Asymptotic behavior of a non-linear rational recurrence I am trying of determining the asymptotic behavior of the following non-linear recurrence, but I do not know how to face it. For constants $c_1,\ldots,c_6$, the recurrence is defined as
\begin{equation*}
a_t=a_{t-1}\left(\frac{c_2}{c_3 a_{t-1}+(t-1)c_4}+\frac{c_5}{(t-1)c_6-c_3a_{t-1}}+1\right)+c_1
\end{equation*}
with $a_0$ initial condition. The constants are defined by: For $0≤p,q≤1$, and $m$ a natural number, $c_1=mp$, $c_2=mpq$, $c_3=2q-1$, $c_4=2m(1-q)$, $c_5=m(1−p)(1−q)$, and $c_6=2mq$. The initial condition is a positive real number.
 A: I tried an example with all $c_i = i$ and $a_0 = 1$, and it didn't appear to settle down to a straight line.  Here are the first $2 \times 10^5$ terms plotted:

EDIT: Writing $a_t = t x_t$, your recurrence can be written in the form
$$ x_t = x_{t-1} + \frac{P(x_{t-1})}{t (c_3 x_{t-1} - c_6) (c_3 x_{t-1} + c_4)} $$
for a certain cubic polynomial $P(x)$ whose leading term is $-c_3^2 x^3$. Being a cubic, it has at least one real root. Such a root $r$ corresponds to a fixed point of the recurrence for $x_t$, i.e. a solution $a_t = r t$ for the original recurrence.  Under appropriate conditions, the fixed point may be stable, so solutions that start out sufficiently close to this one will approach it asymptotically.
A: Hint
Not having yet some more information about the constants, I can just suggest a possible  approach the problem.
Start from rewriting the FDE as
$$
\eqalign{
  & a_{\,t}  = a_{\,t - 1} \left( {{{c_{\,2} } \over {c_{\,3} a_{\,t - 1}  + c_{\,4} \left( {t - 1} \right)}} + {{c_{\,5} } \over { - c_{\,3} a_{\,t - 1}  + c_{\,6} \left( {t - 1} \right)}} + 1} \right) + c_{\,1}   \cr 
  & \Delta a_{\,t}  = a_{\,t + 1}  - a_{\,t}  = a_{\,t} \left( {{{c_{\,2} } \over {c_{\,3} a_{\,t}  + c_{\,4} t}} + {{c_{\,5} } \over { - c_{\,3} a_{\,t}  + c_{\,6} t}}} \right) + c_{\,1}   \cr 
  & \Delta a_{\,t}  = a_{\,t + 1}  - a_{\,t}  = {{c_{\,2} } \over {c_{\,3}  + c_{\,4} \left( {t/a_{\,t} } \right)}} + {{c_{\,5} } \over { - c_{\,3}  + c_{\,6} \left( {t/a_{\,t} } \right)}} + c_{\,1}  \cr} 
$$
Assume that $a_t$ grows  less faster than $t$ as $ t \to \infty$
$$
\eqalign{
  & a_{\,t}  \prec t\quad \quad  \Rightarrow   \cr 
  &  \Rightarrow \quad \Delta a_{\,t}  = \left( {{{c_{\,2} } \over {c_{\,3}  + c_{\,4} \left( {{t \over {a_{\,t} }}} \right)}} + {{c_{\,5} }
 \over { - c_{\,3}  + c_{\,6} \left( {{t \over {a_{\,t} }}} \right)}}} \right) + c_{\,1}  \sim c_{\,1} \quad  \Rightarrow   \cr 
  &  \Rightarrow \quad a_{\,t}  \sim c_{\,1} \;t \cr} 
$$
you get a contradiction, unless of course of particular values of the c's.
If $a_t$ is asymptotic to $t$ , that's a behaviour compatible with the FDE.
$$
\eqalign{
  & a_{\,t}  \sim t\quad \quad  \Rightarrow \quad a_{\,t}  \sim \alpha \,t  \cr 
  &  \Rightarrow \quad \Delta a_{\,t}  = \left( {{{c_{\,2} } \over {c_{\,3}  + c_{\,4} \left( {{t \over {a_{\,t} }}} \right)}} + {{c_{\,5} }
 \over { - c_{\,3}  + c_{\,6} \left( {{t \over {a_{\,t} }}} \right)}}} \right) + c_{\,1}  \sim C\quad  \Rightarrow   \cr 
  &  \Rightarrow \quad a_{\,t}  \sim C\;t \cr} 
$$
Also contradictory would be if $a_t$ grows faster than $t$.
In fact, putting
$$
a_{\,t}  = \alpha t + \beta 
$$
we get a cubic equation in $\alpha$
$$
\eqalign{
  & \alpha  = {{c_{\,2} } \over {c_{\,3}  + c_{\,4} \left( {t/\left( {\alpha t + \beta } \right)} \right)}}
 + {{c_{\,5} } \over { - c_{\,3}  + c_{\,6} \left( {t/\left( {\alpha t + \beta } \right)} \right)}} + c_{\,1} \; \sim \;  \cr 
  & \; \sim {{c_{\,2} } \over {c_{\,3}  + c_{\,4} /\alpha }} + {{c_{\,5} } \over { - c_{\,3}  + c_{\,6} /\alpha }} + c_{\,1}  \cr} 
$$
and of course much depends on the type of solutions thereto.
Taking $c = [1,2,3,4,5,6]$ as in Robert's answer gives as solutions
$$\approx 1.019, \; -1.176 \pm 1.111 \, i$$
while taking the new information you gave , with e.g. $p =  1/4, \; q= 1/3 , \; m=10$ we get
$$\approx 36.9, \; - 3.4 \pm 6.5 \, i$$
and the graphic of $a_t$ is the following.

