I am new to ODEs. Tring to prove the postivity invariance of the following system $\forall t$:
''' \begin{array}{l} \frac{d X}{d t}=-a X+b, \\ \frac{d Y}{d t}=-Y\left(\frac{p_1 X}{X+a}+p Z\right)+r Y\left(1-\frac{Y}{l}\right), \\ \frac{d Z}{d t}=l+m\frac{ Y X}{X+c}-Z\left(r Y+d\right) . \end{array}. '''
The assumptions:
1.parameters and initial conditions are all nonnegative.
- $0 \leq Y(t) \leq l$
I think that I managed to do this for the first one: $\begin{gathered}X(t)=\frac{b}{a}\left(1-e^{-a t}\right) +X(0) e^{-a t}\end{gathered}$ is the solution for first order linear differential equation, so it's ok.
So as I continue, if I use the constraint on $Y(t)$, and $X(t)>0$ for all $t \in[0, \infty)$, I see that $r Y\left(1-\frac{Y}{l}\right)$ is non-negative. Can I use the following consideration to show that $Y(t)$ is more than or equal an exponent? is it valid?
'''\begin{equation} \frac{d Y}{d t}\geq-Y\left(\frac{p_1 X}{X+a}+p Z\right) \end{equation}'''
'''\begin{equation} Y(t)\geq Y(0) exp(-\left(\frac{p_1 X}{X+a}+p Z\right)) \end{equation}'''
Thank you.