I came accross the following problem during my research:
Let $$c:[0,\infty)\to[0,1],t\mapsto c(t)\tag{1}\label{eq1}$$ be the atomic concentration of a substance at the surface of a material with $c(0)=c_0\in(0,1)$. Further I have two continuously differentiable and bounded functions $$a:[0,1]\to[k_a,K_a],c\mapsto a(c)\tag{2}\label{eq2}$$ and $$b:[0,1]\to[k_b,K_b],c\mapsto b(c)\tag{3}\label{eq3}$$ with $k_a>0$ and $k_b>0$.
In my case the time evolution of the concentration is given by the following nonlinear autonomous ODE: $$\frac{dc}{dt}=-\Bigl[a(c(t))\cdot(1-c_0)+b(c(t))\cdot c_0\Bigr]\cdot c(t)+b(c(t))\cdot c_0\tag{4}\label{eq4}$$
What I am interrested in is the behaviour for $t\to\infty$, i.e. assuming only the above given information on the functions $a$ and $b$, is it possible to conclude that the following limit exists?: $$c_\infty:=\lim_{t\to\infty}{c(t)}\tag{5}\label{eq5}$$
If it exists, is it safe to say that $$\lim_{t\to\infty}{\frac{dc}{dt}}(t)=0\tag{6}\label{eq6}$$ or is it not possible to interchange differentiation and the limit?
If $c_\infty$ and the limit in equation $\eqref{eq6}$ would exist, then I could calculate $c_\infty$ using equation $\eqref{eq4}$ to be: $$c_\infty = \frac{b(c_\infty)\cdot c_0}{a(c_\infty)\cdot(1-c_0)+b(c_\infty)\cdot c_0},\tag{7}\label{eq7}$$i.e. the concentration after infinte time depends only on the start concentration $c_0$ and the values $a(c_\infty)$ and $b(c_\infty)$.
Could you please help me with my problem by giving me hints or literature I could read? I already tried defining $\hat{a}:=a\circ c$ and $\hat{b}:=a\circ c$ and "forget" about the concentration dependency of $a$ and $b$, such that I can continue with this linear ODE: $$\frac{dc}{dt}=-\Bigl[\hat{a}(t)\cdot(1-c_0)+\hat{b}(t)\cdot c_0\Bigr]\cdot c(t)+\hat{b}(t)\cdot c_0\tag{8}\label{eq8}$$
The solution for the above ODE would be (according ot wolframalpha):
$$c(t)=\frac{ \int_0^t \exp{\Biggl( \int_0^z\Bigl[\hat{a}(w)\cdot (1-c_0)+\hat{b}(w)\cdot c_0\Bigr]dw \Biggr)} \cdot c_0 \cdot \hat{b}(z)dz +c_0}{\exp{\Biggl(\int_0^t\Bigl[\hat{a}(z)\cdot (1-c_0)+\hat{b}(z)\cdot c_0\Bigr]dz \Biggr)}}\tag{9}\label{eq9}$$
Then I used the rule of L'Hospital to obtain:
$$\lim_{t\to\infty}{c(t)}=\frac{\lim_{t\to\infty}{\Bigl(c_0\cdot\hat{b}(t)\Bigr)}}{\lim_{t\to\infty}{\Bigr(\hat{a}(t)\cdot (1-c_0)+\hat{b}(t)\cdot c_0\Bigr)}}\tag{10}\label{eq10}$$
This would mean that the left hand side of equation $\eqref{eq10}$ would exists if the limits of $\hat{a}(t)$ and $\hat{b}(t)$ exist. This is however not true if we choose those two functions to be periodic, for example $\hat{a}(t)=\sin^2(t)+k_a$ and $\hat{b}(t)=\cos^2(t)+k_b$.
Furthermore we could resubstitute $\hat{a}=a\circ c$ and $\hat{b}=b\circ c$ (I am very sceptical about that) and obtain:
$$\lim_{t\to\infty}{c(t)}=\frac{c_0\cdot b(\lim_{t\to\infty}{c(t)})}{(1-c_0)\cdot a(\lim_{t\to\infty}{c(t)})+c_0\cdot b(\lim_{t\to\infty}{c(t)})}\tag{11}\label{eq11}$$
However now the existence of the left hand side depends on the existence of itself. A viscious circle...