Hi,
I think that there are a some issues here. I suppose that you want to set the rhs of ODEs to zero. In that case we are looking for fixed values of S,I1,R,N1 and P to solve the equations. If you knew the parameters in the equations the solution is quite straight forward and fast:
Solve[-a*S*P + b*R - c*S + d*(S + I1 + R) == 0 && a*S*P - e*I1 - (I1*(c + P)) == 0 &&
e*I1 - b*R == 0 && -f*N1*I1 + g*(N1 + P)*(1 - (N1 + P)/k) - h*N1 - c*N1 == 0 && f*N1*I1 - h*P - c*P == 0 /. {a -> 1, b -> 1, c -> 1, d -> 1, e -> 1, f -> 1, g -> 1, h -> 1, k -> 1}, {S, I1, R, N1, P}]
(*{{I1 -> 0, R -> 0, N1 -> -1, P -> 0}, {I1 -> 0, R -> 0, N1 -> 0,
P -> 0}, {S -> -3, I1 -> -1, R -> -1, N1 -> -2, P -> 1}, {S -> -6,
I1 -> -2, R -> -2, N1 -> -1, P -> 1}}*)
Note that I removed the time dependence of the variables. For these parameters it correctly identifies the origin as a fixed point; I suppose that the other fixed points are irrelevant in your modelling situation, because they are negative.
You can explicitly add positivity conditions like so:
Solve[(-a*S*P + b*R - c*S + d*(S + I1 + R) == 0 && a*S*P - e*I1 - (I1*(c + P)) == 0 &&
e*I1 - b*R == 0 && -f*N1*I1 + g*(N1 + P)*(1 - (N1 + P)/k) - h*N1 - c*N1 == 0 &&
f*N1*I1 - h*P - c*P == 0 && S >= 0 && I1 >= 0 && R >= 0 && N1 >= 0 && P >= 0) /. {a -> 1, b -> 1, c -> 1, d -> 1, e -> 1,
f -> 1, g -> 1, h -> 1, k -> 1}, {S, I1, R, N1, P}]
(*{{I1 -> ConditionalExpression[0, S >= 0],
R -> ConditionalExpression[0, S >= 0],
N1 -> ConditionalExpression[0, S >= 0],
P -> ConditionalExpression[0, S >= 0]}}*)
In theory you could also try to use functions like Reduce to solve the system. Given the large number of parameters and nonlinear expressions I guess that the general conditions are quite complex.
Reduce[-a*S*P + b*R - c*S + d*(S + I1 + R) == 0 && a*S*P - e*I1 - (I1*(c + P)) == 0 &&
e*I1 - b*R == 0 && -f*N1*I1 + g*(N1 + P)*(1 - (N1 + P)/k) - h*N1 - c*N1 == 0 && f*N1*I1 - h*P - c*P == 0 && S > 0 && I1 > 0 && R > 0 && N1 > 0 && P > 0, {S, I1, R, N1, P}, Reals]
hasn't finished on my machine yet. Do you have a priori values for the parameters?
Cheers,
Marco