2over K_{eq,4}}over K_m+S_1+{S_4S_7^2over K_{eq,4}}} + D(S_{4,in} - S_4)$$

$${dS_1over dt} = -Xv_{max} Bigg[{S_1 - {S_3^2S_7^4over K_{eq,3}}over K_m+S_1+{S_3^2S_7^4over K_{eq,3}}} + {S_1 - {S_4S_7^2over K_{eq,4}}over K_m+S_1+{S_4S_7^2over K_{eq,4}}}Bigg] + D(S_{1,in}-S_1)$$

$${dXover dt} = Xv_{max}Y Bigg[4{S_1 - {S_3^2S_7^4over K_{eq,3}}over K_m+S_1+{S_3^2S_7^4over K_{eq,3}}} + 3{S_1 - {S_4S_7^2over K_{eq,4}}over K_m+S_1+{S_4S_7^2over K_{eq,4}}}Bigg] + D(X_{in}-X)$$

$${dS_7over dt} = Xv_{max} Bigg[4{S_1 - {S_3^2S_7^4over K_{eq,3}}over K_m+S_1+{S_3^2S_7^4over K_{eq,3}}} + 2{S_1 - {S_4S_7^2over K_{eq,4}}over K_m+S_1+{S_4S_7^2over K_{eq,4}}}Bigg] + D(S_{7,in}-S_7)$$

Where S₁, S₃, S₄ and S₇ and X are variables

and

K_m, K_eq,3, K_eq,4, v_max, S_1,in, S_3,in, S_4,in, S_7,in, X_in, D and Y are constants.

This system models the change in the substrate S_n 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.

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

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

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