I have a system of four ordinary differential equation. This is a modelling problem we were also meant to criticize some of the issues with the way the problem was presented. Its meant to describe the nitrogen concentration in the available nutrients for an ecosystem. Where $N_c= N+P+Z+D = constant $
$$ \frac{dN}{dt} = -ulP\frac{N}{K_s + N} +aD + (1- \mu)hPZ\\ \frac{dP}{dt} = ulP\frac{N}{K_s + N} - hPZ - sP\\ \frac{dZ}{dt} = \mu hPZ -eZ \\ \frac{dD}{dt} = eZ +sP -aD $$
Apart from $N,P,Z,D$ we consider all other terms constant. I note we were not given an explanation to all the other terms.
We were told there were four steady solutions, two of which are relatively easy to find. I am struggling to understand how to find steady solutions. Could anybody please help with these ? Thanks not experienced with differentials.
Edit: Helpful Reference for contextualization http://mpe.dimacs.rutgers.edu/2013/11/26/ocean-plankton-and-ordinary-differential-equations/
I'll assume none of the constants is $0$.
From equation 2, either $P=0$ or $Z = ((u l - s) N - K_s s)/((N +K_s) h)$.
From equation 3, either $Z=0$ or $P = e/(\mu h)$.
One case is $P=Z=0$. Then from the last equation $D=0$, and the first equation is automatically true, so $N$ is arbitrary.
Another case is $P = e/(\mu h)$, $Z = ((u l - s) N - K_s s)/((N +K_s) h)$. Then we get $$ D={\frac {e \left( \left( s + \mu u l - \mu\,s \right) N+K_{{s}}\mu \,s-K_{{s}}s \right) }{ah\mu\, \left( K_{{s}}+N \right) }} $$ and $N$ is arbitrary.
The third case is $Z = 0$ and $(u l - s) N - K_s s = 0$, i.e. $N = K_s s/(u l - s)$. Then from the last equation $P = aD/s$, and $D$ is arbitrary.