There is an issue with this formulation in that the ,t at the end should not be there (DSolve needs an independent variable, Solve has no notion of that). Also I1 appears in one place without the t dependency; I assume that was a mistake. But it is a difficult system to handle with all the symbolic parameters. It might be more effective to only solve when given numeric values for {a,b,...}. Could be done as below.
exprs = {-a*S[t]*P[t] + b*R[t] - c*S[t] + d*(S[t] + I1[t] + R[t]),
a*S[t]*P[t] - e*I1[t] - (I1[t]*(c + P[t])),
e*I1[t] - b*R[t], -f*N1[t]*I1[t] +
g*(N1[t] + P[t])*(1 - (N1[t] + P[t])/k) - h*N1[t] - c*N1[t],
f*N1[t]*I1[t] - h*P[t] - c*P[t]};
params = {a, b, c, d, e, f, g, h, k};
vars = {S[t], I1[t], R[t], N1[t], P[t]};
solns[a0_?NumberQ, b0_?NumberQ, c0_?NumberQ, d0_?NumberQ, e0_?NumberQ,
f0_?NumberQ, g0_?NumberQ, h0_?NumberQ, k0_?NumberQ] :=
NSolve[exprs /.
Thread[params -> {a0, b0, c0, d0, e0, f0, g0, h0, k0}]]
Example:
solns[.1, .3, .2, .5, .6, .4, .7, .11, .6]
(* Out[460]= {{I1[t] -> -0.833013435701, N1[t] -> -4.46571428571,
P[t] -> 4.8, R[t] -> -1.6660268714,
S[t] -> -9.71849008317}, {I1[t] -> -0.775, N1[t] -> -4.8,
P[t] -> 4.8, R[t] -> -1.55,
S[t] -> -9.04166666667}, {I1[t] -> -0.775, N1[t] -> 0.5,
P[t] -> -0.5, R[t] -> -1.55,
S[t] -> 4.65}, {I1[t] -> -0.464469178082, N1[t] -> 0.834285714286,
P[t] -> -0.5, R[t] -> -0.928938356164,
S[t] -> 2.78681506849}, {I1[t] -> 0., N1[t] -> 0.334285714286,
P[t] -> 0., R[t] -> 0., S[t] -> 0.}, {I1[t] -> 0., N1[t] -> 0.,
P[t] -> 0., R[t] -> 0., S[t] -> 0.}} *)
