Hi. I solved for the equilibrium points but first I changed some of the variable names since I is for imaginary in Mathematica and also not good to use any variable names starting with capital letters in Mathematica so everywhere a capital letter, I changed it to lower case. Here's the Solve code but since I used the same subscripted names, I have to first change it to raw input form to cut and paste it into this thread so it's hard to read. However if you then cut and paste this code into Mathematica then select it and do a Cell/Convert To/Stanard Form, you can recover all the subscripted formatting.

Code:

myeqtpt = First[Solve[{(-Subscript[l, e])*(Subscript[\[Rho], f]/w) +
Subscript[\[Psi], l]*(Subscript[l, b]/w) == 0,
Subscript[\[Rho], f]*Subscript[l, e] - Subscript[\[Psi], l]*
Subscript[l, b] - Subscript[\[Chi], l]*Subscript[l, b] == 0,
Subscript[\[Chi], l]*Subscript[l, b] - Subscript[\[Omega], l]*
Subscript[l, i] == 0,
Subscript[\[Gamma], rr]*Subscript[\[Rho], i] - Subscript[\[Chi], 0]*
(Subscript[\[Rho], f]/\[Sigma]) -
(m*Subscript[l, e]*Subscript[\[Rho], f] - m*Subscript[\[Psi], l]*
Subscript[l, b] + (m*Subscript[\[Chi], l]*Subscript[l, b]*
Subscript[\[Rho], f])/(1 - Subscript[\[Rho], f]))*(1/\[Sigma]) ==
0, Subscript[\[Gamma], S]/(k + c) + Subscript[\[Chi], 0]*f*
(Subscript[\[Rho], f]/\[Sigma]) +
f*(1 + Subscript[\[Rho], f]/(1 - Subscript[\[Rho], f]))*
((m*Subscript[\[Chi], l]*Subscript[l, b])/\[Sigma]) -
Subscript[\[Gamma], rr]*Subscript[\[Rho], i] == 0,
\[Gamma]*(Subscript[\[Omega], l]*r*Subscript[l, i]) - \[Gamma]*(c - 1) ==
0}, {c, Subscript[\[Rho], i], Subscript[\[Rho], f],
Subscript[l, i], Subscript[l, b], Subscript[l, e]}]]

Mathematica then returns the following solution:

Which I then checked by back-substituting:

Code:

In[240]:=
(-Subscript[l, e])*(Subscript[\[Rho], f]/w) +
Subscript[\[Psi], l]*(Subscript[l, b]/w) /. myeqtpt
Subscript[\[Rho], f]*Subscript[l, e] - Subscript[\[Psi], l]*
Subscript[l, b] - Subscript[\[Chi], l]*Subscript[l, b] /. myeqtpt
Subscript[\[Chi], l]*Subscript[l, b] - Subscript[\[Omega], l]*
Subscript[l, i] /. myeqtpt
Subscript[\[Gamma], rr]*Subscript[\[Rho], i] - Subscript[\[Chi], 0]*
(Subscript[\[Rho], f]/\[Sigma]) -
(m*Subscript[l, e]*Subscript[\[Rho], f] - m*Subscript[\[Psi], l]*
Subscript[l, b] + (m*Subscript[\[Chi], l]*Subscript[l, b]*
Subscript[\[Rho], f])/(1 - Subscript[\[Rho], f]))*(1/\[Sigma]) /.
myeqtpt
FullSimplify[Subscript[\[Gamma], S]/(k + c) + Subscript[\[Chi], 0]*f*
(Subscript[\[Rho], f]/\[Sigma]) +
f*(1 + Subscript[\[Rho], f]/(1 - Subscript[\[Rho], f]))*
((m*Subscript[\[Chi], l]*Subscript[l, b])/\[Sigma]) -
Subscript[\[Gamma], rr]*Subscript[\[Rho], i] /. myeqtpt]
\[Gamma]*(Subscript[\[Omega], l]*r*Subscript[l, i]) - \[Gamma]*(c - 1) /. myeqtpt
Out[240]=
0
Out[241]=
0
Out[242]=
0
Out[243]=
0
Out[244]=
0
Out[245]=
0