This line probably needs to be
y_rep[n, k] = poisson_rng(Z[n,k]);
The syntax error message is saying that y_rep[n,k] is an integer while poisson_rng(Z[,k]) is an array of integers and those can not be assigned to each other.
As a sketch for how I think you can look for non-identifiable parameters see below. But this is probably not a useful way of thinking about it, I just write it down as how I tried to help but probably failed. You really want someone who actually knows how ODEs work. I just put my simplistic economist hat on and look at what would happen if the system stabilises. That is \frac{dP}{dt} = 0 and \frac{dH}{dt} = 0. If you rewrite everything a little bit you can then write H as H = (b - r) \times f(P, \theta, c, h, \mu) where f is not a function of b or r. That means that if(!) most of the data is from observations close to a stable system b and r are poorly identified but their difference is well defined. In my simpler models I would then reparametrise (b, r) to (\Delta = b - r, M = b/2 + r/2) which sometimes works.