Real Positive Eigenvalue, but Stable Dynamics
If you have are trying to discretize the continuous time model
$$
\dot{\textbf{x}} = A\textbf{x},
$$
then in discrete time you will have
$$
\textbf{x}_{t+1} = B \textbf{x}_t
$$
but $A\neq B$, since $A$ describes the change in $\textbf{x}$ while $B$ describes the next value of $\textbf{x}$, not just the change. However
$$
\Delta\textbf{x}_t = \textbf{x}_{t+1} - \textbf{x}_t = B \textbf{x}_t - \textbf{x}_t = (B-I) \textbf{x}_t
$$
This new matrix $B-I$ would correspond to $A$.

(From this you can also see why the stability criterion is different.)

Given your equations
\begin{eqnarray*}
\dot{k} &amp; = &amp;  \frac{1}{1+\frac{1}{\rho}} - \frac{1}{1+\lambda} \\
\\
\dot{\lambda} &amp; = &amp;  \rho \lambda - \frac{1}{k}
\end{eqnarray*}
the equations for the discrete modell would be
\begin{eqnarray*}
k_{t+1} &amp; = &amp;  k_t + \frac{1}{1+\frac{1}{\rho}} - \frac{1}{1+\lambda_t} \\
\\
\lambda_{t+1} &amp; = &amp;  \lambda_t + \rho \lambda_t - \frac{1}{k_t}.
\end{eqnarray*}
You will have to calculate the Jacobian of this system to determine stability.